controllable t hermal conductivity in twisted homogeneous

1

Controllable Thermal Conductivity in Twisted Homogeneous

Interfaces of Graphene and Hexagonal Boron Nitride

Wengen Ouyang1 Huasong Qin2 Michael Urbakh1 and Oded Hod1

1Department of Physical Chemistry School of Chemistry The Raymond and Beverly

Sackler Faculty of Exact Sciences and The Sackler Center for Computational

Molecular and Materials Science Tel Aviv University Tel Aviv 6997801 Israel

2State Key Laboratory for Strength and Vibration of Mechanical Structures School of

Aerospace Xirsquoan Jiaotong University Xirsquoan 710049 China

ABSTRACT

Thermal conductivity of homogeneous twisted stacks of graphite is found to strongly

depend on the misfit angle The underlying mechanism relies on the angle dependence

of phonon-phonon couplings across the twisted interface Excellent agreement between

the calculated thermal conductivity of narrow graphitic stacks and corresponding

experimental results indicates the validity of the predictions This is attributed to the

accuracy of interlayer interactions descriptions obtained by the dedicated registry-

dependent interlayer potential used Similar results for h-BN stacks indicate overall

higher conductivity and reduced misfit angle variation This opens the way for the

design of tunable heterogeneous junctions with controllable heat-transport properties

ranging from substrate-isolation to efficient heat evacuation

Keywords Interfacial thermal conductivity graphite h-BN twisted interface misfit

angle phonon-phonon coupling registry-dependent interlayer potential

2

Graphene is considered to be one of the most promising heat dissipating materials in nanoelectronics

[1] due to its ultrahigh in-plane room-temperature thermal conductivity of ~3000-5000 Wm-1K-1 [23]

This however can be hindered by graphene-substrate interactions that may lead to a substantial

reduction of the heat-transport due to phonon leakage across the graphene-substrate interface and

strong interfacial scattering of flexural phonon modes [4] Such undesirable substrate effects can be

reduced by considering multilayer graphene stacks These are expected to effectively isolate the top

graphene layers from the substrate due to the considerably lower cross-plane thermal conductivity

(~68 Wm-1K-1) [5] while exhibiting high in-plane conductivity that can be tuned via the stack

thickness [6-13] Anisotropic thermal conductivity is also observed for bulk hexagonal boron nitride

(h-BN) with the in-plane and cross-plane thermal conductivity in the range of 390-420 Wm-1K-1 and

25-48 Wm-1K-1 respectively [1415]

Efficient is-situ tuning of the thermal conductivity of such graphitic structures can be achieved by

controlling the twist angle between adjacent layers within the stack This has been recently

computationally demonstrated for finite-sized nanoscale few-layer graphene junctions [1617] Two

factors however limit the applicability of these results (i) the simulations were performed using

simplistic isotropic interlayer potentials that are known to be inaccurate for simulating the interlayer

interactions in layered materials [18-21] and (ii) the relevance of the results for large-scale interfaces

is questionable due to significant edge scattering effects inherent to the small finite-sized model

systems studied

To address these issues we investigate the interlayer thermal conductivity of graphene and h-BN

stacks of varying thicknesses and twist angles This allows us to gain fundamental understanding of

the heat transport mechanisms in layered materials stacks and identify feasible means to control it

Our model system consists of two contacting identical AB (AArsquo)-stacked graphite (h-BN) slabs

whose interfacing graphene (h-BN) layers are twisted with respect to each other to create a stacking

fault of misfit angle θ (see Figure 1) Recent experiments demonstrated fine control over the misfit

angle in such setups [2223] The thickness of the entire construction is varied between 27 nm-35 nm

(8-104 layers) and periodic boundary conditions are applied in all directions Heat transport

simulations are performed using state-of-the-art anisotropic interlayer potentials [18-21] applied to

the twisted stacks These potentials were shown to capture well the structural dynamic heat

dissipation and phonon spectrum of graphitic and h-BN layered systems [24-27] A thermal bias is

induced by applying Langevin thermostats with different temperatures to two layers residing on

opposite sides away from the twisted interface (see Sec 1 of the Supporting Information (SI) for

further details)

3

Figure 1 Schematic representation of the simulation setup Two identical AB-stacked graphite slabs

(gray and orange respectively) are twisted with respect to each other to create a stacking fault of misfit

angle θ A thermal bias is induced by applying Langevin thermostats to the two layers marked by

dashed red (Thot) and green (Tcold) rectangles The arrows indicate the direction of the vertical heat

flux Since periodic boundary conditions are applied also in the vertical direction two twisted

interfaces are shown across which heat flows in opposite directions

We start by studying the effect of the misfit angle on the cross-plane thermal conductivity of the

twisted graphite and h-BN stacks Figure 2 presents the dependence of the cross-plane thermal

conductivity of the entire stack on the misfit angle for model systems consisting of 8 (red circles) and

16 (black triangles) layers for (a) graphite and (b) h-BN A pronounced dependence of the cross-plane

thermal conductivity (120581120581CP) of the entire graphitic stack is clearly evident which above a misfit angle

of sim 5deg 120581120581CP drops by a factor of 3-4 with respect to the value obtained for the aligned contact

Similar misfit-angle dependence of 120581120581CP is obtained for twisted bilayer graphene (tBLG) using the

transient MD simulation approach (see Sec 2 of the SI) We note that this sharp drop for graphite is

steeper and that the overall reduction is higher than those previously obtained using Lennard-Jones

interlayer potentials in finite model systems [1617] The corresponding cross-plane thermal

conductivity of the commensurate h-BN stack is found to be approximately double that of graphite

for the same number of layers Notably it reduces more gradually with the twist angle and saturates

at sim 15deg with an overall two-three fold reduction

The thermal conductivity of both graphite and h-BN stacks is found to increase when doubling their

thickness To identify the source of this thickness dependence we plot in Figure 2(c-d) the interfacial

thermal resistance (ITR) (see Sec 12 of the SI for the definition) associated with the twisted junction

formed between the contacting graphene or h-BN layers of the two optimally-stacked slabs Note that

unlike 120581120581CP which measures the conductivity of the entire stack the ITR corresponds to the heat

transport resistance of the two adjacent layers forming the twisted interface Two important

4

observations can be made (i) the ITR is weakly dependent on the stack thickness indicating that the

thickness dependence arises from the conductivity of the optimally-stacked interfacing slabs

Specifically in the thickness range considered the heat conductivity grows with slab thickness due to

reduction of phonon-phonon interactions and increased contribution of long wave-length phonons

below the mean-free path [28-30] (ii) the ITR strongly depends on the twist angle demonstrating a

~10-fold (4-fold) increase when the twist angle at the graphene (h-BN) interface is varied from 0deg

to 15deg This clearly indicates that the twist angle can be utilized to control the cross-plane thermal

conductivity of hexagonal two-dimensional (2D) materials and to effectively thermally isolate the top

layers from the underlying substrate

Figure 2 Twist-angle dependence of the cross-plane thermal conductivity of the entire stack (a b)

and the interfacial thermal resistance (c d) of the twisted contact formed between the optimally-

stacked slabs of graphite (a c) and bulk h-BN (b d) respectively Red circles and black triangles

correspond to the results obtained using 8 and 16 layer models respectively

The strong dependence of the cross-plane thermal conductivity of graphene and h-BN on the stacking

fault twist angle is related to the degree of coupling between the phonon modes of the two contacting

layers at the twisted interface Note that the term ldquocouplingrdquo used herein is not related to the standard

notion of phonon-phonon couplings due to anharmonic effects Instead we regard to the off-diagonal

terms of the Hessian when represented in the basis of the harmonic phonon modes of the isolated

5

layers To demonstrate this we write the dynamical matrix (the mass-reduced Fourier transform of

the force constant matrix) in block form as follows

120625120625(119954119954) = 12062512062511(119954119954) 12062512062512(119954119954)12062512062521(119954119954) 12062512062522(119954119954) (1)

where 12062512062511(119954119954) and 12062512062522(119954119954) are the block matrices relating to the first and second layer and 12062512062512(119954119954)

and 12062512062521(119954119954) = 12062512062512dagger (119954119954) all evaluated at wave-vector 119954119954 The interlayer phonon-phonon couplings are

obtained by diagonalizing separately 12062512062511(119954119954) and 12062512062522(119954119954) such that 12062512062511(119954119954) =

1199321199321dagger(119954119954)12062512062511(119954119954)1199321199321(119954119954) and 12062512062522(119954119954) = 1199321199322

dagger(119954119954)12062512062522(119954119954)1199321199322(119954119954) are diagonal matrices containing the

frequencies (120596120596119894119894) of the phonon modes of the two layers and 1199321199321(119954119954) and 1199321199322(119954119954) are unitary

matrices of the corresponding eigenvectors We now construct a global block diagonal transformation

matrix of the form

119932119932(119954119954) = 1199321199321(119954119954) 120782120782120782120782 1199321199322(119954119954) (2)

and transform the full dynamical matrix as follows

119932119932dagger(119954119954)120625120625(119954119954)119932119932(119954119954) = 12062512062511(119954119954) 12062512062512(119954119954)12062512062521(119954119954) 12062512062522(119954119954)

(3)

where 12062512062512(119954119954) = 1199321199321dagger(119954119954)12062512062512(119954119954)1199321199322(119954119954) and 12062512062521(119954119954) = 12062512062512

dagger (119954119954) are the interlayer phonon-phonon

coupling blocks Naturally when the two layers are infinitely separated these coupling blocks vanish

and the diagonal blocks converge to those of the isolated layers

The overall coupling between the two layers can be obtained from the individual phonon-phonon

coupling matrix elements via Fermirsquos golden rule [31] which reads as (see Sec 4 of SI for a detailed

derivation)

Γtot = 120587120587ℏ3

2sum 119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582 (4)

where 119885119885 = sum 119890119890minus120573120573119864119864119954119954120582120582119954119954120582120582 is the partition function 119864119864119954119954120582120582 is the energy of phonons at branch 120582120582 with

wave number 119954119954 120588120588119864119864119954119954120582120582 is the density-of-states (DOS) at 119864119864119954119954120582120582 and 119881119881120582120582120582120582+31199031199032(119954119954)

2 is the coupling

matrix element between branches of phonons of similar energy in the two layers whose number of

atoms in one unit cell is 119903119903

Using Eq (4) we can rationalize the misfit angle dependence of the heat flux across the twisted

interface from the calculated inter-phonon coupling To that end we performed room temperature

(300K) simulations (technical details can be found in Sec 3 of SI) for tBLG with different misfit

angles using the Greenrsquos function molecular dynamics (GFMD) developed by Kong et al [32] as

6

implemented in LAMMPS [33] The simulations allow us to evaluate the dynamical matrix from

which the phonon-phonon couplings can be extracted (see details in Sec 3 of SI) and the overall heat

transfer rate calculated Figure 3 shows the resulting heat transfer rate (normalized to is value for the

aligned contact (120579120579 = 0) ) as a function of the misfit angle compared to the interfacial thermal

conductivity defined as the inverse of the ITR presented in Figure 2(c) ITC equiv 1ITR The

remarkable agreement between the calculated interfacial thermal conductivity and Fermirsquos golden

rule results indicate that the dependence of the interlayer phonon-phonon couplings on the misfit

angle is responsible for the strong angle dependence of the interfacial conductivity Notably the sharp

heat conductivity drop at misfit angles in the range of 0deg-5deg as well as the small conductivity for larger

misfit angles are well captured by Fermirsquos golden rule

Figure 3 Comparison between Fermirsquos golden rule results (open blue squares) for the interfacial heat-

transfer rate of a tBLG and the calculated interfacial thermal conductivity at various misfit angles

ITC simulation results are presented for both 8 layers (open red circles) and 16 layers (open black

triangles) showing similar behavior For comparison purposes all data sets are normalized to their

value obtained for the aligned contact

To correlate our results with experimentally measured thermal conductivities that are often obtained

for thick samples we repeated our calculations for increasing stack thicknesses at fixed misfit angles

Figure 4 presents results for the calculated heat conductivity of (a) graphite and (b) h-BN stacks either

aligned (open red circles) or twisted by 120579120579 = 3016deg (open black diamond symbols) as a function of

number of layers in the stack As discussed above for both systems the misoriented stack exhibits

lower heat conductivity compared to the aligned system however its thickness dependence is

considerably stronger This can be attributed to the significantly higher interface resistance of the

twisted interface that when plugged in Eq (S2) of the SI for the overall conductivity induces stronger

7

thickness dependence

Comparing our calculated heat conductivities for the aligned contact (open red circles) to available

experimental data for ~35 nm thick graphite slabs [34] (dashed green line) we find that at the thickest

model system considered of 104 graphene layers (~34 nm thick) the calculated value of 085plusmn005

W m sdot Kfrasl is in remarkable agreement with the measured value of ~07 W m sdot Kfrasl Furthermore

experimental values for bulk graphite [5] indicate that the thermal conductivity continues to grow up

to ~68 W m sdot Kfrasl (black dash-dotted line) which is consistent with the general trend of the calculated

heat conductivity that does not saturate for the thickest model system considered These results

strongly enforce the validity of our force-field and model systems to model the heat conductivity of

twisted layered materials interfaces Available experimental results for the heat conductivity of bulk

h-BN are marked by the dashed-dotted black and dashed-green lines in Figure 4(b) In line with our

findings for the graphitic interface our calculated finite slab heat conductivities for the aligned

interface (open red circles) continue to grow with the number of layers and are consistently below the

bulk value

Figure 4 Thickness dependence of the thermal conductivity 120581120581CP of aligned (open red circles) and

twisted by 3016deg (open black diamond symbols) graphite (a) and h-BN (b) stacks Blue squares

represent results obtained using the isotropic Lennard-Jones potential for the aligned contacts The

green dashed and black dash-dotted lines represent experimental results measured for graphite (a)

(Refs [534]) and bulk h-BN (b) (Refs [1415]) Note that both axis scales are logarithmic Error bar

estimation procedure is discussed in Sec 1 of the SI

Another important factor that may affect the interlayer thermal transport properties of 2D material

stacks is the average temperature of the system which was taken to be ~300K in all abovementioned

simulations To evaluate the sensitivity of our results towards this parameter we repeated the heat

conductivity and interfacial resistance calculations of optimally stacked graphite and h-BN stacks for

an average temperature of 400 K The results presented in see Sec 4 of the SI indicate that 120581120581CP and

8

ITR weakly depend on the average temperature over the entire thickness range considered This

conclusion is consistent with recent experimental findings [11] We may attribute this to the fact that

the thickness of the graphite and h-BN stack model considered is much smaller than their phonon

mean-free path (~200 nm for graphite [1011133536] and ~100 nm for bulk h-BN [15]) such that

the phonon transport is dominated by phonon-boundary scattering which weakly depends on

temperature in the range considered Therefore temperature dependent Umklapp processes have only

marginal contribution to our results [11]

Finally we note that previous calculations of the heat conductivity of twisted graphitic interfaces

relied on Lennard-Jones (LJ) potentials describing the interlayer interactions [8-1013] To

demonstrate the importance of using registry-dependent interlayer potentials we have repeated our

calculations of the heat conductivity of graphitic slabs with the REBO intralayer potential augmented

by LJ interlayer interactions [37] (120576120576 = 284 meV120590120590 = 34 Å ) We find that the calculated heat

conductivities obtained using the LJ interlayer potential are consistently higher than those obtained

by our ILP and that the difference between them grows with the model system thickness Notably the

heat conductivity obtained using the LJ potential for a graphitic slab of thickness ~34 nm is 154

W m sdot Kfrasl overestimating the experimental value by more than a factor of 2

The excellent agreement of our ILP calculations with experimental data of nanoscale graphitic stacks

therefore demonstrates the reliability of our predictions for the strong interfacial misfit angle

dependence of cross-layer thermal conductivity in graphite and h-BN The observed sharp

conductivity decrease of twisted graphitic interfaces at misfit angles lt 5deg opens the way to control

the thermal evacuation rate and thermal isolation of active layers in graphene-based electronic and

mechanical devices The revealed underlying mechanism suggests that design rules can be obtained

by carefully tailoring the phonon-phonon couplings across the twisted interface While the misfit

angle dependence of h-BN is found to be weaker than that of graphite the overall thermal

conductivity of the former is found to be higher This may be utilized to achieve higher conductivity

and controllability in twisted heterogeneous junctions of layered materials

ASSOCIATED CONTENT

Supporting Information

The supporting Information section includes a description of the methodology thermal conductivity

of twisted bilayer graphene theory for calculating the phonon coupling of twisted bilayer graphene

9

derivation of Fermirsquos golden rule and temperature dependence of the cross-plane thermal

conductivity

AUTHOR INFORMATION

Corresponding Author

E-mail urbakhtauextauacil

Notes

The authors declare no competing financial interest

ACKNOWLEDGMENTS

The authors would like to thank Prof Abraham Nitzan Dr Guy Cohen and Dr Yiming Pan for

helpful discussions WO acknowledges the financial support from a fellowship program for

outstanding postdoctoral researchers from China and India in Israeli Universities and the support from

the National Natural Science Foundation of China (Nos 11890673 and 11890674) HQ

acknowledges the financial support from the National Natural Science Foundation of China (No

11890674) MU acknowledges the financial support of the Israel Science Foundation Grant

No 114118 and the ISF-NSFC joint grant 319119 OH is grateful for the generous financial

support of the Israel Science Foundation under grant no 158617 and the Naomi Foundation for

generous financial support via the 2017 Kadar Award This work is supported in part by COST Action

MP1303

10

References

[1] A A Balandin Thermal properties of graphene and nanostructured carbon materials Nat Mater 10 569 (2011) [2] S Ghosh I Calizo D Teweldebrhan E P Pokatilov D L Nika A A Balandin W Bao F Miao and C N Lau Extremely high thermal conductivity of graphene Prospects for thermal management applications in nanoelectronic circuits Appl Phys Lett 92 151911 (2008) [3] A A Balandin S Ghosh W Bao I Calizo D Teweldebrhan F Miao and C N Lau Superior thermal conductivity of single-layer graphene Nano Lett 8 902 (2008) [4] J H Seol et al Two-Dimensional Phonon Transport in Supported Graphene Science 328 213 (2010) [5] R Taylor The thermal conductivity of pyrolytic graphite J Philosophical Magazine 13 157 (1966) [6] S Ghosh W Bao D L Nika S Subrina E P Pokatilov C N Lau and A A Balandin Dimensional crossover of thermal transport in few-layer graphene Nat Mater 9 555 (2010) [7] Z Wang R Xie C T Bui D Liu X Ni B Li and J T L Thong Thermal Transport in Suspended and Supported Few-Layer Graphene Nano Lett 11 113 (2011) [8] Z Wei Z Ni K Bi M Chen and Y Chen Interfacial thermal resistance in multilayer graphene structures Phys Lett A 375 1195 (2011) [9] Y Ni Y Chalopin and S Volz Significant thickness dependence of the thermal resistance between few-layer graphenes Appl Phys Lett 103 061906 (2013) [10] J Chen J H Walther and P Koumoutsakos Strain Engineering of Kapitza Resistance in Few-Layer Graphene Nano Lett 14 819 (2014) [11] Q Fu J Yang Y Chen D Li and D Xu Experimental evidence of very long intrinsic phonon mean free path along the c-axis of graphite Appl Phys Lett 106 031905 (2015) [12] G Fugallo A Cepellotti L Paulatto M Lazzeri N Marzari and F Mauri Thermal Conductivity of Graphene and Graphite Collective Excitations and Mean Free Paths Nano Lett 14 6109 (2014) [13] Z Wei J Yang W Chen K Bi D Li and Y Chen Phonon mean free path of graphite along the c-axis Appl Phys Lett 104 081903 (2014) [14] E K Sichel R E Miller M S Abrahams and C J Buiocchi Heat capacity and thermal conductivity of hexagonal pyrolytic boron nitride Phys Rev B 13 4607 (1976) [15] P Jiang X Qian R Yang and L Lindsay Anisotropic thermal transport in bulk hexagonal boron nitride Phys Rev Mater 2 064005 (2018) [16] M-H Wang Y-E Xie and Y-P Chen Thermal transport in twisted few-layer graphene Chinese Physics B 26 116503 (2017) [17] X Nie L Zhao S Deng Y Zhang and Z Du How interlayer twist angles affect in-plane and cross-plane thermal conduction of multilayer graphene A non-equilibrium molecular dynamics study Int J Heat Mass Tran 137 161 (2019) [18] A N Kolmogorov and V H Crespi Registry-dependent interlayer potential for graphitic systems Phys Rev B 71 235415 (2005) [19] M Reguzzoni A Fasolino E Molinari and M C Righi Potential energy surface for graphene on graphene Ab initio derivation analytical description and microscopic interpretation Phys Rev B 86 (2012) [20] M M van Wijk A Schuring M I Katsnelson and A Fasolino Moire Patterns as a Probe of Interplanar Interactions for Graphene on h-BN Phys Rev Lett 113 135504 (2014) [21] D Mandelli W Ouyang M Urbakh and O Hod The Princess and the Nanoscale Pea Long-Range Penetration of Surface Distortions into Layered Materials Stacks ACS Nano 13 7603 (2019) [22] E Koren E Loumlrtscher C Rawlings A W Knoll and U Duerig Adhesion and friction in mesoscopic graphite contacts Science 348 679 (2015) [23] E Koren I Leven E Loumlrtscher A Knoll O Hod and U Duerig Coherent commensurate electronic states at the interface between misoriented graphene layers Nat Nanotechnol 11 752 (2016)

11

[24] W Ouyang I Azuri D Mandelli A Tkatchenko L Kronik M Urbakh and O Hod Mechanical and Tribological Properties of Layered Materials under High Pressure Assessing the Importance of Many-Body Dispersion Effects J Chem Theory Comput 16 666 (2020) [25] W Ouyang D Mandelli M Urbakh and O Hod Nanoserpents Graphene Nanoribbon Motion on Two-Dimensional Hexagonal Materials Nano Lett 18 6009 (2018) [26] I Leven T Maaravi I Azuri L Kronik and O Hod Interlayer Potential for Grapheneh-BN Heterostructures J Chem Theory Comput 12 2896 (2016) [27] T Maaravi I Leven I Azuri L Kronik and O Hod Interlayer Potential for Homogeneous Graphene and Hexagonal Boron Nitride Systems Reparametrization for Many-Body Dispersion Effects J Phys Chem C 121 22826 (2017) [28] L H Liang and B Li Size-dependent thermal conductivity of nanoscale semiconducting systems Phys Rev B 73 153303 (2006) [29] P K Schelling S R Phillpot and P Keblinski Comparison of atomic-level simulation methods for computing thermal conductivity Phys Rev B 65 144306 (2002) [30] J Shiomi Nonequilirium molecular dynamics methods for lattice heat conduction calculations Annual Review of Heat Transfer 17 (2014) [31] G E Stedman Fermis Golden RulemdashAn Exercise in Quantum Field Theory Am J Phys 39 205 (1971) [32] L T Kong G Bartels C Campantildeaacute C Denniston and M H Muumlser Implementation of Greens function molecular dynamics An extension to LAMMPS Comput Phys Commun 180 1004 (2009) [33] S Plimpton Fast Parallel Algorithms for Short-Range Molecular Dynamics J Comput Phys 117 1 (1995) [34] M Harb et al The c-axis thermal conductivity of graphite film of nanometer thickness measured by time resolved X-ray diffraction Appl Phys Lett 101 233108 (2012) [35] H Zhang X Chen Y-D Jho and A J Minnich Temperature-Dependent Mean Free Path Spectra of Thermal Phonons Along the c-Axis of Graphite Nano Lett 16 1643 (2016) [36] D P Sellan E S Landry J E Turney A J H McGaughey and C H Amon Size effects in molecular dynamics thermal conductivity predictions Phys Rev B 81 214305 (2010) [37] S J Stuart A B Tutein and J A Harrison A reactive potential for hydrocarbons with intermolecular interactions J Chem Phys 112 6472 (2000)

S1

Supporting information for ldquoControllable Thermal Conductivity of

Twisted Graphene and Hexagonal Boron Nitride Interfacesrdquo







The Supporting information includes following sections

1 Methodology

2 Thermal conductivity of twisted bilayer graphene

3 Theory for calculating the phonon coupling of twisted bilayer graphene

4 Derivation of Fermirsquos golden rule

5 Temperature dependence of cross-plane thermal conductivity

S2

1 Methodology

11 Model system

The initial interlayer distance across the layered stack was set equal to 34 Aring and 33 Aring for graphite

and bulk h-BN respectively Periodic boundary conditions were applied in all directions It should be

noted that the lattice structure is rigorously periodic only at some specific twist angles the values of

which are listed in Table S1 in section 3 below While the cross-sectional area for each misfit angle

θ is different all systems considered have a contact area exceeding 12 nm2 which was shown to

provide converged results with respect to unit-cell dimensions [1] The intralayer interactions within

each graphene and h-BN layer were modeled via the second generation REBO potential [2] and

Tersoff potential [3] respectively The interlayer interactions between the layers of graphite and bulk

h-BN were described via our dedicated interlayer potential (ILP) [4] which is implemented in the

LAMMPS [5] suite of codes [6]

Fig S1 Schematic representation of the simulation setup (a) and steady-state temperature profile (b) respectively In panel (a) two identical AB-stacked graphite slabs (gray and orange respectively) are twisted with respect to each other to create a stacking fault of misfit angle θ A thermal bias is induced by applying Langevin thermostats to the two layers marked by dashed red (Thot) and green (Tcold) rectangles The arrows indicate the direction of the vertical heat flux Since periodic boundary conditions are applied also in the vertical direction two twisted interfaces are shown across which heat flows in opposite directions The steady-state temperature profiles are illustrated in panel (b) where N is the total number of layers in the model system and RAB dAB and Rθ and dθ mark the interfacial Kapitza resistance [89] and interlayer distance for contacting graphene layers with AB-stacking and misfit angle θ respectively The red lines in panel (b) mark the temperature variation across the twisted interface where the vertical axis corresponds to the position of the various layers along the stack and the horizontal axis marks the temperature of the various layers

S3

12 Simulation Protocol

All MD simulations were performed with the LAMMPS simulation package [5] The velocity-Verlet

algorithm with a time-step of 05 fs was used to propagate the equations of motion A Noseacute-Hoover

thermostat with a time constant of 025 ps was used for constant temperature simulations To maintain

a specified hydrostatic pressure the three translational vectors of the simulation cell were adjusted

independently by a Noseacute-Hoover barostat with a time constant of 10 ps [7] To relax the box we first

equilibrated the systems in the NPT ensemble at a temperature of T = 300 K and zero pressure for

250 ps (see Fig S2) After equilibration Langevin thermostats with damping coefficients 10 ps-1

were applied to the bottom and middle layer of the graphene stack (see Fig S1) with target

temperature Thot = 375 K (hot reservoir) and Tcold = 225 K (cold reservoir) respectively Then the

system was allowed to reach steady-state over a subsequent simulation period of 750 ps (see Fig S2)

during which the dynamics of all non-thermostated layers followed the NVE ensemble For the larger

model systems the length of the NPT and Langevin stages was doubled (for the 32 and 48 layers

systems) or tripled (for the 104 layers graphitic system) to ensure convergence of the obtained steady-

state Once steady-state was obtained the last 500 ps were used to calculate the thermal conductivity

of the twisted graphite and bulk h-BN The statistical errors were estimated using ten different data

sets each calculated over a time interval of 50 ps

Fig S2 Time evolution of the temperature of thermostated layers for 16 layers twisted graphite with misfit angle (a) θ = 0deg (b) θ = 509deg (c) θ = 1518deg and (d) θ = 3016deg Note that the thermal fluctuations increase with increasing the misfit angle due to the growing interfacial thermal resistance that enhances phonon back scattering at the twisted junction

S4

13 Calculation of the interfacial thermal resistance

According to Fourierrsquos law the cross-plane thermal conductivity (120581120581CP) of a twisted graphitic interface

of misfit angle θ can be calculated as

120581120581CP = 119876119860119860Δ119879119879Δ119911119911

(S1)

where 119876 is the heat flux A is the cross-section area and Δ119879119879Δ119911119911 is the temperature gradient along

the direction of heat flux (perpendicular to the basal plane in our case) Fig S1(b) shows a schematic

temperature profile along the z-direction where the vertical axis corresponds to the position of the

various layers along the stack and the horizontal axis marks the temperature of the various layers The

actual temperature profiles extracted from the NEMD simulations for twisted graphite with different

number of layers can be found in Fig S3 For Bernal-stacked graphite (ie θ = 0deg red circles in Fig

S3) only the linear region of the temperature profile was used to calculate 120581120581CP and the points

corresponding to the layers where the thermostats were applied were omitted (marked with green

triangle in Fig S3) The 120581120581CP of the system was calculated using Eq (S1) by averaging over the two

linear regions of the temperature profiles For the twisted case (θ ne 0deg) we found a sudden

temperature decrease Δ119879119879120579120579 at the position of the twisted interface (see black squares in Fig S3) 120581120581CP

in this case was calculated using the temperature gradient calculated for the same layer range as that

for θ = 0deg To characterize the thermal properties of the twisted interface the concept of interfacial

thermal resistance (ITR) ie Kapitza resistance [89] was introduced According the definition of

the Kapitza resistance [8] 119877119877 = 119860119860Δ119879119879119876 and noticing that Δ119879119879tot = (1198731198732 minus 3)Δ119879119879AB + Δ119879119879120579120579 and

Δ119911119911 = (1198731198732 minus 3)119889119889AB + 119889119889120579120579 Eq (1) can be rewritten considering two-resistors in series as [see Fig

S1(b)]

(1198731198732 minus 3)119877119877AB + 119877119877120579120579 = [(1198731198732 minus 3)119889119889AB + 119889119889120579120579] 120581120581CPfrasl (S2)

Here 119877119877AB 119889119889AB Δ119879119879AB and 119877119877120579120579 119889119889120579120579 Δ119879119879120579120579 are the ITR interlayer distance and temperature

difference for adjacent AB-stacked and twisted graphene layers respectively For the aligned contact

(120579120579 = 0deg) the ITR can be simply calculated as 119877119877AB = 119889119889AB120581120581AB Once 119877119877AB is known 119877119877120579120579 can be

calculated from Eq (S2) We note that the sharp temperature drop at the twisted interface indicates

that 119877119877120579120579 should be much larger than 119877119877AB

S5

Fig S3 Temperature profiles for graphitic stacks consisting of (a) 8 and (b) 16 layers The red circles and black squares represent the temperature profiles for the aligned (120579120579 = 0deg) and twisted (120579120579 =3016deg) junctions respectively Green triangles represent data points that were omitted in the 120581120581CP calculation


As comparison we also calculated the interfacial thermal conductivity (ITC) and ITR of twisted

bilayer graphene (tBLG) with the transient MD simulation approach [15-17] since the NEMD

simulation protocol used in the main text becomes invalid in this case In this protocol the system

was first equilibrated within the NPT ensemble at T = 200 K and zero pressure for 100 ps which was

followed by a 100 ps NVT ensemble equilibration stage and a 100 ps of NVE ensemble equilibration

stage After the system reached steady-state an ultrafast heat impulse was imposed on the top layer

of the t-BLG for 50 fs to increase the temperature of the top layer from 200 K to 400 K while that of

bottom layer of tBLG remained unchanged After the external heat source was removed thermal

energy flowed from the top layer to the bottom layer due to the temperature difference and the

temperature of both layers approached 300 K when quasi-steady-state was reached During the

thermal relaxation time interval (500 ps) the temperature and energy of the system sections were

recorded The ITR could then be extracted using the following equation [15-17]

part119864119864t120597120597120597120597

= 119860119860119877119877119879119879bot(119905119905) minus 119879119879top(119905119905) (S3)

where 119864119864t is the total energy of the top graphene layer R is the ITR of the tBLG A is the interfacial

cross-section area and 119879119879bot and 119879119879top are the instantaneous temperatures measured for the bottom

and top layers of the tBLG respectively Note that in Eq S3 we assume a linear dependence of the

heat flux on the temperature difference between the layers The ITC of the tBLG is simply defined as

ITC equiv 119889119889ITR where d is the average interlayer distance

S6

The ITC and ITR of tBLG as functions of misfit angle calculated with the transient MD simulation

protocol are illustrated in Fig S4 demonstrating similar misfit-angle dependence as that for the

NEMD protocol with Langevin thermostat exercised to obtain the results presented in the main text

This further validates the reliability of the simulation protocol adopted in the main text which is more

suitable to treat thick slabs and allows to obtain a true stead-state

Fig S4 Misfit-angle dependence of (a) ITC and (b) ITR for a twisted bilayer graphene obtained using the transient MD simulation approach

3 Theory for calculating the phonon-phonon coupling of twisted bilayer graphene

31 Brillouin Zone of supercell in tBLG

For tBLG the lattice structure is rigorously periodic only at some specific misfit angles θ where the

lattice vector 1199231199231 = 11989911989911199381199381 + 11989911989921199381199382 in the bottom layer equals the vector 1199231199232 = 11989811989811199381199381 + 11989811989821199381199382 in the

top layer with certain integers m1 m2 and n1 n2 Here 1199381199381 = 119886119886(10) and 1199381199382 = 11988611988612radic32 are

the primitive lattice vectors of the bottom layer and a is the lattice constant of monolayer graphene

Thus the exact superlattice period is then given by [18]

119871119871 = |11989911989911199381199381 + 11989911989921199381199382| = 11988611988611989911989912 + 11989911989922 + 11989911989911198991198992 = |1198991198991minus1198991198992|1198861198862 sin(1205791205792) (S4)

where θ is the angle between two lattice vectors 1199231199231 and 1199231199232 In the simulations below we always

rotated the supercell such that its lattice vector is 1199231199231 = 119871119871(10) and 1199231199232 = 11987111987112radic32 In this case

the corresponding reciprocal lattice vector of the moireacute superlattice satisfies the relation 119918119918119894119894 ∙ 119923119923119895119895 =

2120587120587120575120575119894119894119895119895 such that

1199181199181 = 4120587120587radic3119871119871

radic32

minus12 1199181199182 = 4120587120587

radic3119871119871(01) (S5)

S7

Both the lattice vectors and the corresponding reciprocal lattice vectors of the superlattice of the tBLG

are presented in Fig S5 Table S1 reports the parameters used to construct rhombus periodic

supercells of different misfit angles that can be duplicated to construct a rectangular periodic supercell

Table S1 The parameters used to construct periodic supercells of various misfit angles

120579120579 (deg) 119860119860 (nm2) 1198991198991 1198991198992 1198981198981 1198981198982

0 60383688 24 0 24 0

0696407 709613169 48 47 47 48

1121311 273718420 30 29 29 30

2000628 85648391 17 16 16 17

3006558 152322047 23 21 21 23

4048894 189013524 26 23 23 26

5085849 53255058 14 12 12 14

7926470 49376245 14 11 11 14

9998709 86277388 19 14 14 19

15178179 31554670 15 4 4 15

19932013 42876612 15 8 8 15

25039660 13942761 9 4 4 9

30158276 18974735 11 4 4 11

32204228 21805221 12 4 4 12

32 Special points for Brillouin Zone integration

The calculation of the sum over wave vector q in Eq (4) in the main text can be transformed to an

integral using the relation sum (⋯ )119954119954 = 1119881119881119887119887int (⋯ )BZ d119954119954 where 119881119881119887119887 = (2120587120587)3119881119881 is the volume of the

Brillouin Zone (BZ) and V is volume of the real-space unit-cell The calculation of integral is usually

inefficient since it requires calculating the value of the function over a large set of k points in the first

BZ To calculate such integrations more efficiently simple k-point meshes can be replaced by a

carefully selected set of special points in the BZ 119954119954119894119894 [19-22] over which the function is evaluated

The integral can then be estimated via

119868119868 = 1119881119881119887119887int 119891119891(119954119954)BZ d119954119954 asymp 1

119873119873sum 119908119908119894119894119891119891(119954119954119894119894)119894119894 (S6)

S8

where 119881119881119887119887 is the volume of the BZ 119908119908119894119894 is the weight of the ith data point and N normalizes the

weighting factors to unity 119873119873 = sum 119908119908119894119894119894119894 The set of selected 119954119954119894119894 forms a grid in the irreducible

Brillouin zone (IBZ) as is illustrated by the red points in Fig S5(b) The coordinates of these points

for a hexagonal lattice are presented in Eq (S7)

Fig S5 (a) Twisted bilayer graphene of misfit angle θ = 509deg L1 and L2 are the superlattice vectors (b) The corresponding first Brillouin Zone of (a) G1 and G2 are the reciprocal lattice vectors of the superlattice The triangle ΔΓMK represents the irreducible Brillouin zone Red circles mark the position of the special points used to evaluate the integral over the first Brillouin Zone

⎩⎪⎨

⎪⎧ 1199541199541 = 1

9 1205971205973 1199541199542 = 2

9 1205971205973 1199541199543 = 1

18 512059712059718

1199541199544 = 518

512059712059718 1199541199545 = 1

9 21205971205979 1199541199546 = 2

9 21205971205979

1199541199547 = 118

312059712059718 1199541199548 = 1

9 1205971205979 1199541199549 = 1

18 12059712059718

(S7)

Here 119905119905 = radic3 and the unit of the coordinates is 2120587120587119871119871 The weighting factors 119908119908119894119894 are [19]

119908119908124689 = 112

119908119908357 = 16 (S8)

Using Eqs (S6-S8) Eq (4) in the main text can be evaluated as follows

Γtot = 120587120587ℏ3

2sum 119908119908119896119896 sum

119890119890minus120573120573119864119864119954119954119948119948120582120582

119885119885

120588120588119864119864119954119954119948119948120582120582119881119881120582120582120582120582+31199031199032(119954119954119948119948)

2

1198641198641199541199541199481199481205821205822120582120582

9119896119896=1 (S9)

This equation was used to calculate the transition rate presented in Fig 3 in the main text


The derivation of Fermirsquos golden rule is provided in a separate file (Fermirsquos Golden Rule for

phononspdf) due to its extent

S9

5 Temperature dependence of interfacial thermal conductivity

In the main text the target temperatures of the Langevin thermostats for the bottom and middle layers

of graphene and h-BN were set to 225 K and 375 K respectively After reaching the steady-state the

average temperature of the system was found to be ~300 K To check the effect of average temperature

on our results we calculated the cross-plane thermal conductivity (120581120581CP ) and the corresponding

interfacial thermal resistance (ITR) at a different temperature gradient (325 K ndash 475 K) resulting the

average steady-state temperature of ~400K The protocol described in Section 1 above was used to

perform these calculations as well Both ILP and Lennard-Jones (LJ) potential were tested for

graphite whereas for the bulk h-BN simulations only the ILP was used The results for graphite and

bulk h-BN are illustrated in Fig S6 and Fig S7 respectively For the ILP we find that the overall

values of 120581120581CP (ITR) decrease (increase) slightly with increasing average temperature which is

consistent with a recent experiment [10] The LJ potential calculations as well exhibit very week

dependence on average temperature within the range studied Altogether the layer dependences of

both quantities remain mostly insensitive to the average temperature The reason is that the thickness

of the graphite and h-BN model systems was chosen to be much smaller than their phonon mean free

path (~200 nm for graphite [110-13] and ~100 nm for bulk h-BN [14]) such that phonon transport is

dominated by phonon-boundary scattering and the Umklapp process only makes marginal

contribution [10]

S10

Fig S6 Layer dependence of 120581120581CP (a c) and ITR (b d) for Bernal-stacked graphite at average steady-state temperatures of 300 K (red circles) and 400 K (black squares) The left and right columns correspond to the 120581120581CP and ITR calculated with ILP and LJ potential respectively

Fig S7 Layer dependence of 120581120581CP (a) and ITR (b) for AArsquo-stacked h-BN at average temperatures of 300 K (red circles) and 400 K (blue squares)

S11

References [1] Z Wei J Yang W Chen K Bi D Li and Y Chen Phonon mean free path of graphite along the c-axis Appl Phys Lett 104 081903 (2014) [2] D W Brenner O A Shenderova J A Harrison S J Stuart B Ni and S B Sinnott A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons J Phys Condens Matter 14 783 (2002) [3] A Kınacı J B Haskins C Sevik and T Ccedilağın Thermal conductivity of BN-C nanostructures Phys Rev B 86 (2012) [4] W Ouyang I Azuri D Mandelli A Tkatchenko L Kronik M Urbakh and O Hod Mechanical and Tribological Properties of Layered Materials under High Pressure Assessing the Importance of Many-Body Dispersion Effects J Chem Theory Comput 16 666 (2020) [5] S Plimpton Fast Parallel Algorithms for Short-Range Molecular Dynamics J Comput Phys 117 1 (1995) [6] W Ouyang D Mandelli M Urbakh and O Hod Nanoserpents Graphene Nanoribbon Motion on Two-Dimensional Hexagonal Materials Nano Lett 18 6009 (2018) [7] W Shinoda M Shiga and M Mikami Rapid estimation of elastic constants by molecular dynamics simulation under constant stress Phys Rev B 69 134103 (2004) [8] G L Pollack Kapitza Resistance Rev Mod Phys 41 48 (1969) [9] H-S Yang G R Bai L J Thompson and J A Eastman Interfacial thermal resistance in nanocrystalline yttria-stabilized zirconia Acta Mater 50 2309 (2002) [10] Q Fu J Yang Y Chen D Li and D Xu Experimental evidence of very long intrinsic phonon mean free path along the c-axis of graphite Appl Phys Lett 106 031905 (2015) [11] J Chen J H Walther and P Koumoutsakos Strain Engineering of Kapitza Resistance in Few-Layer Graphene Nano Lett 14 819 (2014) [12] H Zhang X Chen Y-D Jho and A J Minnich Temperature-Dependent Mean Free Path Spectra of Thermal Phonons Along the c-Axis of Graphite Nano Lett 16 1643 (2016) [13] D P Sellan E S Landry J E Turney A J H McGaughey and C H Amon Size effects in molecular dynamics thermal conductivity predictions Phys Rev B 81 214305 (2010) [14] P Jiang X Qian R Yang and L Lindsay Anisotropic thermal transport in bulk hexagonal boron nitride Phys Rev Mater 2 064005 (2018) [15] J Zhang Y Hong and Y Yue Thermal transport across graphene and single layer hexagonal boron nitride J Appl Phys 117 134307 (2015) [16] B Liu F Meng C D Reddy J A Baimova N Srikanth S V Dmitriev and K Zhou Thermal transport in a graphenendashMoS2 bilayer heterostructure a molecular dynamics study RSC Advances 5 29193 (2015) [17] Z Ding Q-X Pei J-W Jiang W Huang and Y-W Zhang Interfacial thermal conductance in grapheneMoS2 heterostructures Carbon 96 888 (2016) [18] N N T Nam and M Koshino Lattice relaxation and energy band modulation in twisted bilayer graphene Phys Rev B 96 075311 (2017)

S12

[19] R Ramiacutearez and M C Boumlhm Simple geometric generation of special points in brillouin-zone integrations Two-dimensional bravais lattices Int J Quantum Chem 30 391 (1986) [20]S L Cunningham Special points in the two-dimensional Brillouin zone Phys Rev B 10 4988 (1974) [21] H J Monkhorst and J D Pack Special points for Brillouin-zone integrations Phys Rev B 13 5188 (1976) [22] D J Chadi Special points for Brillouin-zone integrations Phys Rev B 16 1746 (1977)

1

Fermirsquos golden rule for phonons

1 Basic theory for phonons 11 Basic notations

Let us consider a 3D crystal with a total of 119873119873 = 119873119873111987311987321198731198733 unit cells and periodic boundary conditions

To be specific let 119938119938119894119894 119894119894 = 123 be the lattice vectors that define the unit cell We index unit cells

with n = (n1n2n3) where each ni = 12 ∙∙∙ Ni and their locations are 119929119929119899119899 = sum 1198991198991198941198941199381199381198941198943119894119894=1 Assume that

there are r atoms in each unit cell which are indexed with 119904119904 = 1⋯ 119903119903 The mass and the equilibrium

distance of the sth atom are notated as 119872119872s and 119929119929s0 respectively Then the location of the sth atom in

the nth unit cell at time t can be expressed as

119955119955119899119899119899119899(119905119905) = 119929119929119899119899 + 119929119929s0 + 119958119958119899119899119899119899(119905119905) (11)

where 119958119958119899119899119899119899(119905119905) is its displacement from its equilibrium position

The Lagrangian for this classical problem can be written as

ℒ = sum sum 119872119872119904119904|119955119899119899119904119904|2(119905119905)2

119903119903119899119899=1

119873119873119899119899=1 minus 119881119881 (12)

where the second term is the sum of interactions between all pairs of atoms

Under the harmonic approximation ie expanding the total potential energy 119881119881 around the

equilibrium positions The Lagrangian can be simplified as

ℒ = sum sum 119872119872119904119904|119958119899119899119904119904|2(119905119905)2

119903119903119899119899=1

119873119873119899119899=1 minus 1

2sum sum sum 120601120601120572120572120572120572prime

119899119899119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime (13)

where 119906119906119899119899119899119899120572120572 120572120572 = 123 are the Cartesian coordinates of the displacement 119958119958119899119899119899119899(119905119905) and

120601120601120572120572120572120572prime 119899119899119899119899prime119904119904 119904119904prime = 1205971205972119881119881

120597120597119903119903119899119899119904119904119899119899119903119903119899119899prime119904119904prime119899119899primeeq

= 120601120601120572120572prime120572120572 119899119899prime119899119899119904119904prime 119904119904 = 120601120601120572120572120572120572prime

119899119899 minus 119899119899prime119904119904 119904119904prime (14)

Note that the first order term vanishes because we are expanding around the equilibrium positions

120601120601120572120572120572120572prime 119899119899119899119899prime119904119904 119904119904prime represents the component of the force acted on the sth atom in the nth unit cell along 120572120572

direction when the atom 119904119904prime in the unit cell 119899119899prime moves a unit displacement along 120572120572prime direction The

symmetries of 120601120601120572120572120572120572prime 119899119899 119899119899prime119904119904 119904119904prime appearing in Eq (14) arise from the intechangability of the second

derivative and the translational invariance of the interactions

12 Dynamical matrix

The equation of motion of the sth atom in the nth unit cell can be derived using the EulerndashLagrange

equation as follows

2

119872119872119899119899119906119899119899119899119899120572120572 = minussum 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119899119899prime119899119899prime120572120572prime 119906119906119899119899prime119899119899prime120572120572prime (15)

If we displace all atoms equally ie shifting 119906119906119899119899prime119899119899prime120572120572prime to 119906119906119899119899prime119899119899prime120572120572prime + 120575120575 the total force on the sth atom in

the nth unit cell does not change From the above equation we have

sum 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119899119899prime120572120572prime = 0 (16)

We are looking for normal modes (because any general solution can be written as a linear combination

of them) these are solutions where all atoms oscillate with the same frequency Moreover because

of the lattice structure we expect solutions to reflect this periodicity So we guess solutions of the

form

119906119906119899119899119899119899120572120572(119905119905) = 1119872119872119904119904

119890119890119899119899120572120572119890119890119894119894(119954119954∙119929119929119899119899minus120596120596119905119905) (17)

where 119890119890119899119899120572120572 are real-space solutions that will be determined later and 119954119954 is the wave-vector in

reciprocal space Substituting Eq (17) into Eq (15) we can get

1205961205962119890119890119899119899120572120572(119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)119890119890119899119899prime120572120572prime(119954119954)119899119899prime120572120572prime (18)

where

119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = sum 1

119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime

119899119899 minus 119899119899prime119904119904 119904119904prime 119890119890

minus119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime119899119899prime = sum 1

119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime 119890119890

minus119894119894119954119954∙119929119929119897119897119897119897 (19)

is called dynamical matrix (dimension 3119903119903 times 3119903119903) Note that we have defined the relative cell distance

vector 119929119929119897119897 equiv 119929119929119899119899 minus 119929119929119899119899prime and number index 119897119897 equiv 119899119899 minus 119899119899prime where the infinite sum over 119899119899prime can be

replaced by the sum over 119897119897 for any value of the index 119899119899 Note that dynamical matrix is Hermitian

symmetric (ie 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)

lowast= 119863119863119899119899prime120572120572prime

119899119899120572120572 (minus119954119954)) because ϕ is symmetric so all the eigenvalues of Eq (18)

(120596120596120582120582(119954119954) 120582120582 = 12⋯ 3119903119903) are real for each 119954119954 in the Brillouin zone (BZ) which is determined by

det1205961205961205821205822(119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = 0 (110)

Taking the conjugate of this equation we have

0 = det 1205961205961205821205822(119954119954)lowast120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899

prime120572120572prime(119954119954)lowast = det1205961205961205821205822(119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899

prime120572120572prime(minus119954119954) (111)

while replacing 119954119954 by minus119954119954 in Eq (110) we have det1205961205961205821205822(minus119954119954)120575120575119899119899120572120572120575120575119899119899prime120572120572prime minus 119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954) = 0

Itrsquos clear that 1205961205961205821205822(119954119954) and 1205961205961205821205822(minus119954119954) obey the same equation thus we have

120596120596120582120582(minus119954119954) = 120596120596120582120582(119954119954) (112)

The corresponding eigenvectors are orthonormal

sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582

prime119899119899120572120572 = 120575120575120582120582120582120582prime (113)

The complex conjugate of Eq (18) gives

1205961205962119890119890119899119899120572120572lowast (119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(119954119954)

lowast119890119890119899119899prime120572120572primelowast (119954119954)119899119899prime120572120572prime = sum 119863119863119899119899prime120572120572prime

119899119899120572120572 (minus119954119954)119890119890119899119899prime120572120572primelowast (119954119954)119899119899prime120572120572prime (114)

3

While replacing 119954119954 by minus119954119954 in Eq (18) we get the following equation

1205961205962119890119890119899119899120572120572(minus119954119954) = sum 119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime(minus119954119954)119899119899prime120572120572prime (115)

From Eq (114) and Eq (115) we see that eigenvectors 119890119890119899119899120572120572120582120582 (minus119954119954)lowast and 119890119890119899119899120572120572120582120582 (119954119954) obey the same

eigenvalue equation Since the eigenvectors are normalized we get the following property

119890119890119899119899120572120572120582120582 (minus119954119954)lowast

= 119890119890119899119899120572120572120582120582 (119954119954) (116)

2 Second quantization The general solution is a linear combination of all these normal modes thus we have

119906119906119899119899119899119899120572120572(119905119905) = sum 119862119862120582120582(119954119954)119873119873119872119872119904119904

119890119890119899119899120572120572120582120582 (119954119954)119890119890minus119894119894120596120596119954119954120582120582119905119905119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 = sum 119876119876120582120582(119954119954119905119905)119873119873119872119872119904119904

119890119890119899119899120572120572120582120582 (119954119954)119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 (21)

where the we define the normal coordinates as 119876119876120582120582(119954119954 119905119905) equiv 119862119862120582120582(119954119954)119890119890minus119894119894120596120596120582120582(119954119954)119905119905 in the eigenvectors

representation To ensure that the displacements 119906119906119899119899119899119899120572120572(119905119905) are real (namely 119906119906119899119899119899119899120572120572lowast (119905119905) = 119906119906119899119899119899119899120572120572(119905119905)) the

following relation on 119876119876120582120582(119954119954 119905119905) and 119890119890119899119899120572120572120582120582 is enforced

119876119876120582120582(119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)lowast

= 119876119876120582120582(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (minus119954119954) (22)

where we used the fact that the sum over 119954119954 runs symmetrically over both negative and positive

values Using Eq (116) we have

[119876119876120582120582(119954119954 119905119905)]lowast = 119876119876120582120582(minus119954119954 119905119905) (23)

21 Kinetic energy term

Using Eq (21) the kinetic energy of the system can be expressed as

119879119879 = 12sum 1198721198721198991198991199061198991198991198991198991205721205722 (119905119905)119899119899119899119899120572120572 = 1

2sum 119872119872119904119904

119873119873119872119872119904119904sum sum 119876120582120582(119954119954 119905119905)119876120582120582prime(119954119954prime 119905119905)119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582

prime(119954119954prime) sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899119954119954prime120582120582prime119954119954120582120582119899119899120572120572 =

12sum sum 119876120582120582(119954119954 119905119905)119876120582120582prime(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582

prime(minus119954119954)119954119954119954119954prime120582120582prime =119899119899120572120572

12sum 119876120582120582(119954119954 119905119905)119876120582120582prime

lowast (119954119954 119905119905) sum 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime(119954119954)

lowast119899119899120572120572 =119954119954120582120582120582120582prime

12sum 119876120582120582(119954119954 119905119905)119876120582120582lowast(119954119954 119905119905)119954119954120582120582 =

12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582

ie

119879119879 = 12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582 (24)

To derive Eq (24) we used Eqs (113) and (116) as well as the following equations

1119873119873sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899 = 120575120575119954119954+119954119954prime120782120782 (25)

4

22 Potential energy term

Similarity the potential energy can be rewritten in terms of the normal coordinates as follows

119880119880 =12120601120601119899119899119899119899120572120572119899119899prime119899119899prime120572120572prime119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime

120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime=

12120601120601120572120572120572120572prime

119899119899 minus 119899119899prime119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119899119899+119954119954prime∙119929119929119899119899prime

119873119873119872119872119899119899119872119872119899119899prime

[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)

119954119954prime120582120582prime119954119954120582120582119899119899119899119899prime120572120572120572120572prime119899119899119899119899prime

=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894(119954119954+119954119954prime)∙119929119929119899119899119890119890minus119894119894119954119954prime∙119929119929119897119897

119873119873119872119872119899119899119872119872119899119899prime


119954119954prime120582120582prime119954119954120582120582119899119899119897119897120572120572120572120572prime119899119899119899119899prime

=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954prime∙119929119929119897119897

119872119872119899119899119872119872119899119899prime[119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(119954119954prime 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime

120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899

119899119899=120575120575119902119902+119902119902prime=0


=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954119929119929119897119897

119872119872119899119899119872119872119899119899prime [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)] 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime

120582120582prime (minus119954119954)119954119954120582120582120582120582prime119897119897120572120572120572120572prime119899119899119899119899prime

=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)

119899119899120572120572

119863119863119899119899120572120572119899119899prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime


=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)

119899119899120572120572

1205961205961205821205822(minus119954119954)119890119890119899119899120572120572120582120582prime (minus119954119954)

119954119954120582120582120582120582prime

=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]1205961205961205821205822(119954119954)120575120575120582120582120582120582prime =119954119954120582120582120582120582prime

121205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)

119954119954120582120582

ie

119880119880 = 12sum 1205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)119954119954120582120582 (26)

where in above deriviation the orthogonality of its eigenvectores and the symmetry of its eigenvalues

are used Thus the Lagrangian reads as

ℒ = 119879119879 minus 119881119881 = 12sum 119876120582120582(minus119954119954 119905119905)119876120582120582(119954119954 119905119905) minus 120596120596120582120582

2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)119954119954120582120582 (27)

Using the relation 119875119875120582120582(119954119954 119905119905) = 120597120597ℒ part119876120582120582(119954119954 119905119905) the Hamiltonian of the system then can be written as

119867119867 = 119879119879 + 119881119881 = 12sum [119875119875120582120582(minus119954119954 119905119905)119875119875120582120582(119954119954 119905119905) + 120596120596120582120582

2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)]119954119954120582120582 (28)

Now we quantize H by asking the momenta and coordinates to be operators

119867119867 = 12sum 119875119875minus119954119954120582120582119875119875119954119954120582120582 + 120596120596119954119954120582120582

2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582 (29)

here 119875119875119954119954120582120582 and 119876119876119954119954120582120582 obey the following commutation relations

119876119876119954119954120582120582119875119875119954119954prime120582120582prime = 119894119894ℏ120575120575119954119954119954119954prime120575120575120582120582120582120582prime

119876119876119954119954120582120582119876119876119954119954prime120582120582prime = 0 119875119875119954119954120582120582119875119875119954119954prime120582120582prime = 0 (210)

Similar to the case of ordinary quantum harmonic oscillators it is convenient to define ladder

operators for each mode as follows

119876119876119954119954120582120582 = ℏ

2120596120596119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582

dagger 119875119875119954119954120582120582 = 119894119894ℏ120596120596119954119954120582120582

2119887119887119954119954120582120582

dagger minus 119887119887minus119954119954120582120582 (211)

where 119887119887119954119954120582120582dagger and 119887119887119954119954120582120582 are Bosonic creation and annihilation operators for phonons with momentum

119954119954 branch index 120582120582 and frequency 120596120596119954119954120582120582 which obeys the Bosonic commutation relation

5

119887119887119954119954120582120582 119887119887119954119954prime120582120582prime

dagger = 120575120575119954119954119954119954prime120575120575120582120582120582120582prime

119887119887119954119954120582120582 119887119887119954119954prime120582120582prime = 0 119887119887119954119954120582120582dagger 119887119887119954119954prime120582120582prime

dagger = 0 (212)

Substituting Eqs (211) into (29) and using the properties Eq (210) and (212) we have

119867119867 =12119875119875minus119954119954120582120582119875119875119954119954120582120582 +120596120596119954119954120582120582

2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582

=12minus

ℏ120596120596119954119954120582120582

2119887119887minus119954119954120582120582

dagger minus 119887119887119954119954120582120582119887119887119954119954120582120582dagger minus 119887119887minus119954119954120582120582+ 120596120596119954119954120582120582

2 ℏ2120596120596119954119954120582120582

119887119887minus119954119954120582120582 + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 119887119887minus119954119954120582120582

dagger 119954119954120582120582

=ℏ4120596120596119954119954120582120582minus119887119887minus119954119954120582120582

dagger 119887119887119954119954120582120582dagger minus 119887119887minus119954119954120582120582

dagger 119887119887minus119954119954120582120582 minus 119887119887119954119954120582120582119887119887119954119954120582120582dagger + 119887119887119954119954120582120582119887119887minus119954119954120582120582

119954119954120582120582

+ 119887119887minus119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582119887119887minus119954119954120582120582dagger + 119887119887119954119954120582120582

dagger 119887119887119954119954120582120582 + 119887119887119954119954120582120582dagger 119887119887minus119954119954120582120582

dagger

=ℏ4120596120596119954119954120582120582119887119887minus119954119954120582120582

dagger 119887119887minus119954119954120582120582 + 119887119887minus119954119954120582120582119887119887minus119954119954120582120582dagger + 119887119887119954119954120582120582119887119887119954119954120582120582

dagger + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582

119954119954120582120582

=ℏ41205961205961199541199541205821205822119887119887minus119954119954120582120582

dagger 119887119887minus119954119954120582120582 + 1+ 2119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1

119954119954120582120582

= ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 +

12

119954119954120582120582

Therefore we finally get the quantized representation of non-interacting phonons

119867119867 = sum ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1

2119954119954120582120582 (213)

The operator of atom displacements (Eq 114) is expressed in terms of the phonon operators by

119906119906119899119899119899119899120572120572 = sum ℏ

2119873119873119872119872119904119904120596120596119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119887119887119954119954120582120582 + 119887119887minus119954119954120582120582

dagger 119954119954120582120582 (214)

These equations will be used below

3 Inter-phonon coupling within harmonic approximation 31 Hamiltonian with inter-phonon coupling

In this section we consider systems that consist of two (or more) covalently bonded units that are

weakly coupled between them Unlike previous studies that considered phonon-phonon couplings

resulting from anharmonicity effects [1] here all phonon mode considered are Harmonic and the

couplings arise from the division of the entire system into subunits The Hamiltonian of the whole

system can be written as

119867119867 = 1198671198671 + 1198671198672 + 11986711986712 (31)

To derive the expressions of 1198671198671 1198671198672 and 11986711986712 in Eq (31) we consider the Hamiltonian of the

whole system written as the function of the atomic displacements

119867119867 = 119879119879 + 119881119881 = sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2119899119899119899119899120572120572 + 1

2sum sum sum 120601120601120572120572120572120572prime

119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime120572120572120572120572prime119899119899119899119899prime119899119899119899119899prime (32)

6

Letrsquos assume that subsystem I and II contain atoms with indeices ranging from 119904119904 = 12⋯ 1199031199032 and

119904119904 = 1199031199032

+ 1 1199031199032

+ 2⋯ 119903119903 respectively Then the kinetic energy term in Eq (32) can be rewritten as

119879119879 = sum sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2

1199031199032119899119899=1 + sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)

2119903119903119899119899=1199031199032+1

119899119899120572120572 (33)

While the potential energy term is

119881119881 =12 120601120601120572120572120572120572prime

119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime

1199031199032

119899119899119899119899prime=1

+ 120601120601120572120572120572120572prime 119899119899 minus 119899119899prime119904119904 119904119904prime 119906119906119899119899119899119899120572120572119906119906119899119899prime119899119899prime120572120572prime

119903119903

119899119899119899119899prime=1199031199032+1


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime119899119899119899119899prime


1199031199032

119899119899prime=1

119903119903

119899119899=1199031199032+1

= 11988111988111 + 11988111988122 + 11988111988112 + 11988111988121

Or equivalently

⎩⎪⎪⎨

⎪⎪⎧ 11988111988111 = 1

2sum sum sum 120601120601120572120572120572120572prime


1199031199032119899119899119899119899prime=1120572120572120572120572prime119899119899119899119899prime

11988111988122 = 12sum sum sum 120601120601120572120572120572120572prime


119903119903119899119899119899119899prime=1199031199032+1120572120572120572120572prime119899119899119899119899prime

11988111988112 = 12sum sum sum sum 120601120601120572120572120572120572prime


119903119903119899119899prime=1199031199032+1

1199031199032119899119899=1120572120572120572120572prime119899119899119899119899prime



1199031199032119899119899prime=1

119903119903119899119899=1199031199032+1120572120572120572120572prime119899119899119899119899prime

(34)

32 Second quantization

We may now quantize this Hamiltonian and write it in the basis of the eigenstates of the coupled

subunits In second quantization the atomic displacement operators of the two subunits are given in

the following form

119906119906119899119899119899119899120572120572 =

⎩⎨

⎧ sum ℏ

2119873119873119872119872119904119904ω119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger 119954119954120582120582 119904119904 isin 1 1199031199032

sum ℏ

2119873119873119872119872119904119904120596120596119954119954prime120582120582prime119890119899119899120572120572120582120582

prime119890119890119894119894119954119954prime∙119929119929119899119899 119886119886 119954119954prime120582120582prime + 119886119886minus119954119954prime120582120582primedagger 119954119954prime120582120582prime 119904119904 isin 119903119903

2+ 1 119903119903

(35)

Here we use different notations for the creation and annihilation operators (119886119886119954119954120582120582119886119886minus119954119954120582120582dagger and 119886119886119954119954120582120582119886119886minus119954119954120582120582

dagger )

eigenvalues (ω119954119954120582120582120596120596119954119954120582120582) and eigenvectors (119890119890119899119899120572120572120582120582 119890119899119899120572120572120582120582 ) for the two subunits Note that 119890119890119899119899120572120572120582120582 and 119890119899119899120572120572120582120582 are

of the dimensions of the whole system however their non-zero elements appear only on the relevant

subunits such that they obey the following relations sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582

prime119899119899120572120572 = 120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582

lowast119890119899119899120572120572120582120582

prime119899119899120572120572 =

120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582

prime119899119899120572120572 = sum 119890119890119899119899120572120572120582120582

lowast119890119899119899120572120572120582120582

prime119899119899120572120572 = 0 Defining the normal coordinates of the two subunits as

119876119876119954119954120582120582 equiv ℏ

2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger 119876119876119954119954120582120582 equiv ℏ

2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger (36)

and the corresponding momenta operators as

7

119875119875119954119954120582120582 = 119894119894ℏω119954119954120582120582

2119886119886119954119954120582120582

dagger minus 119886119886minus119954119954120582120582 119875119875119954119954120582120582 = 119894119894ℏω1199541199541205821205822

119886119886119954119954120582120582dagger minus 119886119886minus119954119954120582120582 (37)

Eq (35) reads

119906119906119899119899119899119899120572120572 = sum 119890119890119904119904119899119899120582120582 (119954119954)

119873119873119872119872119904119904119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1 119903119903

2

sum 119890119904119904119899119899120582120582 (119954119954)119873119873119872119872119904119904

119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1199031199032

+ 1 119903119903 (38)

The second quantized kinetic energy operator is then written as

119879119879 = 1198791198791 + 1198791198792 = 12sum 119875119875119954119954120582120582119875119875minus119954119954120582120582119954119954120582120582 + 1

2sum 119875119875119954119954prime120582120582prime119875119875minus119954119954prime120582120582prime119954119954prime120582120582prime (39)

Correspondingly the various potential energy terms are obtained by substituting Eq (38) in Eq (34)

11988111988111 =12 120601120601120572120572120572120572prime


119899119899119899119899prime

119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime120582120582prime (119954119954prime)

119872119872119899119899119872119872119899119899prime

1119873119873119890119890119894119894119954119954∙119929119929119899119899+119894119894119954119954prime∙119929119929119899119899prime119876119876119954119954120582120582119876119876119954119954prime120582120582prime

119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime

119954119954119954119954prime120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime

120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime


120582120582prime (119954119954prime) 120601120601120572120572120572120572prime 119897119897

119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime

119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime

120582120582prime (minus119954119954)120601120601120572120572120572120572prime 119897119897

119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime

119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954) 119863119863119899119899120572120572119899119899

prime120572120572prime(minus119954119954)119890119890119899119899prime120572120572prime120582120582prime (minus119954119954)

120572120572prime

1199031199032

119899119899prime=1

1199031199032

119899119899=1120572120572

=12 ω119954119954120582120582prime

2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582120582120582prime

119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime (119954119954)

lowast

1199031199032

119899119899=1

=120572120572

12ω119954119954120582120582

2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582

ie

11988111988111 = 12sum ω119954119954120582120582

2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582 (310)

Going from the second line to the third line we used the fact that the sum over 119899119899prime runs between plusmninfin

and the summand depends only on the difference between 119899119899 and 119899119899prime hence the sum is independent

of the value of the index 119899119899 Therefore we can replace the sum over 119899119899prime by a sum over 119897119897 equiv 119899119899 minus 119899119899prime

amd define 119929119929119897119897 equiv 119929119929119899119899 minus 119929119929119899119899prime Following the same procedure we can get the corresponding

expressions for the second diagonal term

11988111988122 = 12sum 1205961205961199541199541205821205822 119876119876119954119954120582120582119876119876minus119954119954120582120582119954119954120582120582 (311)

Similarly for the off-diagonal terms we get

8

11988111988112 =12 120601120601120572120572120572120572prime


119899119899119899119899prime

119903119903

119899119899prime=1199031199032+1

119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime

120582120582prime (119954119954prime)

119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954

119890119899119899prime120572120572prime120582120582prime (119954119954)

lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119890119890119899119899120572120572120582120582 (119954119954)119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954

Where we have defined

119881119881120582120582120582120582prime(119954119954) equiv sum sum sum 119890119899119899prime120572120572prime120582120582prime (119954119954)

lowast119881119881119899119899120572120572119899119899

prime120572120572prime(119954119954)119890119890119899119899120572120572120582120582 (119954119954)119903119903119899119899prime=1199031199032+1

1199031199032119899119899=1120572120572120572120572prime (312)

where

119881119881119899119899120572120572119899119899prime120572120572prime(119954119954) equiv sum 119890119890119894119894119954119954∙119929119929119897119897

119872119872119904119904119872119872119904119904prime119897119897 120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime (313)

Itrsquos easy to show that 119881119881120582120582120582120582prime(119954119954) has the following property

119881119881120582120582120582120582primelowast (119954119954) = 119881119881120582120582120582120582prime(minus119954119954) (314)

Following the same procedure we have

9

11988111988121 =12 120601120601120572120572120572120572prime


119899119899119899119899prime

119903119903

119899119899=1199031199032+1

119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime

120582120582prime (119954119954prime)

119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899prime=1120572120572120572120572prime

=12 120601120601120572120572prime120572120572

119899119899prime minus 119899119899119904119904prime 119904119904

119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1

119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954)

119872119872119899119899prime119872119872119899119899

1119873119873 119890119890

119894119894119954119954prime∙119929119929119899119899prime+119894119894119954119954∙119929119929119899119899119876119876119954119954prime120582120582prime119876119876119954119954120582120582119954119954prime120582120582prime119954119954120582120582

1199031199032

119899119899=1120572120572prime120572120572

=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582

119954119954119954119954prime120582120582120582120582prime119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954)

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582

119954119954119954119954prime120582120582120582120582prime119890119899119899prime120572120572prime120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (119954119954) 120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954prime120582120582prime119876119876minus119954119954prime120582120582119890119899119899prime120572120572prime

120582120582prime (119954119954prime)119890119890119899119899120572120572120582120582 (minus119954119954prime)120601120601120572120572120572120572prime 119897119897

119904119904 119904119904prime119890119890minus119894119894119954119954prime∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897119954119954prime120582120582120582120582prime

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954prime120582120582prime119876119876119954119954prime120582120582

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954prime 119890119890119899119899120572120572120582120582 (119954119954prime)

lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954prime∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119890119899119899prime120572120572prime120582120582prime (119954119954prime)

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582prime119876119876119954119954120582120582

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954

119890119890119899119899120572120572120582120582 (119954119954)lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger dagger

3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954


lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119890119890119899119899120572120572120582120582 (119954119954)119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

lowast

=

⎩⎨

⎧12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954⎭⎬

⎫dagger

= 11988111988112dagger

Define

1198671198671 = 11987911987911 + 119881119881111198671198672 = 11987911987922 + 1198811198812211986711986712 = 11988111988112 + 11988111988121

(315)

we finally get the expressions of 1198671198671 1198671198672 and 11986711986712 in Eq (31) as follows

⎩⎪⎨

⎪⎧ 1198671198671 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582

dagger 119886119886119954119954120582120582 + 1231199031199032

120582120582=1119954119954

1198671198672 = sum sum ℏ120596120596119954119954120582120582prime 119886119886119954119954120582120582primedagger 119886119886119954119954120582120582prime + 1

23119903119903

120582120582prime=1+31199031199032119954119954

11986711986712 = 12sum sum sum 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903120582120582prime=31199031199032 +1

31199031199032120582120582=1 + h c 119954119954

(316)

Here hc means the Hermitian conjugate Since the indices of 119886119886119954119954120582120582 119886119886119954119954120582120582prime and 119876119876119954119954120582120582 119876119876119954119954120582120582prime belong to

the two system sections we can define an abbreviated notation 119886119886119954119954120582120582 and 119876119876119954119954120582120582 using the index 120582120582 to

identify which subsystem they belong to In this case the Hamiltonian of the coupled system can be

10

simplified as follows

119867119867 = 1198671198670 + 119867119867119862119862 (317)

where

1198671198670 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582

dagger 119886119886119954119954120582120582 + 123119903119903

120582120582=1119954119954


dagger3119903119903120582120582prime=31199031199032 +1

31199031199032120582120582=1 + ℎ 119888119888 119954119954

(318)

4 Greenrsquos function 41 Dynamics of the ladder operators

In order to describe the dynamics of the ladder operators appearing in the Hamiltonian of Eq (317)

we express them in the Heisenberg picture as follows

119886119886119901119901(120591120591) = 119890119890119894119894ℏ119867119867119905119905119886119886119901119901119890119890

minus119894119894ℏ119867119867119905119905 = 119890119890

119867119867120591120591ℏ 119886119886119901119901119890119890

minus119867119867120591120591ℏ (41)

where we define the imaginary time 120591120591 equiv 119894119894119905119905 and introduce the notation 119953119953 equiv (119954119954 120582120582) and 119953119953 equiv (minus119954119954 120582120582)

The corresponding equation of motion for the ladder operators is given by

ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119867119867119886119886119953119953(120591120591) (42)

For the uncoupled system (119867119867119862119862 = 0) this gives

ℏpart119886119886119953119953(120591120591)120597120597120591120591

= 1198671198670119886119886119953119953(120591120591) = 1198671198670 1198901198901198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ = 1198671198670119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ minus 119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ 1198671198670

= 1198901198901198671198670120591120591ℏ 1198671198670119886119886119953119953 minus 1198861198861199531199531198671198670119890119890

minus1198671198670120591120591ℏ = 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ

ie

ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ (43)

The commutator on the right-hand-side of Eq (43) can be evaluated as follows

1198671198670119886119886119953119953prime = sum ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 + 1

2119953119953 119886119886119953119953prime = sum ℏ120596120596119953119953119886119886119953119953

dagger119886119886119953119953119886119886119953119953prime119953119953 = sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 119886119886119953119953prime119886119886119953119953

dagger119886119886119953119953119953119953 =

sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 120575120575119953119953prime119953119953 + 119886119886119953119953

dagger119886119886119953119953prime119886119886119953119953119953119953 = minusℏ120596120596119953119953prime119886119886119953119953prime + sum ℏ120596120596119953119953119886119886119953119953dagger119886119886119953119953119886119886119953119953prime minus 119886119886119953119953

dagger119886119886119953119953119886119886119953119953prime119953119953 = minusℏ120596120596119953119953prime119886119886119953119953prime

ie

1198671198670119886119886119953119953prime = minusℏ120596120596119953119953prime119886119886119953119953prime (44)

Therefore we have

ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119886119886119953119953(120591120591) (45)

11

The solution of Eq (45) is 119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591 For 119886119886119953119953dagger(120591120591) we have

1198671198670119886119886119953119953primedagger (0) = 1198671198670119886119886119953119953prime

dagger = ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 +

12

119953119953

119886119886119953119953primedagger = ℏ120596120596119953119953119886119886119953119953

dagger119886119886119953119953119886119886119953119953primedagger

119953119953

= ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953119886119886119953119953prime

dagger minus 119886119886119953119953primedagger 119886119886119953119953

dagger119886119886119953119953119953119953

= ℏ120596120596119953119953 119886119886119953119953dagger 120575120575119953119953119953119953prime + 119886119886119953119953prime

dagger 119886119886119953119953 minus 119886119886119953119953primedagger 119886119886119953119953

dagger119886119886119953119953119953119953

= ℏ120596120596119953119953prime119886119886119953119953primedagger + ℏ120596120596119953119953 119886119886119953119953

dagger119886119886119953119953primedagger 119886119886119953119953 minus 119886119886119953119953prime

dagger 119886119886119953119953dagger119886119886119953119953

119953119953

= ℏ120596120596119953119953prime119886119886119953119953primedagger

In summary we have

119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591

119886119886119953119953dagger(120591120591) = 119886119886119953119953

dagger(0)119890119890120596120596119953119953120591120591 = 119886119886119953119953dagger119890119890120596120596119953119953120591120591 (46)

42 Greenrsquos function

To describe thermal transport properties between different system sections we use the formalism of

thermal (or imaginary time) phonon Greenrsquos function [2] To this end we define the thermal Greenrsquos

function for phonons as

119866119866119953119953119953119953prime(120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)rang = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime

dagger (120591120591prime) (47)

where 119879119879120591120591 is the time ordering operator and 120588120588119867119867 is the statistical operator for the grand canonical

ensemble (note that the chemical potential for phonons is zero)

120588120588119867119867 = 119890119890minus120573120573119867119867119885119885119867119867 (48)

where the partition function is given by 119885119885119867119867 equiv Tr119890119890minus120573120573119867119867 and 120573120573 = 1119896119896119861119861119879119879 with 119896119896119861119861 being

Boltzmannrsquos constant and 119879119879 the temperature For time independent Hamiltonians 119866119866119953119953119953119953prime(120591120591 120591120591prime)

depends only on 120591120591 minus 120591120591prime ie

119866119866119953119953119953119953prime(120591120591 120591120591prime) = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0) (49)

To show this we shall first assume that 120591120591 gt 120591120591prime such that


dagger (120591120591prime) = minusTr 120588120588119867119867119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime)

= minusTr 120588120588119867119867119890119890119867119867120591120591ℏ 119886119886119953119953119890119890

minus119867119867120591120591ℏ 119890119890

119867119867120591120591primeℏ 119886119886119953119953prime

dagger 119890119890minus119867119867120591120591primeℏ = minusTr 120588120588119867119867119890119890

minus119867119867120591120591primeℏ 119890119890

119867119867120591120591ℏ 119886119886119953119953119890119890

minus119867119867120591120591minus120591120591prime

ℏ 119886119886119953119953primedagger

= minusTr 120588120588119867119867119890119890119867119867120591120591minus120591120591prime

ℏ 119886119886119953119953119890119890minus119867119867120591120591minus120591120591prime

ℏ 119886119886119953119953primedagger = minusTr 120588120588119867119867119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime

dagger (0)

= minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953primedagger (0) = minus lang119879119879120591120591119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime

dagger (0)rang = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0)

Where we have used the commutativity of 120588120588119867119867 and 119890119890plusmn119867119867120591120591prime

ℏ and the invariance of the trace operation

12

towards cyclic permutations Similarly for 120591120591 lt 120591120591prime we have


dagger (120591120591prime) = minusTr 120588120588119867119867119886119886119953119953primedagger (120591120591prime)119886119886119953119953(120591120591)

= minusTr 120588120588119867119867119890119890119867119867120591120591primeℏ 119886119886119953119953prime

dagger 119890119890minus119867119867120591120591primeℏ 119890119890

119867119867120591120591ℏ 119886119886119953119953119890119890

minus119867119867120591120591ℏ = minusTr 120588120588119867119867119886119886119953119953prime

dagger 119890119890119867119867120591120591minus120591120591prime

ℏ 119886119886119953119953119890119890minus119867119867120591120591ℏ 119890119890

119867119867120591120591primeℏ

= minusTr 120588120588119867119867119886119886119953119953primedagger 119890119890

119867119867120591120591minus120591120591primeℏ 119886119886119953119953119890119890


ℏ = minusTr 120588120588119867119867119886119886119953119953primedagger (0)119886119886119953119953(120591120591 minus 120591120591prime)

= minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953primedagger (0)119886119886119953119953(120591120591 minus 120591120591prime) = minusTr 120588120588119867119867119879119879120591120591 119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953prime

dagger (0)

= minuslang119879119879120591120591119886119886119953119953(120591120591 minus 120591120591prime)119886119886119953119953primedagger (0)rang = 119866119866119953119953119953119953prime(120591120591 minus 120591120591prime 0)

For simplicity we will introduce the notation 119866119866119953119953119953119953prime(120591120591 0) equiv 119866119866119953119953119953119953prime(120591120591) We further note that when using

imaginary time 119866119866119953119953119953119953prime(120591120591) is a periodic functions in the domain [minus120573120573ℏ120573120573ℏ] with a period of 120573120573ℏ (see

Page 236 of Ref [2])

119866119866119953119953119953119953prime(120591120591 + 120573120573ℏ 0) = 119866119866119953119953119953119953prime(120591120591 0) 120591120591 lt 0119866119866119953119953119953119953prime(120591120591 minus 120573120573ℏ 0) = 119866119866119953119953119953119953prime(120591120591 0) 120591120591 gt 0 (410)

To show this we shall again assume first that minus120573120573ℏ lt 120591120591 lt 0 to write that is

119866119866119953119953119953119953prime(120591120591 + 120573120573ℏ 0) = minus lang119879119879120591120591119886119886119953119953(120591120591 + 120573120573ℏ)119886119886119953119953primedagger (0)rang = minuslang119886119886119953119953(120591120591 + 120573120573ℏ)119886119886119953119953prime

dagger (0)rang

= minusTr 120588120588119867119867119890119890119867119867(120591120591+120573120573ℏ)

ℏ 119886119886119953119953119890119890minus119867119867(120591120591+120573120573ℏ)

ℏ 119886119886119953119953prime = minusTr 120588120588119867119867119890119890120573120573119867119867119890119890

119867119867120591120591ℏ 119886119886119953119953119890119890

minus119867119867120591120591ℏ 119890119890minus120573120573119867119867119886119886119953119953prime

= minusTr120588120588119867119867119890119890120573120573119867119867119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0) = minus

1119885119885119867119867

Tr119890119890minus120573120573119867119867119890119890120573120573119867119867119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0)

= minus1119885119885119867119867

Tr119886119886119953119953(120591120591)119890119890minus120573120573119867119867119886119886119953119953prime(0) = minus1119885119885119867119867

Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119886119886119953119953(120591120591)

= minus1119885119885119867119867

Tr119890119890minus120573120573119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime(0) = minusTr120588120588119867119867119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime = 119866119866119953119953119953119953prime(120591120591 0)

Similarly for 120573120573ℏ gt 120591120591 gt 0 we have

119866119866119953119953119953119953prime(120591120591 minus 120573120573ℏ 0) = minus lang119879119879120591120591119886119886119953119953(120591120591 minus 120573120573ℏ)119886119886119953119953primedagger (0)rang = minuslang119886119886119953119953prime

dagger (0)119886119886119953119953(120591120591 minus 120573120573ℏ)rang

= minusTr 120588120588119867119867119886119886119953119953prime119890119890119867119867(120591120591minus120573120573ℏ)

ℏ 119886119886119953119953119890119890minus119867119867(120591120591minus120573120573ℏ)

ℏ = minusTr 120588120588119867119867119886119886119953119953prime119890119890minus120573120573119867119867119890119890

119867119867120591120591ℏ 119886119886119953119953119890119890

minus119867119867120591120591ℏ 119890119890120573120573119867119867

= minusTr120588120588119867119867119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591)119890119890120573120573119867119867 = minus1119885119885119867119867

Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591)119890119890120573120573119867119867

= minus1119885119885119867119867

Tr119886119886119953119953prime(0)119890119890minus120573120573119867119867119886119886119953119953(120591120591) = minus1119885119885119867119867

Tr119890119890minus120573120573119867119867119886119886119953119953(120591120591)119886119886119953119953prime(0)

= minus1119885119885119867119867


13

Therefore 119866119866119953119953119953119953prime(120591120591) can be expanded as a Fourier series in the domain [0120573120573ℏ] as follows

119866119866119953119953119953119953prime(120591120591) = 1120573120573ℏsum 119890119890minus119894119894120596120596119899119899120591120591119866119866119953119953119953119953prime(119894119894120596120596119899119899)infinminusinfin (411)

where 120596120596119899119899 = 2120587120587119899119899120573120573ℏ

and the associated Fourier coefficient is given by

119866119866119953119953119953119953prime(119894119894120596120596119899119899) = int d120591120591119890119890119894119894120596120596119899119899120591120591119866119866119953119953119953119953prime(120591120591)120573120573ℏ0 120596120596119899119899 = 2119899119899120587120587

120573120573ℏ (412)

Having proven the translational time invariance and the periodicity of the Greenrsquos functios we can

now calculate it for the uncoupled system

119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = minuslang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime

dagger (120591120591prime)rang0 == minus lang119886119886119953119953(0)119890119890minus120596120596119953119953120591120591119886119886119953119953prime

dagger (0)119890119890120596120596119953119953prime120591120591primerang 120591120591 minus 120591120591prime gt 0

minus lang119886119886119953119953primedagger (0)119890119890120596120596119953119953prime120591120591

prime119886119886119953119953(0)119890119890minus120596120596119953119953120591120591rang 120591120591 minus 120591120591prime lt 0

= minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591

primelang119886119886119953119953(0)119886119886119953119953prime

dagger (0)rang 120591120591 minus 120591120591prime gt 0

minus119890119890minus120596120596119953119953120591120591119890119890120596120596119953119953prime120591120591primelang119886119886119953119953prime

dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0


primelang119886119886119953119953prime

dagger (0)119886119886119953119953(0) + 119886119886119953119953(0)119886119886119953119953primedagger (0)rang 120591120591 minus 120591120591prime gt 0




primelang119886119886119953119953prime

dagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime 120591120591 minus 120591120591prime gt 0



ie

119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =




dagger (0)119886119886119953119953(0)rang 120591120591 minus 120591120591prime lt 0 (413)

where we have used Eq (46) for 119886119886119953119953(120591120591)119886119886119953119953primedagger (120591120591prime) and Commutation relation 119886119886119953119953(0)119886119886119953119953prime

dagger (0) = 120575120575119953119953119953119953prime

To calculate lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang we first prove following equation

119890119890119860119860119861119861119890119890minus119860119860 = sum 1119899119899119860(119899119899)119861119861infin

119899119899=0 (414)

where 119860(119896119896)119861119861 equiv 119860 119860(119896119896minus1)119861119861 To prove this identity we define the operator 119891119891(119905119905) = 119890119890119905119905A119861119861119890119890minus119905119905119860119860

and taylor expand it around 119905119905 = 0

119891119891(119905119905) = 119891119891(0) + 119905119905119891119891prime(0) + 1199051199052

2119891119891primeprime(0) + ⋯ = sum 119905119905119899119899

119899119899119889119889119899119899119891119889119889119905119905119899119899

119905119905=0

infin119899119899=0 (415)

The corresponding derivatives are given by

⎩⎪⎨

⎪⎧ 119891119891prime(119905119905) = 119890119890119905119905119860119860119860119861119861119890119890minus119905119905119860119860 minus 119890119890119905119905119860119860119861119861119860119890119890minus119905119905119860119860 = 119890119890119905119905119860119860119860119861119861119890119890minus119905119905119860119860

119891119891primeprime(119905119905) = 119890119890119905119905119860119860119860119860119861119861 minus 119860119861119861119860119890119890minus119905119905119860119860 = 119890119890119905119905119860119860119860(2)119861119861119890119890minus119905119905119860119860

⋮119891119891(119899119899)(119905119905) = 119890119890119905119905119860119860119860(119899119899)119861119861119890119890minus119905119905119860119860

(416)

14

Substituting Eq (416) into Eq (415) we have

119890119890119905119905A119861119861119890119890minus119905119905119860119860 = sum 119905119905119899119899

119899119899119860(119899119899)119861119861infin

119899119899=0 (417)

Itrsquos clear that Eq (413) is a spectial case of Eq (416) with 119905119905 = 1 With this we can proceed as

follows

lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang =

11198851198851198671198670

Tr119890119890minus1205731205731198671198670119886119886119953119953primedagger (0)119886119886119953119953(0) =

11198851198851198671198670

Tr119890119890minus1205731205731198671198670119886119886119953119953primedagger (0)1198901198901205731205731198671198670119890119890minus1205731205731198671198670119886119886119953119953(0)

=11198851198851198671198670

Tr (minus120573120573)119899119899

1198991198991198671198670

(119899119899)119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)

Note that 1198671198670119886119886119953119953primedagger (0) = ℏ120596120596119953119953prime119886119886119953119953prime

dagger (0) we have

1198671198670(119899119899)119886119886119953119953prime

dagger (0) = ℏ120596120596119953119953prime119899119899119886119886119953119953primedagger (0) (418)

such that


11198851198851198671198670

Tr (minus120573120573)119899119899

1198991198991198671198670

(119899119899)119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)

=11198851198851198671198670

Tr minus120573120573ℏ120596120596119953119953prime

119899119899

119899119899119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)

= 119890119890minus120573120573ℏ120596120596119953119953prime11198851198851198671198670

Tr119890119890minus1205731205731198671198670119886119886119953119953(0)119886119886119953119953primedagger (0) = 119890119890minus120573120573ℏ120596120596119953119953primelang119886119886119953119953(0)119886119886119953119953prime

dagger (0)rang

= 119890119890minus120573120573ℏ120596120596119953119953primelang119886119886119953119953primedagger (0)119886119886119953119953(0)rang + 120575120575119953119953119953119953prime

therefore we obtain lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang 119890119890120573120573ℏ120596120596119953119953prime minus 1 = 120575120575119953119953119953119953prime For 119953119953prime ne 119953119953 and general 119890119890120573120573ℏ120596120596119953119953prime we have

lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang = 0 For 119953119953prime = 119953119953 we obtain lang119886119886119953119953

dagger(0)119886119886119953119953(0)rang = 119890119890120573120573ℏ120596120596119953119953 minus 1minus1

Hence we have

lang119886119886119953119953primedagger (0)119886119886119953119953(0)rang = 120575120575119953119953prime119953119953

1119890119890120573120573ℏ120596120596119953119953minus1

equiv 120575120575119953119953prime119953119953119899119899119861119861120596120596119953119953 (419)

where 119899119899119861119861120596120596119953119953 is the Bose-Einstein distribution for phonons Substituting Eq (418) in Eq (413)

we obtain

119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =

minus120575120575119953119953119953119953prime1 + 119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591minus120591120591prime 120591120591 minus 120591120591prime gt 0

minus120575120575119953119953119953119953prime119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591minus120591120591prime 120591120591 minus 120591120591prime lt 0 (420)

Note that only those Greenrsquos functions of the form 119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = 1198661198661199531199530(120591120591 minus 120591120591prime)120575120575119953119953119953119953prime are non-zero due

to the orthogonality of normal modes Looking at the non-vanishing terms and setting 120591120591prime = 0

without loss of generality we can calculate the Fourier transform of 1198661198661199531199530(120591120591) from Eq (412)

1198661198661199531199530(119894119894120596120596119899119899) = int d1205911205911198901198901198941198941205961205961198991198991205911205911198661198661199531199530(120591120591)120573120573ℏ0 = minusint d120591120591119890119890119894119894120596120596119899119899120591120591120573120573ℏ

0 1 + 119899119899119861119861120596120596119953119953119890119890minus120596120596119953119953120591120591 = minus1+119899119899119861119861120596120596119953119953119894119894120596120596119899119899minus120596120596119953119953

119890119890119894119894120596120596119899119899minus120596120596119953119953120573120573ℏ minus

15

1 = minus1+ 1

119890119890120573120573ℏ120596120596119953119953minus1

119894119894120596120596119899119899minus120596120596119953119953119890119890119894119894

2120587120587119899119899120573120573ℏ 120573120573ℏ119890119890minus120573120573ℏ120596120596119953119953 minus 1 = minus 119890119890120573120573ℏ120596120596119953119953

119894119894120596120596119899119899minus120596120596119953119953

119890119890minus120573120573ℏ120596120596119953119953minus1

119890119890120573120573ℏ120596120596119953119953minus1= minus 1

119894119894120596120596119899119899minus120596120596119953119953

1minus119890119890120573120573ℏ120596120596119953119953

119890119890120573120573ℏ120596120596119953119953minus1= 1

119894119894120596120596119899119899minus120596120596119953119953

ie

1198661198661199531199530(119894119894120596120596119899119899) == 1119894119894120596120596119899119899minus120596120596119953119953

(421)

5 Fermirsquos golden rule 51 Interaction picture

Next we can proceed with calculating the Greenrsquos function of the coupled system To this end we

define the coupling Hamiltonian operator term in the interaction picture as

119867119867119862119862119868119868 (120591120591) = 1198901198901198671198670120591120591ℏ 119867119867119862119862119890119890

minus1198671198670120591120591ℏ (51)

The time evolution of 119867119867119862119862(120591120591) is then given by

ℏ part119867119867119862119862119868119868 (120591120591)120597120597120591120591

= 1198671198670119867119867119862119862119868119868 (120591120591) (52)

Note that the Greenrsquos function defined above was given in the Heisenberg picture To proceed we

need to transform it to the interaction picture

119866119866119953119953119953119953prime(120591120591 0) = minus lang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang = minusTr 120588120588119867119867119879119879120591120591 119890119890

119867119867120591120591ℏ 119886119886119953119953(0)119890119890minus

119867119867120591120591ℏ 119886119886119953119953prime

dagger (0)

= minusTr 120588120588119867119867119879119879120591120591 119890119890119867119867120591120591ℏ 119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119890119890minus

119867119867120591120591ℏ 119886119886119953119953prime

dagger (0)

= minusTr 120588120588119867119867119879119879120591120591 119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953primedagger (0)

where

119886119886119953119953119868119868 (120591120591) equiv 1198901198901198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus

1198671198670120591120591ℏ (53)

is the operator in interaction picture and the operator 119880119880 is defined by

119880119880(1205911205911 1205911205912) equiv 11989011989011986711986701205911205911ℏ 119890119890minus

119867119867 (1205911205911minus1205911205912)ℏ 119890119890minus

11986711986701205911205912ℏ (54)

Note that while 119880119880 is not unitary it satisfies the following group property

119880119880(1205911205911 1205911205912)119880119880(1205911205912 1205911205913) = 119880119880(1205911205911 1205911205913) (55)

and the boundary condition 119880119880(1205911205911 1205911205911) = 1 In addition the 120591120591 derivative of 119880119880 is simply

ℏpart119880119880(120591120591 120591120591prime)

120597120597120591120591= 1198671198670119890119890

1198671198670120591120591ℏ 119890119890minus

119867119867120591120591minus120591120591primeℏ minus 119890119890

1198671198670120591120591ℏ 119867119867119890119890minus

119867119867120591120591minus120591120591primeℏ 119890119890minus

1198671198670120591120591primeℏ

= 1198901198901198671198670120591120591ℏ 1198671198670 minus 119867119867119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119890119890minus


1198671198670120591120591primeℏ = 119890119890

1198671198670120591120591ℏ minus119867119867119862119862119890119890

minus1198671198670120591120591ℏ 119880119880(120591120591 120591120591prime)

= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime)

16

ie

ℏ part119880119880120591120591120591120591prime120597120597120591120591

= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime) (56)

The solution of Eq (56) is (see page 235 of Ref [2])

119880119880(120591120591 120591120591prime) = summinus1ℏ

119899119899

119899119899 int 1198891198891205911205911120591120591120591120591prime ⋯int 119889119889120591120591119899119899

120591120591120591120591prime 119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119899119899)infin

119899119899=0 (57)

The exact thermal Greens function now may be rewritten in the interaction picture as

119866119866119953119953119953119953prime(120591120591 0) = minus1119885119885119867119867

Tr 119890119890minus120573120573119867119867119879119879120591120591119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953primedagger (0)

= minusTr 119890119890minus120573120573119867119867119879119879120591120591 119880119880(0 120591120591)119886119886119953119953119868119868 (120591120591)119880119880(120591120591 0)119886119886119953119953prime

dagger (0)

Tr119890119890minus120573120573119867119867

= minusTr 119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)119879119879120591120591 119886119886119953119953119868119868 (120591120591)119880119880(0 120591120591)119880119880(120591120591 0)119886119886119953119953prime

dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)

= minusTr 119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)119879119879120591120591 119886119886119953119953119868119868 (120591120591)119880119880(00)119886119886119953119953prime

dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)=

= minusTr 119890119890minus1205731205731198671198670119879119879120591120591 119880119880(120573120573ℏ 0)119886119886119953119953119868119868 (120591120591)119886119886119953119953prime

dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)

Where we used the fact that we are free to change the order of the operators within the time ordering

operation (see pages 241-242 of Ref [2])

52 Wickrsquos theorem

Thus the Greenrsquos function can be expanded as [2]

119866119866119953119953119953119953prime(120591120591 0) = minussum 1

119898119898minus1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)119886119886119953119953119868119868 (120591120591)119886119886

119953119953primedagger (0)rang0infin

119898119898=0

sum 1119898119898minus

1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0infin

119898119898=0 (58)

where lang⋯ rang0 represents the ensemble average with respect to the non-interacting basis

Tr119890119890minus1205731205731198671198670(⋯ ) Or explicitly

119866119866119953119953119953119953prime(120591120591 0) = minus119866119866119953119953119953119953prime0 (120591120591)minus1ℏint 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang0+

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0+⋯

1minus1ℏint 1198891198891205911205911120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)rang0+12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0+⋯

(59)

For consiceness we introduce the following notation

119863119863119898119898 equiv 1119898119898minus 1

ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0 (510)

To simplify the calculation in Eq (58) we adopt Wickrsquos theorem (see pages 237-241 of Ref

[2])which can be expressed as follows

lang119879119879120591120591119860119861119861119862119863119863 ⋯119884119884119885rang0 = lang119879119879120591120591119860119861119861rang0lang119879119879120591120591119862119863119863rang0⋯ lang119879119879120591120591119884119884119885rang0 + lang119879119879120591120591119860119862rang0lang119879119879120591120591119861119861119863119863rang0⋯ lang119879119879120591120591119883119883119885rang0 + ⋯ (511)

17

Here 119860119861119861 hellip 119884119884 119885 represent 119886119886119953119953(120591120591) or 119886119886119953119953dagger(120591120591) In Eq (511) each term corresponds to a particular

pairing of the operators 119860119861119861119862119863119863 ⋯119884119884119885 and all possible pairings are taken into account Here the only

non-vanishing propagators will have the form [see Eq (420)]

lang119879119879120591120591 119886119886119953119953primedagger (120591120591prime)119886119886119953119953(120591120591)rang0 = minuslang119879119879120591120591 119886119886119953119953(120591120591)119886119886119953119953prime

dagger (120591120591prime)rang0 = 119866119866119953119953119953119953prime0 (120591120591 120591120591prime) = 1198661198661199531199530(120591120591 minus 120591120591prime)120575120575119953119953119953119953prime (512)

Therefore the second term in the numerator of Eq (59) can be calculated as

1198681198682 = minus1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0

= minus1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119886119886119953119953prime

dagger (0)119886119886119953119953(120591120591)rang0 minus1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

= 119866119866119953119953119953119953prime0 (120591120591 0) minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

The first term in above equation is called disconnected part since the pairing is performed separately

on 119867119867119862119862(1205911205911) and 119886119886119953119953(120591120591)119886119886119953119953primedagger (0) All other terms have pairs that mix creation and annihilation operators

of the Hamiltonian with 119886119886119953119953(120591120591) or 119886119886119953119953primedagger (0) and are said to have connected party For simplicity we

include all these terms in the notation lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953primedagger (0)rang0119888119888 Furthermore we define 119866119866119953119953119953119953prime

(1) (120591120591) equiv

minus1ℏ int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 Then we have

1198681198683 = minus1ℏint 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0 = 119866119866119953119953119953119953prime0 (120591120591 0)1198631198631 + 119866119866119953119953119953119953prime

(1) (120591120591) (513)

Using this method the third term in the numerator of Eq (59) can be calculated as

1198681198683 = 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0 = 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 lang119879119879120591120591119886119886119953119953prime

dagger (0)119886119886119953119953(120591120591)rang0 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 = 119866119866119953119953119953119953prime0 (120591120591 0) 1

2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 +

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

Here the last term is denoted as 119866119866119953119953119953119953prime(2) (120591120591)

Using Eq (513) the second and third terms on the right hand above equation can be simplified as

follows

18

12ℏ2

1198891198891205911205912120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 +1

2ℏ2 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

=12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

+12 minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0

minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

=12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

+12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

= minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 = 1198631198631119866119866119953119953119953119953prime(1) (120591120591)

Where we performed the following integration variables interchange 1205911205911 ⟷ 1205911205912 Thus we have

1198681198683 = 119866119866119953119953119953119953prime(2) (120591120591) + 119866119866119953119953119953119953prime

(1) (120591120591)1198631198631 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198632 (514)

Similarity we can calculate the fourth term of Eq (59) as

1198681198684 = minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0 =

minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0 lang119879119879120591120591119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0 + 3 times

minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime


minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 +

minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 =

119866119866119953119953119953119953prime0 (120591120591 0) minus1

6ℏ3 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0+

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0

minus1ℏ int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888+

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 minus1ℏ int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0+ 119866119866119953119953119953119953prime

(3) (120591120591) = 119866119866119953119953119953119953prime(3) (120591120591) +

119866119866119953119953119953119953prime(2) (120591120591)1198631198631 + 119866119866119953119953119953119953prime

(1) (120591120591)1198631198632 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198633

Higher order terms can be treated in the same manner (see pages 95-96 of Ref [2]) Then when

substituting 1198681198682 1198681198683 1198681198684 and all higher order terms into Eq (59) we can simplify the numerator as

follows

119866119866119953119953119953119953prime0 (120591120591) minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0

+1

2ℏ2 1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0 + ⋯

= 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime

(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (1 + 1198631198631 + 1198631198632 + ⋯ )

Noting that the expression (1 + 1198631198631 + 1198631198632 + ⋯ ) is exactly canceled with the denominator the

Greenrsquos function can be simplified as

19

119866119866119953119953119953119953prime(120591120591) = 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime

(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (515)

where

119866119866119953119953119953119953prime

(1) (120591120591) = minus1ℏ int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime

dagger (0)rang0119888119888

119866119866119953119953119953119953prime(2) (120591120591) = 1

2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime

dagger (0)rang0119888119888 (516)

53 Calculation of Greenrsquos function

531 First order appriximation In Eq (516) we go back to the full notation 119886119886119954119954120582120582 to distinguish phonons of different branches

then 119866119866119953119953119953119953prime(1) (120591120591) is calculated as

119866119866119953119953119953119953prime(1) (120591120591) = minus

1ℏ

12 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591 119881119881119895119895119895119895prime(119948119948)119876119876119948119948119895119895(1205911205911)119876119876119948119948119895119895prime

dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948

+ 119881119881119895119895119895119895primelowast (119948119948)119876119876119948119948119895119895prime(1205911205911)119876119876119948119948119895119895

dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948

119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888

= minus12ℏ

1198891198891205911205911120573120573ℏ

0lang119879119879120591120591

ℏ119881119881119895119895119895119895prime(119948119948)2120596120596119948119948119895119895120596120596119948119948119895119895prime

119886119886119948119948119895119895(1205911205911) + 119886119886minus119948119948119895119895dagger (1205911205911) 119886119886minus119948119948119895119895prime(1205911205911) + 119886119886119948119948119895119895prime

dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948

+ ℏ119881119881119895119895119895119895prime

lowast (119948119948)

2120596120596119948119948119895119895120596120596119948119948119895119895prime119886119886119948119948119895119895prime(1205911205911) + 119886119886minus119948119948119895119895prime

dagger (1205911205911) 119886119886minus119948119948119895119895(1205911205911) + 119886119886119948119948119895119895dagger (1205911205911)

3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


According to Wickrsquos theorem [2] the terms that contain 119886119886119948119948119895119895119886119886minus119948119948119895119895prime and 119886119886minus119948119948119895119895dagger 119886119886119948119948119895119895prime

dagger are equal to zero

thus the first term of above equation are calculated as

11989211989211 = minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886minus119948119948119895119895

dagger (1205911205911)119886119886minus119948119948119895119895prime(1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888

3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948

+ lang119879119879120591120591119886119886119948119948119895119895(1205911205911)119886119886119948119948119895119895primedagger (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime

dagger (0)rang0119888119888

= minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886minus119948119948119895119895

dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus119948119948119895119895prime(1205911205911)rang0

3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948

+ lang119879119879120591120591119886119886119948119948119895119895primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime

dagger (0)119886119886119948119948119895119895(1205911205911)rang0

= minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

01198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954minus119948119948120575120575120582120582119895119895119866119866119954119954prime120582120582prime

0 (1205911205911 0)120575120575119954119954primeminus119948119948120575120575120582120582prime119895119895prime3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948

+ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119948119948120575120575120582120582119895119895prime119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954prime119948119948120575120575120582120582prime119895119895

20

Simplification of above equation gives

11989211989211 =

⎩⎪⎨

⎪⎧minus119881119881120582120582120582120582prime(minus119954119954)

4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582

0 (120591120591 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)

4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582

0 (120591120591 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903

2lt 120582120582 le 3119903119903

0 else

(517)

Similarity the second term of 119866119866119953119953119953119953prime(1) (120591120591) is calculated as

11989211989212 = minus14

minus119881119881119895119895119895119895primelowast (119948119948)

4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886119948119948119895119895prime(1205911205911)119886119886119948119948119895119895

dagger (1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888

3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948

+ lang119879119879120591120591119886119886minus119948119948119895119895primedagger (1205911205911)119886119886minus119948119948119895119895(1205911205911)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime

dagger (0)rang0119888119888

= minus14


4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886119954119954prime120582120582prime

dagger (0)119886119886119948119948119895119895prime(1205911205911)rang0 lang119879119879120591120591119886119886119948119948119895119895dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0

3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948

+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus119948119948119895119895(1205911205911)rang0 lang119879119879120591120591119886119886minus119948119948119895119895prime

dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0

ie

11989211989212 =

⎩⎪⎨

⎪⎧


4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954prime120582120582prime

0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582lowast (minus119954119954)


0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903

2lt 120582120582 le 3119903119903

0 else

(518)

Note that since 119881119881120582120582120582120582primelowast (119954119954) = 119881119881120582120582120582120582prime(minus119954119954) [Eq (314)] the first order approximation of Greenrsquos function

can be written as

119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591) =

⎩⎪⎨

⎪⎧minus119881119881

120582120582120582120582prime(minus119954119954)

2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ

0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le3119903119903

2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582

(119954119954)

2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ

0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le3119903119903

2lt 120582120582 le 3119903119903

0 else

(519)

To derive the Greenrsquos function in frequency domain we use the expression for Fourier series then

we have

1198891198891205911205911120573120573ℏ

01198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0) = 1198891198891205911205911120573120573ℏ

0

1120573120573ℏ

119890119890minus119894119894120596120596119899119899(120591120591minus1205911205911)1198661198661199541199541205821205820 (119894119894120596120596119899119899)

119899119899

1120573120573ℏ

119890119890minus119894119894120596120596119899119899prime1205911205911119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899prime)

119899119899prime

=1

(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591 1198891198891205911205911120573120573ℏ

0119890119890119894119894120596120596119899119899minus120596120596119899119899prime1205911205911 119866119866119954119954120582120582

0 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899prime)

119899119899119899119899prime=

=1

(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591[120573120573ℏ120575120575119899119899119899119899prime]1198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime

0 (119894119894120596120596119899119899prime)119899119899119899119899prime

=1120573120573ℏ

119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954prime120582120582prime

0 (119894119894120596120596119899119899)119899119899

According to the definition of Fourier series [see Eq (412)] we get the Fourier coefficient of

21

119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591)

119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =

⎩⎪⎨



0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582 le 31199031199032

lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)


0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582prime le 31199031199032

lt 120582120582 le 3119903119903

0 else

(520)

where 119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) = 119866119866119954119954120582120582

0 (119894119894120596120596119899119899) times Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) times 119866119866119954119954prime120582120582prime

0 (119894119894120596120596119899119899) and Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) is defined as

Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =

⎩⎪⎨


2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)

2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903

2lt 120582120582 le 3119903119903

0 else

(521)

532 second order approximation The second order approximation of the Greenrsquos function in Eq (516) 119866119866119954119954120582120582119954119954prime120582120582prime

(2) (120591120591) is calculated as

119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =

18ℏ2 1198891198891205911205911

120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0lang119879119879120591120591

⎣⎢⎢⎡ 11988111988111989511989511198951198951prime(119948119948120783120783)1198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951prime

dagger (1205911205911)3119903119903

1198951198951prime=31199031199032 +1

31199031199032

1198951198951=1119948119948120783120783


dagger (1205911205911)3119903119903

1198941198941prime=31199031199032 +1

31199031199032

1198941198941=1119948119948120783120783 ⎦⎥⎥⎤

⎣⎢⎢⎡ 11988111988111989511989521198951198952prime(119948119948120784120784)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime

dagger (1205911205912)3119903119903

1198951198952prime=31199031199032 +1

31199031199032

1198951198952=1119948119948120784120784


dagger (1205911205912)3119903119903

1198941198942prime=31199031199032 +1

31199031199032

1198941198942=1119948119948120784120784 ⎦⎥⎥⎤119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime

dagger (0)rang0119888119888 = 1198921198921 + 1198921198922 + 1198921198923 + 1198921198924

where

1198921198921 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0

sum sum 11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989511989521198951198952prime(119948119948120784120784)1le1198951198952le

31199031199032 lt1198951198952

primele311990311990311994811994812078412078411989511989521198951198952prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime

times lang1198791198791205911205911198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951primedagger (1205911205911)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime


(522)

1198921198922 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0

sum sum 11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942primelowast (119948119948120784120784)

1le1198941198942le31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime

times lang1198791198791205911205911198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951primedagger (1205911205911)1198761198761199481199481207841207841198941198942prime (1205911205912)1198761198761199481199481207841207841198941198942


(523)

1198921198923 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0

sum sum 11988111988111989411989411198941198941primelowast (119948119948120783120783)11988111988111989511989521198951198952prime(119948119948120784120784)

1le1198951198952le31199031199032 lt1198951198952

primele311990311990311994811994812078412078411989511989521198951198952prime

1le1198941198941le31199031199032 lt1198941198941

primele311990311990311994811994812078312078311989411989411198941198941prime

times lang1198791198791205911205911198761198761199481199481207831207831198941198941prime (1205911205911)1198761198761199481199481207831207831198941198941dagger (1205911205911)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime


(524)

22

1198921198924 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0

sum sum 11988111988111989411989411198941198941primelowast (119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)1le1198941198942le

31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198941198941le31199031199032 lt1198941198941

primele311990311990311994811994812078312078311989411989411198941198941prime

times lang1198791198791205911205911198761198761199481199481207831207831198941198941prime (1205911205911)1198761198761199481199481207831207831198941198941dagger (1205911205911)1198761198761199481199481207841207841198941198942prime (1205911205912)1198761198761199481199481207841207841198941198942


(525)

In what follows we will expand the term 1198921198922 The expansion of all other terms follows the same lines

and eventually results in the same expression Expanding the product 11987611987611994811994812078312078311989511989511198761198761199481199481207831207831198951198951primedagger 1198761198761199481199481207841207841198941198942prime1198761198761199481199481207841207841198941198942

dagger we can

rewrite Eq (523) as follows

1198921198922 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)119881119881

11989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime

times 119868119868 (526)

Where

119868119868 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911) + 119886119886minus1199481199481207831207831198951198951dagger (1205911205911) 119886119886minus1199481199481207831207831198951198951prime(1205911205911) + 1198861198861199481199481207831207831198951198951prime

dagger (1205911205911) 1198861198861199481199481207841207841198941198942prime (1205911205912) + 119886119886minus1199481199481207841207841198941198942primedagger (1205911205912) 119886119886minus1199481199481207841207841198941198942(1205911205912) + 1198861198861199481199481207841207841198941198942

dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0c

= lang119879119879120591120591 1198861198861199481199481207831207831198951198951119886119886minus1199481199481207831207831198951198951prime + 11988611988611994811994812078312078311989511989511198861198861199481199481207831207831198951198951primedagger + 119886119886minus1199481199481207831207831198951198951

dagger 119886119886minus1199481199481207831207831198951198951prime + 119886119886minus1199481199481207831207831198951198951dagger 1198861198861199481199481207831207831198951198951prime

dagger 12059112059111198861198861199481199481207841207841198941198942prime119886119886minus1199481199481207841207841198941198942 + 1198861198861199481199481207841207841198941198942prime1198861198861199481199481207841207841198941198942

dagger + 119886119886minus1199481199481207841207841198941198942primedagger 119886119886minus1199481199481207841207841198941198942

+ 119886119886minus1199481199481207841207841198941198942primedagger 1198861198861199481199481207841207841198941198942

dagger 1205911205912119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime

dagger (0)rang0c

= lang119879119879120591120591 11988611988611994811994812078312078311989511989511198861198861199481199481207831207831198951198951primedagger + 119886119886minus1199481199481207831207831198951198951

dagger 119886119886minus1199481199481207831207831198951198951prime12059112059111198861198861199481199481207841207841198941198942prime1198861198861199481199481207841207841198941198942



Where the symbol [hellip ]1205911205911 signifies that the operators in the brackets are given in the interaction

picture The last equality in above equation comes from the fact that the contractions of the product

of the operators are equal to zero when the number of creation and annihilation operators are not the

same (see Wickrsquos theorem in Ref [2]) Using Wickrsquos theorem [2] we are able to calculate the above

equation term by term as follows

⎩⎪⎨

⎪⎧ 1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951prime

dagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime

dagger (0)rang0119888119888

1198681198682 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942prime

dagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582primedagger (0)rang0119888119888

1198681198683 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942


1198681198684 = lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)119886119886minus1199481199481207841207841198941198942prime


(527)

We shall now calculate them term by term

23

1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)1198861198861199481199481207841207841198941198942


= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942

dagger (1205911205912)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0

+ lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime


+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951prime

dagger (1205911205911)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0

+ lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207841207841198941198942prime(1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime


= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199481199481207841207841198951198952

0 (1205911205912 1205911205912)12057512057511989411989421198941198942prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582

+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime

0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582

+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199481199481207831207831198951198951

0 (1205911205911 1205911205911)12057512057511989511989511198951198951prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582

+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙ 119866119866119954119954prime120582120582prime

0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582

= 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime



0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582

The last equality results from the fact that 1198951198951 and 1198951198951prime are indices of different ranges (belonging to

different subsystems) and the same holds for 1198941198942 and 1198941198942prime Similarity



= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951

dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0

+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951

dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0


dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0

+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942prime

dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0

= 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582

0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus11994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime

+ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582

0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime


0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime

+ 119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 119866119866119954119954120582120582

0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942

= 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 119866119866119954119954120582120582




Where again the last equality results from the fact that 1198951198951 and 1198951198951prime are indices of different ranges

(belonging to different subsystems) and the same holds for 1198941198942 and 1198941198942prime Similarly we obtain 1198681198682 =

1198681198683 = 0 for the same reason as shown below

24



= lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951prime


+ lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime

dagger (0)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0


dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0 lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)1198861198861199481199481207831207831198951198951(1205911205911)rang0

+ lang1198791198791205911205911198861198861199481199481207831207831198951198951primedagger (1205911205911)119886119886minus1199481199481207841207841198941198942(1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime

dagger (0)1198861198861199481199481207831207831198951198951(1205911205911)rang0 lang119879119879120591120591119886119886minus1199481199481207841207841198941198942primedagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0


0 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 ∙ 119866119866minus11994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime

+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)120575120575minus11994811994812078412078411994811994812078312078312057512057511989511989511198941198942prime ∙ 119866119866119954119954prime120582120582prime

0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582


0 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime

+ 1198661198661199481199481207831207831198951198951prime0 (1205911205912 1205911205911)120575120575minus11994811994812078412078411994811994812078312078312057512057511989411989421198951198951prime ∙ 119866119866119954119954prime120582120582prime

0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 = 0




dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0

+ lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime

dagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0

+ lang119879119879120591120591119886119886119954119954prime120582120582primedagger (0)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942

dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0

+ lang119879119879120591120591119886119886minus1199481199481207831207831198951198951dagger (1205911205911)1198861198861199481199481207841207841198941198942prime (1205911205912)rang0 lang119879119879120591120591119886119886119954119954prime120582120582prime

dagger (0)119886119886minus1199481199481207831207831198951198951prime(1205911205911)rang0 lang1198791198791205911205911198861198861199481199481207841207841198941198942dagger (1205911205912)119886119886119954119954120582120582(120591120591)rang0


0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 11986611986611994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime

+ 11986611986611994811994812078412078411989411989420 (1205911205911 1205911205912)120575120575minus1199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942 ∙ 119866119866119954119954prime120582120582prime

0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582


0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime

+ 1198661198661199481199481207841207841198941198942prime0 (1205911205912 1205911205911)120575120575minus1199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951 ∙ 119866119866119954119954prime120582120582prime

0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 = 0

The two non-vanishing terms (1198681198681 and 1198681198684) produce the following contributions to 1198921198922 of Eq (526)

11989211989221 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198681198681 =

132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951

0 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime0 (1205911205912 0)120575120575119948119948120784120784119954119954prime1205751205751198941198942prime120582120582prime ∙

1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 + 1

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951prime

0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙

119866119866119954119954prime120582120582prime0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 119866119866119954119954120582120582

0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 =

⎩⎪⎨

⎪⎧

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum

1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast (119954119954prime)

1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912) ∙ 119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum

119881119881120582120582prime1198951198951prime(119954119954prime)1198811198811205821205821198951198951prime

lowast (119954119954)

1205961205961199541199541198951198951prime 120596120596119954119954120582120582120596120596119954119954prime120582120582prime

31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205912 1205911205911) ∙ 119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582

0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

0 else

and

25

11989211989224 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198681198684 =

132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903


0 (1205911205912 0)120575120575minus119948119948120784120784119954119954prime1205751205751198941198942120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙

119866119866minus1199481199481207831207831198951198951prime 0 (1205911205911 1205911205912)1205751205751199481199481207831207831199481199481207841207841205751205751198951198951prime1198941198942prime + 1

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903


0 (1205911205911 0)120575120575minus119948119948120783120783119954119954prime1205751205751198951198951prime120582120582prime ∙

1198661198661199541199541205821205820 (120591120591 1205911205912)120575120575minus1199481199481207841207841199541199541205751205751198941198942prime120582120582 ∙ 119866119866minus1199481199481207831207831198951198951

0 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 =

⎩⎪⎨

⎪⎧120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime0 (1205911205912 0)119866119866119954119954120582120582

0 (120591120591 1205911205911) 120582120582 120582120582prime isin 1 31199031199032

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

0 else

Here we used the following properties 120596120596minus119954119954120582120582 = 120596120596119954119954120582120582 and 1198811198811205821205821198951198951prime(minus119954119954) = 1198811198811205821205821198951198951primelowast (119954119954) [see Eqs (112) and

(314)]

The equivalence of 11989211989221 and 11989211989224 can be seen by changing the integration variables 1205911205912 harr 1205911205911

Substituting 11989211989221 and 11989211989224 into Eq (526) 1198921198922 is given by

1198921198922 =

⎩⎪⎨

⎪⎧

120575120575119954119954119954119954prime

16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

120575120575119954119954119954119954prime

16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582

0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

0 else

(528)

By changing the indices and integration variables itrsquos straightforward to show that the other three

terms (119892119892111989211989231198921198924 ) are identical to 1198921198922 thus the final expression of 119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) [Eq (516)) can be

calculated as

119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) = 41198921198922 =

⎩⎪⎨

⎪⎧120575120575119954119954119954119954prime

4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum

119881119881120582120582prime1198951198951prime119954119954prime119881119881

1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

120575120575119954119954119954119954prime

4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum

1198811198811198951198951120582120582(119954119954)1198811198811198951198951120582120582primelowast 119954119954prime

1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 31199031199032

+ 13119903119903

0 else

(529)

To calculate 119866119866119954119954120582120582119954119954prime120582120582prime(2) in frequency space we expand 119866119866119954119954120582120582119954119954prime120582120582prime

(2) (120591120591) in a Fourier series

119866119866119954119954120582120582119954119954prime120582120582prime

(2) (120591120591) = 1120573120573ℏsum 119890119890minus119894119894120596120596119899119899120591120591119866119866119954119954120582120582119954119954prime120582120582prime

(2) (119894119894120596120596119899119899)119899119899

119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = int d120591120591119890119890119894119894120596120596119899119899120591120591119866119866119954119954120582120582119954119954prime120582120582prime

(2) (120591120591)120573120573ℏ0

(530)

Using Eq (421) we get for 120582120582 120582120582prime isin 1 31199031199032

26

119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =

120575120575119954119954119954119954prime4

1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954120582120582prime

0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912)

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0

1120573120573ℏ

119890119890minus1198941198941205961205961198991198992(1205911205912minus1205911205911)1198661198661199541199541198951198951prime0 1198941198941205961205961198991198992

1198991198992

∙1120573120573ℏ

119890119890minus11989411989412059612059611989911989911205911205911119866119866119954119954120582120582prime0 1198941198941205961205961198991198991

1198991198991

∙1120573120573ℏ

119890119890minus119894119894120596120596119899119899(120591120591minus1205911205912)1198661198661199541199541205821205820 (119894119894120596120596119899119899)

119899119899

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1120573120573ℏ

119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954120582120582prime

0 11989411989412059612059611989911989911198661198661199541199541198951198951prime0 1198941198941205961205961198991198992

11989911989911989911989911198991198992⎩⎪⎨

⎪⎧ 1120573120573ℏ

1198891198891205911205911120573120573ℏ

0119890119890minus1198941198941205961205961198991198991minus12059612059611989911989921205911205911

times1120573120573ℏ

1198891198891205911205912120573120573ℏ

0119890119890minus1198941198941205961205961198991198992minus1205961205961198991198991205911205912

⎭⎪⎬

⎪⎫

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1120573120573ℏ

119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)

11989911989911989911989911198991198992

119866119866119954119954120582120582prime0 11989411989412059612059611989911989911198661198661199541199541198951198951prime

0 1198941198941205961205961198991198992120575120575119899119899111989911989921205751205751198991198991198991198992

=120575120575119954119954119954119954prime120573120573ℏ

119890119890minus119894119894120596120596119899119899120591120591

119899119899

1198661198661199541199541205821205820 (119894119894120596120596119899119899)

14

119881119881120582120582prime1198951198951prime(119954119954

prime)1198811198811205821205821198951198951primelowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899)119866119866119954119954120582120582prime

0 (119894119894120596120596119899119899)

Comparing above expression with Eq (530) the Fourier transform of 119866119866119954119954120582120582119954119954120582120582prime(2) (120591120591) reads as

119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582

0 (119894119894120596120596119899119899) 120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 119866119866119954119954120582120582prime

0 (119894119894120596120596119899119899) (531)

where we have introduced the notation

Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) equiv

120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) (532)

and 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582

0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954120582120582prime(2) (119894119894120596120596119899119899) ∙ 119866119866119954119954120582120582prime

0 (119894119894120596120596119899119899) Similarly for 120582120582 120582120582prime isin 31199031199032

+ 13119903119903 we

have

Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) (533)

and all together we have for Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899)

Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =

⎩⎪⎪⎨

⎪⎪⎧120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903

2

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

0 else

(534)

27

533 Dysonrsquos equation Substituting Eqs (521) and (534) into Eq (516) and performing a Fourier transform we have

119866119866119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) = 119866119866119954119954120582120582119954119954prime120582120582prime0 (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime

(1) (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) + ⋯

= 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582

0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime

0 (119894119894120596120596119899119899) + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime

(2) (119894119894120596120596119899119899)

∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) + ⋯ = 119866119866119954119954120582120582

0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime

0 (119894119894120596120596119899119899)

or

119866119866119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) = 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582

0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) ∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) (535)

Eq (535) is the so-called Dysonrsquos equation (see Ref [2]) where

Σ119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899) = Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) + Σ119954119954120582120582119954119954prime120582120582prime

(2) (119894119894120596120596119899119899) + ⋯

is called self-energy and Σ119954119954120582120582119954119954120582120582prime(119899119899) (119894119894120596120596119899119899)119899119899 = 12⋯ is the self-energy of nth order approximation Up

to the second order approximation the self-energy is written as

Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) =

⎩⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎧ minus

120575120575119954119954119954119954prime

2

119881119881120582120582120582120582primelowast (119954119954)

120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903

minus120575120575119954119954119954119954prime

2

119881119881120582120582prime120582120582(119954119954)

120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582prime le 3119903119903

2lt 120582120582 le 3119903119903

120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903

2

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

(536)

54 Fermirsquos golden Rule

To get the Fermirsquos golden Rule we need to calculate the retarded Greenrsquos function which can be

calculated from 119866119866119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899) by analytic continuation to the real axis via 119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894 with an

infinitesimal positive 119894119894 [12]

119866119866119954119954120582120582119954119954120582120582prime119903119903 (120596120596) equiv 119866119866119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894) 119894119894 rarr 0 (537)

Similarity the retarded self-energy is defined as

Σ119954119954120582120582119954119954120582120582prime119903119903 (120596120596) equiv Σ119954119954120582120582119954119954120582120582prime(119894119894120596120596119899119899 rarr 120596120596 + 119894119894119894119894) 119894119894 rarr 0 (538)

The transition rate of phonons of branch 120582120582 at 119954119954 Γ119954119954120582120582(120596120596) is related to the retarded self energy

Σ119954119954120582120582119903119903 (120596120596) via [13]

Γ119954119954120582120582(120596120596) = minus2ImΣ119954119954120582120582119954119954120582120582119903119903 (120596120596) (539)

Using the expression for 1198661198661199541199541205821205820 (119894119894120596120596119899119899) [Eq (421)] and focused on the second order approximation [13]

we have

28

Σ119954119954120582120582119954119954120582120582119903119903 (119894119894120596120596119899119899) =

⎩⎪⎨

⎪⎧14sum

1198811198811205821205821198951198951prime(119954119954)

2

1205961205961199541199541198951198951prime120596120596119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

1

120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894

120582120582 isin 1 31199031199032

14sum 1198811198811198951198951120582120582(119954119954)2

1205961205961199541199541198951198951120596120596119954119954120582120582

311990311990321198951198951=1

1120596120596minus1205961205961199541199541198951198951+119894119894119894119894

120582120582 isin 31199031199032

+ 13119903119903

(540)

Using the relation

Im 1

120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894

= minus120587120587120575120575120596120596 minus 1205961205961199541199541198951198951prime (541)

The transition rate reads as

Γ119954119954120582120582(120596120596) =

⎩⎪⎨

⎪⎧1205871205872sum

1198811198811205821205821198951198951prime(119954119954)

2

1205961205961199541199541198951198951prime120596120596119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

120575120575120596120596 minus 1205961205961199541199541198951198951prime 120582120582 isin 1 31199031199032

1205871205872sum 1198811198811198951198951120582120582(119954119954)2

1205961205961199541199541198951198951120596120596119954119954120582120582

311990311990321198951198951=1

120575120575120596120596 minus 1205961205961199541199541198951198951 120582120582 isin 31199031199032

+ 13119903119903

(542)

Or equivalently

Γ119954119954120582120582(119864119864 = ℏ120596120596) =

⎩⎪⎨

⎪⎧120587120587ℏ3

2sum

1198811198811205821205821198951198951prime(119954119954)

2

1198641198641199541199541198951198951prime119864119864119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

120575120575119864119864 minus 1198641198641199541199541198951198951prime 120582120582 isin 1 31199031199032

120587120587ℏ3

2sum 1198811198811198951198951120582120582(119954119954)2

1198641198641199541199541198951198951119864119864119954119954120582120582

311990311990321198951198951=1

120575120575119864119864 minus 1198641198641199541199541198951198951 120582120582 isin 31199031199032

+ 13119903119903

(543)

Γ119954119954120582120582(119864119864) represents the probability per unit time of a transition with energy 119864119864 from phonon branch 120582120582

at wave number 119954119954 in subsystem I (II) to a set of phonon branches in subsystem II (I) with the same

wave number Here state 119954119954120582120582 in one subsystem is coupled to state 1199541199541198951198951prime in the other subsystem via

1198811198811205821205821198951198951prime(119954119954) Since the two expressions in Eq (543) are completely equivalent just representing

transitions of opposite directions we consider only the first one in what follows Assuming that

subsystem II sufficiently large such that its density of states is nearly continuous the sum over its

states in the above expression can be replaced by an integral over the energy sum (⋯ )119895119895 rarr

int(⋯ )120588120588119864119864119954119954d119864119864119954119954 giving

Γ119954119954120582120582(119864119864) = 120587120587ℏ3

2 intd119864119864119954119954prime120588120588119864119864119954119954prime 119881119881120582120582119895119895prime(119954119954)

2119864119864119954119954119895119895prime=119864119864119954119954prime

119864119864119954119954prime119864119864119954119954120582120582120575120575119864119864 minus 119864119864119954119954prime = 120587120587ℏ3

2120588120588(119864119864)

119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864

119864119864119864119864119954119954120582120582 (544)

Eq (544) represents the transition rate of the phonons with energy 119864119864 from branch 120582120582 and wave

number 119954119954 in subsystem I to the continuous manifold of states in subsystem II The total transition

rate between the two subsystems is then given by the sum of transition rates from all states in

subsystem I weighted by their phonon populations to the manifold of states in subsystem II

Assuming that the two subsystems are weakly coupled such that the width of state 119954119954120582120582 in subsystem

I is small and the probability to leave it at an energy other than 119864119864119954119954120582120582 is negligible we can now write

29

total transition rate as

Γtot = 1119885119885119890119890minus120573120573119864119864119954119954120582120582Γ119954119954120582120582119864119864119954119954120582120582

119954119954120582120582

=120587120587ℏ3

2

119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582 119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864119954119954120582120582

1198641198641199541199541205821205822119954119954120582120582

=120587120587ℏ3

2

119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582 119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582

Finally we get

Γtot = Γ119954119954120582120582(119864119864) =120587120587ℏ3

2sum 119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582 (545)

where 119885119885 = sum 119890119890minus120573120573119864119864119954119954120582120582119954119954120582120582 is the partition function Here we choose the index of phonon branches 120582120582

such that the phonon branch with lower energy has a smaller index ie 1198641198641199541199541205821205821 lt 1198641198641199541199541205821205822 for index 1205821205821 lt

1205821205822 Note that phonon branches 120582120582 and 1198951198951prime belong to the subsystem I (with index from 1 to 31199031199032) and

subsystem II (with index from 1 to 1 + 31199031199032 to 3119903119903) respectively we choose 119895119895prime = 120582120582 + 31199031199032

to make

sure that 119864119864119954119954119895119895prime = 119864119864119954119954120582120582 since phonon energy of two uncoupled system is identical Eq (545) is the

Fermirsquos golden rule for inter-phonon coupling

References

[1] Y Xu J-S Wang W Duan B-L Gu and B Li Nonequilibrium Greens function method for phonon-phonon interactions and ballistic-diffusive thermal transport Phys Rev B 78 224303 (2008) [2] A L Fetter and J D Walecka Quantum theory of many-particle systems (Courier Corporation 2012) [3] G E Stedman Fermis Golden RulemdashAn Exercise in Quantum Field Theory Am J Phys 39 205 (1971)

MainText

Controllable Thermal Conductivity in Twisted Homogeneous Interfaces of Graphene and Hexagonal Boron Nitride

SI

1 Methodology

11 Model system









Fermis-Golden-Rule-for-PhononsFinal


1 Basic theory for phonons

11 Basic notations

12 Dynamical matrix




3 Inter-phonon coupling within harmonic approximation

31 Hamiltonian with inter-phonon coupling



41 Dynamics of the ladder operators


5 Fermirsquos golden rule

51 Interaction picture



531 First order appriximation

532 second order approximation

533 Dysonrsquos equation


2

Graphene is considered to be one of the most promising heat dissipating materials in nanoelectronics

[1] due to its ultrahigh in-plane room-temperature thermal conductivity of ~3000-5000 Wm-1K-1 [23]

This however can be hindered by graphene-substrate interactions that may lead to a substantial

reduction of the heat-transport due to phonon leakage across the graphene-substrate interface and

strong interfacial scattering of flexural phonon modes [4] Such undesirable substrate effects can be

reduced by considering multilayer graphene stacks These are expected to effectively isolate the top

graphene layers from the substrate due to the considerably lower cross-plane thermal conductivity

(~68 Wm-1K-1) [5] while exhibiting high in-plane conductivity that can be tuned via the stack

thickness [6-13] Anisotropic thermal conductivity is also observed for bulk hexagonal boron nitride

(h-BN) with the in-plane and cross-plane thermal conductivity in the range of 390-420 Wm-1K-1 and

25-48 Wm-1K-1 respectively [1415]

Efficient is-situ tuning of the thermal conductivity of such graphitic structures can be achieved by

controlling the twist angle between adjacent layers within the stack This has been recently

computationally demonstrated for finite-sized nanoscale few-layer graphene junctions [1617] Two

factors however limit the applicability of these results (i) the simulations were performed using

simplistic isotropic interlayer potentials that are known to be inaccurate for simulating the interlayer

interactions in layered materials [18-21] and (ii) the relevance of the results for large-scale interfaces

is questionable due to significant edge scattering effects inherent to the small finite-sized model

systems studied

To address these issues we investigate the interlayer thermal conductivity of graphene and h-BN

stacks of varying thicknesses and twist angles This allows us to gain fundamental understanding of

the heat transport mechanisms in layered materials stacks and identify feasible means to control it

Our model system consists of two contacting identical AB (AArsquo)-stacked graphite (h-BN) slabs

whose interfacing graphene (h-BN) layers are twisted with respect to each other to create a stacking

fault of misfit angle θ (see Figure 1) Recent experiments demonstrated fine control over the misfit

angle in such setups [2223] The thickness of the entire construction is varied between 27 nm-35 nm

(8-104 layers) and periodic boundary conditions are applied in all directions Heat transport

simulations are performed using state-of-the-art anisotropic interlayer potentials [18-21] applied to

the twisted stacks These potentials were shown to capture well the structural dynamic heat

dissipation and phonon spectrum of graphitic and h-BN layered systems [24-27] A thermal bias is

induced by applying Langevin thermostats with different temperatures to two layers residing on

opposite sides away from the twisted interface (see Sec 1 of the Supporting Information (SI) for

further details)

3


























4



















5



120625120625(119954119954) = 12062512062511(119954119954) 12062512062512(119954119954)12062512062521(119954119954) 12062512062522(119954119954) (1)




1199321199321dagger(119954119954)12062512062511(119954119954)1199321199321(119954119954) and 12062512062522(119954119954) = 1199321199322




matrix of the form

119932119932(119954119954) = 1199321199321(119954119954) 120782120782120782120782 1199321199322(119954119954) (2)


119932119932dagger(119954119954)120625120625(119954119954)119932119932(119954119954) = 12062512062511(119954119954) 12062512062512(119954119954)12062512062521(119954119954) 12062512062522(119954119954)

(3)

where 12062512062512(119954119954) = 1199321199321dagger(119954119954)12062512062512(119954119954)1199321199322(119954119954) and 12062512062521(119954119954) = 12062512062512






derivation)

Γtot = 120587120587ℏ3

2sum 119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582 (4)



2 is the coupling







6
























7














bulk value












8



























ASSOCIATED CONTENT




9


conductivity

AUTHOR INFORMATION



Notes


ACKNOWLEDGMENTS










MP1303

10

References


11


S1










1 Methodology





S2

1 Methodology

11 Model system












S3



















S4




120581120581CP = 119876119860119860Δ119879119879Δ119911119911

(S1)

















S1(b)]







S5















part119864119864t120597120597120597120597

= 119860119860119877119877119879119879bot(119905119905) minus 119879119879top(119905119905) (S3)






S6














119871119871 = |11989911989911199381199381 + 11989911989921199381199382| = 11988611988611989911989912 + 11989911989922 + 11989911989911198991198992 = |1198991198991minus1198991198992|1198861198862 sin(1205791205792) (S4)




2120587120587120575120575119894119894119895119895 such that

1199181199181 = 4120587120587radic3119871119871

radic32

minus12 1199181199182 = 4120587120587

radic3119871119871(01) (S5)

S7





120579120579 (deg) 119860119860 (nm2) 1198991198991 1198991198992 1198981198981 1198981198982

0 60383688 24 0 24 0

0696407 709613169 48 47 47 48

1121311 273718420 30 29 29 30

2000628 85648391 17 16 16 17

3006558 152322047 23 21 21 23

4048894 189013524 26 23 23 26

5085849 53255058 14 12 12 14

7926470 49376245 14 11 11 14

9998709 86277388 19 14 14 19

15178179 31554670 15 4 4 15

19932013 42876612 15 8 8 15

25039660 13942761 9 4 4 9

30158276 18974735 11 4 4 11

32204228 21805221 12 4 4 12









119868119868 = 1119881119881119887119887int 119891119891(119954119954)BZ d119954119954 asymp 1

119873119873sum 119908119908119894119894119891119891(119954119954119894119894)119894119894 (S6)

S8






⎩⎪⎨

⎪⎧ 1199541199541 = 1

9 1205971205973 1199541199542 = 2

9 1205971205973 1199541199543 = 1

18 512059712059718

1199541199544 = 518

512059712059718 1199541199545 = 1

9 21205971205979 1199541199546 = 2

9 21205971205979

1199541199547 = 118

312059712059718 1199541199548 = 1

9 1205971205979 1199541199549 = 1

18 12059712059718

(S7)


119908119908124689 = 112

119908119908357 = 16 (S8)


Γtot = 120587120587ℏ3

2sum 119908119908119896119896 sum

119890119890minus120573120573119864119864119954119954119948119948120582120582

119885119885

120588120588119864119864119954119954119948119948120582120582119881119881120582120582120582120582+31199031199032(119954119954119948119948)

2

1198641198641199541199541199481199481205821205822120582120582

9119896119896=1 (S9)





S9


















contribution [10]

S10



S11


S12


1









119955119955119899119899119899119899(119905119905) = 119929119929119899119899 + 119929119929s0 + 119958119958119899119899119899119899(119905119905) (11)



ℒ = sum sum 119872119872119904119904|119955119899119899119904119904|2(119905119905)2

119903119903119899119899=1

119873119873119899119899=1 minus 119881119881 (12)




ℒ = sum sum 119872119872119904119904|119958119899119899119904119904|2(119905119905)2

119903119903119899119899=1

119873119873119899119899=1 minus 1

2sum sum sum 120601120601120572120572120572120572prime












12 Dynamical matrix


equation as follows

2








form

119906119906119899119899119899119899120572120572(119905119905) = 1119872119872119904119904

119890119890119899119899120572120572119890119890119894119894(119954119954∙119929119929119899119899minus120596120596119905119905) (17)




where

119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = sum 1

119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime



119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime 119890119890

minus119894119894119954119954∙119929119929119897119897119897119897 (19)















120596120596120582120582(minus119954119954) = 120596120596120582120582(119954119954) (112)


sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582

prime119899119899120572120572 = 120575120575120582120582120582120582prime (113)





3





119890119890119899119899120572120572120582120582 (minus119954119954)lowast

= 119890119890119899119899120572120572120582120582 (119954119954) (116)


119906119906119899119899119899119899120572120572(119905119905) = sum 119862119862120582120582(119954119954)119873119873119872119872119904119904

119890119890119899119899120572120572120582120582 (119954119954)119890119890minus119894119894120596120596119954119954120582120582119905119905119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 = sum 119876119876120582120582(119954119954119905119905)119873119873119872119872119904119904

119890119890119899119899120572120572120582120582 (119954119954)119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 (21)




119876119876120582120582(119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)lowast

= 119876119876120582120582(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (minus119954119954) (22)



[119876119876120582120582(119954119954 119905119905)]lowast = 119876119876120582120582(minus119954119954 119905119905) (23)



119879119879 = 12sum 1198721198721198991198991199061198991198991198991198991205721205722 (119905119905)119899119899119899119899120572120572 = 1

2sum 119872119872119904119904





12sum 119876120582120582(119954119954 119905119905)119876120582120582prime

lowast (119954119954 119905119905) sum 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime(119954119954)

lowast119899119899120572120572 =119954119954120582120582120582120582prime

12sum 119876120582120582(119954119954 119905119905)119876120582120582lowast(119954119954 119905119905)119954119954120582120582 =

12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582

ie

119879119879 = 12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582 (24)


1119873119873sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899 = 120575120575119954119954+119954119954prime120782120782 (25)

4





12120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899+119954119954prime∙119929119929119899119899prime

119873119873119872119872119899119899119872119872119899119899prime



=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime


119873119873119872119872119899119899119872119872119899119899prime



=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954prime∙119929119929119897119897


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899

119899119899=120575120575119902119902+119902119902prime=0


=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954119929119929119897119897



=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)

119899119899120572120572



=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)

119899119899120572120572


119954119954120582120582120582120582prime


121205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)

119954119954120582120582

ie

119880119880 = 12sum 1205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)119954119954120582120582 (26)




2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)119954119954120582120582 (27)


119867119867 = 119879119879 + 119881119881 = 12sum [119875119875120582120582(minus119954119954 119905119905)119875119875120582120582(119954119954 119905119905) + 120596120596120582120582

2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)]119954119954120582120582 (28)


119867119867 = 12sum 119875119875minus119954119954120582120582119875119875119954119954120582120582 + 120596120596119954119954120582120582

2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582 (29)






119876119876119954119954120582120582 = ℏ

2120596120596119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582

dagger 119875119875119954119954120582120582 = 119894119894ℏ120596120596119954119954120582120582

2119887119887119954119954120582120582




5

119887119887119954119954120582120582 119887119887119954119954prime120582120582prime



dagger = 0 (212)


119867119867 =12119875119875minus119954119954120582120582119875119875119954119954120582120582 +120596120596119954119954120582120582

2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582

=12minus

ℏ120596120596119954119954120582120582

2119887119887minus119954119954120582120582


2 ℏ2120596120596119954119954120582120582


dagger 119954119954120582120582

=ℏ4120596120596119954119954120582120582minus119887119887minus119954119954120582120582



119954119954120582120582



dagger

=ℏ4120596120596119954119954120582120582119887119887minus119954119954120582120582


dagger + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582

119954119954120582120582

=ℏ41205961205961199541199541205821205822119887119887minus119954119954120582120582


119954119954120582120582

= ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 +

12

119954119954120582120582


119867119867 = sum ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1

2119954119954120582120582 (213)


119906119906119899119899119899119899120572120572 = sum ℏ

2119873119873119872119872119904119904120596120596119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119887119887119954119954120582120582 + 119887119887minus119954119954120582120582

dagger 119954119954120582120582 (214)








119867119867 = 1198671198671 + 1198671198672 + 11986711986712 (31)



119867119867 = 119879119879 + 119881119881 = sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2119899119899119899119899120572120572 + 1

2sum sum sum 120601120601120572120572120572120572prime


6


119904119904 = 1199031199032

+ 1 1199031199032


119879119879 = sum sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2

1199031199032119899119899=1 + sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)

2119903119903119899119899=1199031199032+1

119899119899120572120572 (33)


119881119881 =12 120601120601120572120572120572120572prime


1199031199032

119899119899119899119899prime=1


119903119903

119899119899119899119899prime=1199031199032+1


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime119899119899119899119899prime


1199031199032

119899119899prime=1

119903119903

119899119899=1199031199032+1

= 11988111988111 + 11988111988122 + 11988111988112 + 11988111988121

Or equivalently

⎩⎪⎪⎨

⎪⎪⎧ 11988111988111 = 1

2sum sum sum 120601120601120572120572120572120572prime



11988111988122 = 12sum sum sum 120601120601120572120572120572120572prime





119903119903119899119899prime=1199031199032+1

1199031199032119899119899=1120572120572120572120572prime119899119899119899119899prime



1199031199032119899119899prime=1

119903119903119899119899=1199031199032+1120572120572120572120572prime119899119899119899119899prime

(34)




the following form

119906119906119899119899119899119899120572120572 =

⎩⎨

⎧ sum ℏ

2119873119873119872119872119904119904ω119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger 119954119954120582120582 119904119904 isin 1 1199031199032

sum ℏ

2119873119873119872119872119904119904120596120596119954119954prime120582120582prime119890119899119899120572120572120582120582


2+ 1 119903119903

(35)


dagger )




prime119899119899120572120572 = 120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582

lowast119890119899119899120572120572120582120582

prime119899119899120572120572 =

120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582

prime119899119899120572120572 = sum 119890119890119899119899120572120572120582120582

lowast119890119899119899120572120572120582120582


119876119876119954119954120582120582 equiv ℏ

2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger 119876119876119954119954120582120582 equiv ℏ

2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger (36)


7

119875119875119954119954120582120582 = 119894119894ℏω119954119954120582120582

2119886119886119954119954120582120582



Eq (35) reads

119906119906119899119899119899119899120572120572 = sum 119890119890119904119904119899119899120582120582 (119954119954)

119873119873119872119872119904119904119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1 119903119903

2

sum 119890119904119904119899119899120582120582 (119954119954)119873119873119872119872119904119904

119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1199031199032

+ 1 119903119903 (38)


119879119879 = 1198791198791 + 1198791198792 = 12sum 119875119875119954119954120582120582119875119875minus119954119954120582120582119954119954120582120582 + 1



11988111988111 =12 120601120601120572120572120572120572prime


119899119899119899119899prime


119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime

119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954) 119863119863119899119899120572120572119899119899


120572120572prime

1199031199032

119899119899prime=1

1199031199032

119899119899=1120572120572

=12 ω119954119954120582120582prime


119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime (119954119954)

lowast

1199031199032

119899119899=1

=120572120572

12ω119954119954120582120582

2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582

ie

11988111988111 = 12sum ω119954119954120582120582

2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582 (310)






11988111988122 = 12sum 1205961205961199541199541205821205822 119876119876119954119954120582120582119876119876minus119954119954120582120582119954119954120582120582 (311)


8

11988111988112 =12 120601120601120572120572120572120572prime


119899119899119899119899prime

119903119903

119899119899prime=1199031199032+1

119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime

120582120582prime (119954119954prime)

119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954


lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119890119890119899119899120572120572120582120582 (119954119954)119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954



lowast119881119881119899119899120572120572119899119899


1199031199032119899119899=1120572120572120572120572prime (312)

where


119872119872119904119904119872119872119904119904prime119897119897 120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime (313)




9

11988111988121 =12 120601120601120572120572120572120572prime


119899119899119899119899prime

119903119903

119899119899=1199031199032+1

119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime

120582120582prime (119954119954prime)

119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899prime=1120572120572120572120572prime

=12 120601120601120572120572prime120572120572


119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1


119872119872119899119899prime119872119872119899119899

1119873119873 119890119890


1199031199032

119899119899=1120572120572prime120572120572

=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582


1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582


119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime





119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime


dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954prime 119890119890119899119899120572120572120582120582 (119954119954prime)

lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954prime∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582prime119876119876119954119954120582120582

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954

119890119890119899119899120572120572120582120582 (119954119954)lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger dagger

3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954


lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119890119890119899119899120572120572120582120582 (119954119954)119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

lowast

=

⎩⎨

⎧12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954⎭⎬

⎫dagger

= 11988111988112dagger

Define

1198671198671 = 11987911987911 + 119881119881111198671198672 = 11987911987922 + 1198811198812211986711986712 = 11988111988112 + 11988111988121

(315)


⎩⎪⎨

⎪⎧ 1198671198671 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582

dagger 119886119886119954119954120582120582 + 1231199031199032

120582120582=1119954119954


23119903119903

120582120582prime=1+31199031199032119954119954


dagger3119903119903120582120582prime=31199031199032 +1

31199031199032120582120582=1 + h c 119954119954

(316)




10


119867119867 = 1198671198670 + 119867119867119862119862 (317)

where

1198671198670 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582

dagger 119886119886119954119954120582120582 + 123119903119903

120582120582=1119954119954


dagger3119903119903120582120582prime=31199031199032 +1

31199031199032120582120582=1 + ℎ 119888119888 119954119954

(318)




119886119886119901119901(120591120591) = 119890119890119894119894ℏ119867119867119905119905119886119886119901119901119890119890

minus119894119894ℏ119867119867119905119905 = 119890119890

119867119867120591120591ℏ 119886119886119901119901119890119890

minus119867119867120591120591ℏ (41)



ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119867119867119886119886119953119953(120591120591) (42)


ℏpart119886119886119953119953(120591120591)120597120597120591120591

= 1198671198670119886119886119953119953(120591120591) = 1198671198670 1198901198901198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ = 1198671198670119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ minus 119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ 1198671198670

= 1198901198901198671198670120591120591ℏ 1198671198670119886119886119953119953 minus 1198861198861199531199531198671198670119890119890

minus1198671198670120591120591ℏ = 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ

ie

ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ (43)



2119953119953 119886119886119953119953prime = sum ℏ120596120596119953119953119886119886119953119953


dagger119886119886119953119953119953119953 =




ie


Therefore we have

ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119886119886119953119953(120591120591) (45)

11



dagger = ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 +

12

119953119953

119886119886119953119953primedagger = ℏ120596120596119953119953119886119886119953119953


119953119953

= ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953119886119886119953119953prime


dagger119886119886119953119953119953119953



dagger119886119886119953119953119953119953



dagger 119886119886119953119953dagger119886119886119953119953

119953119953


In summary we have

119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591

119886119886119953119953dagger(120591120591) = 119886119886119953119953

dagger(0)119890119890120596120596119953119953120591120591 = 119886119886119953119953dagger119890119890120596120596119953119953120591120591 (46)






dagger (120591120591prime) (47)



120588120588119867119867 = 119890119890minus120573120573119867119867119885119885119867119867 (48)








= minusTr 120588120588119867119867119890119890119867119867120591120591ℏ 119886119886119953119953119890119890

minus119867119867120591120591ℏ 119890119890

119867119867120591120591primeℏ 119886119886119953119953prime


minus119867119867120591120591primeℏ 119890119890

119867119867120591120591ℏ 119886119886119953119953119890119890


ℏ 119886119886119953119953primedagger




dagger (0)





12






119867119867120591120591ℏ 119886119886119953119953119890119890



ℏ 119886119886119953119953119890119890minus119867119867120591120591ℏ 119890119890

119867119867120591120591primeℏ


119867119867120591120591minus120591120591primeℏ 119886119886119953119953119890119890




dagger (0)




36 of Ref [2])




dagger (0)rang

= minusTr 120588120588119867119867119890119890119867119867(120591120591+120573120573ℏ)

ℏ 119886119886119953119953119890119890minus119867119867(120591120591+120573120573ℏ)

ℏ 119886119886119953119953prime = minusTr 120588120588119867119867119890119890120573120573119867119867119890119890

119867119867120591120591ℏ 119886119886119953119953119890119890



1119885119885119867119867


= minus1119885119885119867119867


Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119886119886119953119953(120591120591)

= minus1119885119885119867119867






ℏ 119886119886119953119953119890119890minus119867119867(120591120591minus120573120573ℏ)


119867119867120591120591ℏ 119886119886119953119953119890119890

minus119867119867120591120591ℏ 119890119890120573120573119867119867



= minus1119885119885119867119867


Tr119890119890minus120573120573119867119867119886119886119953119953(120591120591)119886119886119953119953prime(0)

= minus1119885119885119867119867


13



where 120596120596119899119899 = 2120587120587119899119899120573120573ℏ



120573120573ℏ (412)









primelang119886119886119953119953(0)119886119886119953119953prime





primelang119886119886119953119953prime





primelang119886119886119953119953prime




ie

119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =






dagger (0) = 120575120575119953119953119953119953prime


119890119890119860119860119861119861119890119890minus119860119860 = sum 1119899119899119860(119899119899)119861119861infin

119899119899=0 (414)



119891119891(119905119905) = 119891119891(0) + 119905119905119891119891prime(0) + 1199051199052

2119891119891primeprime(0) + ⋯ = sum 119905119905119899119899

119899119899119889119889119899119899119891119889119889119905119905119899119899

119905119905=0

infin119899119899=0 (415)


⎩⎪⎨



⋮119891119891(119899119899)(119905119905) = 119890119890119905119905119860119860119860(119899119899)119861119861119890119890minus119905119905119860119860

(416)

14


119890119890119905119905A119861119861119890119890minus119905119905119860119860 = sum 119905119905119899119899

119899119899119860(119899119899)119861119861infin

119899119899=0 (417)


follows


11198851198851198671198670


11198851198851198671198670


=11198851198851198671198670

Tr (minus120573120573)119899119899

1198991198991198671198670

(119899119899)119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)


dagger (0) we have

1198671198670(119899119899)119886119886119953119953prime


such that


11198851198851198671198670

Tr (minus120573120573)119899119899

1198991198991198671198670

(119899119899)119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)

=11198851198851198671198670

Tr minus120573120573ℏ120596120596119953119953prime

119899119899

119899119899119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)

= 119890119890minus120573120573ℏ120596120596119953119953prime11198851198851198671198670


dagger (0)rang





Hence we have


1119890119890120573120573ℏ120596120596119953119953minus1

equiv 120575120575119953119953prime119953119953119899119899119861119861120596120596119953119953 (419)


we obtain

119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =








119890119890119894119894120596120596119899119899minus120596120596119953119953120573120573ℏ minus

15

1 = minus1+ 1

119890119890120573120573ℏ120596120596119953119953minus1

119894119894120596120596119899119899minus120596120596119953119953119890119890119894119894


119894119894120596120596119899119899minus120596120596119953119953

119890119890minus120573120573ℏ120596120596119953119953minus1

119890119890120573120573ℏ120596120596119953119953minus1= minus 1

119894119894120596120596119899119899minus120596120596119953119953

1minus119890119890120573120573ℏ120596120596119953119953

119890119890120573120573ℏ120596120596119953119953minus1= 1

119894119894120596120596119899119899minus120596120596119953119953

ie

1198661198661199531199530(119894119894120596120596119899119899) == 1119894119894120596120596119899119899minus120596120596119953119953

(421)




119867119867119862119862119868119868 (120591120591) = 1198901198901198671198670120591120591ℏ 119867119867119862119862119890119890

minus1198671198670120591120591ℏ (51)


ℏ part119867119867119862119862119868119868 (120591120591)120597120597120591120591

= 1198671198670119867119867119862119862119868119868 (120591120591) (52)




119867119867120591120591ℏ 119886119886119953119953(0)119890119890minus

119867119867120591120591ℏ 119886119886119953119953prime

dagger (0)

= minusTr 120588120588119867119867119879119879120591120591 119890119890119867119867120591120591ℏ 119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119890119890minus

119867119867120591120591ℏ 119886119886119953119953prime

dagger (0)


where

119886119886119953119953119868119868 (120591120591) equiv 1198901198901198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus

1198671198670120591120591ℏ (53)


119880119880(1205911205911 1205911205912) equiv 11989011989011986711986701205911205911ℏ 119890119890minus

119867119867 (1205911205911minus1205911205912)ℏ 119890119890minus

11986711986701205911205912ℏ (54)


119880119880(1205911205911 1205911205912)119880119880(1205911205912 1205911205913) = 119880119880(1205911205911 1205911205913) (55)


ℏpart119880119880(120591120591 120591120591prime)

120597120597120591120591= 1198671198670119890119890

1198671198670120591120591ℏ 119890119890minus


1198671198670120591120591ℏ 119867119867119890119890minus


1198671198670120591120591primeℏ

= 1198901198901198671198670120591120591ℏ 1198671198670 minus 119867119867119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119890119890minus


1198671198670120591120591primeℏ = 119890119890

1198671198670120591120591ℏ minus119867119867119862119862119890119890

minus1198671198670120591120591ℏ 119880119880(120591120591 120591120591prime)

= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime)

16

ie

ℏ part119880119880120591120591120591120591prime120597120597120591120591

= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime) (56)


119880119880(120591120591 120591120591prime) = summinus1ℏ

119899119899

119899119899 int 1198891198891205911205911120591120591120591120591prime ⋯int 119889119889120591120591119899119899

120591120591120591120591prime 119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119899119899)infin

119899119899=0 (57)


119866119866119953119953119953119953prime(120591120591 0) = minus1119885119885119867119867



dagger (0)

Tr119890119890minus120573120573119867119867


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)=


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)





119866119866119953119953119953119953prime(120591120591 0) = minussum 1

119898119898minus1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)119886119886119953119953119868119868 (120591120591)119886119886


119898119898=0

sum 1119898119898minus

1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0infin

119898119898=0 (58)




120573120573ℏ0 lang119879119879120591120591119867119867119862119862


12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0+⋯

1minus1ℏint 1198891198891205911205911120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)rang0+12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0+⋯

(59)


119863119863119898119898 equiv 1119898119898minus 1

ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0 (510)




17







1198681198682 = minus1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0

= minus1ℏ 1198891198891205911205911120573120573ℏ



0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

= 119866119866119953119953119953119953prime0 (120591120591 0) minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888





(1) (120591120591) equiv

minus1ℏ int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime


1198681198683 = minus1ℏint 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime


(1) (120591120591) (513)


1198681198683 = 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0 = 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)119886119886119953119953(120591120591)rang0 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime


2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 +

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888



follows

18

12ℏ2

1198891198891205911205912120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 +1

2ℏ2 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

=12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

+12 minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0

minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

=12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

+12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

= minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime



1198681198683 = 119866119866119953119953119953119953prime(2) (120591120591) + 119866119866119953119953119953119953prime

(1) (120591120591)1198631198631 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198632 (514)


1198681198684 = minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0 =

minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913



minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913



minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913


dagger (0)rang0119888119888 +

minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 =

119866119866119953119953119953119953prime0 (120591120591 0) minus1

6ℏ3 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0+

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0

minus1ℏ int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888+

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime


120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0+ 119866119866119953119953119953119953prime

(3) (120591120591) = 119866119866119953119953119953119953prime(3) (120591120591) +

119866119866119953119953119953119953prime(2) (120591120591)1198631198631 + 119866119866119953119953119953119953prime

(1) (120591120591)1198631198632 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198633



follows

119866119866119953119953119953119953prime0 (120591120591) minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0

+1

2ℏ2 1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime


= 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime

(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (1 + 1198631198631 + 1198631198632 + ⋯ )



19


(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (515)

where

119866119866119953119953119953119953prime

(1) (120591120591) = minus1ℏ int 1198891198891205911205911


dagger (0)rang0119888119888

119866119866119953119953119953119953prime(2) (120591120591) = 1

2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 (516)




119866119866119953119953119953119953prime(1) (120591120591) = minus

1ℏ

12 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591 119881119881119895119895119895119895prime(119948119948)119876119876119948119948119895119895(1205911205911)119876119876119948119948119895119895prime

dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


= minus12ℏ

1198891198891205911205911120573120573ℏ

0lang119879119879120591120591

ℏ119881119881119895119895119895119895prime(119948119948)2120596120596119948119948119895119895120596120596119948119948119895119895prime


dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948

+ ℏ119881119881119895119895119895119895prime

lowast (119948119948)



3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948





11989211989211 = minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886minus119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)rang0119888119888

= minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886minus119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)119886119886119948119948119895119895(1205911205911)rang0

= minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ



119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


20


11989211989211 =

⎩⎪⎨


4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582


2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)

4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582


2lt 120582120582 le 3119903119903

0 else

(517)


11989211989212 = minus14


4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886119948119948119895119895prime(1205911205911)119886119886119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)rang0119888119888

= minus14


4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886119954119954prime120582120582prime


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0

ie

11989211989212 =

⎩⎪⎨

⎪⎧



0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903



0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903

2lt 120582120582 le 3119903119903

0 else

(518)


can be written as

119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591) =

⎩⎪⎨

⎪⎧minus119881119881

120582120582120582120582prime(minus119954119954)

2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ

0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le3119903119903

2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582

(119954119954)

2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ

0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le3119903119903

2lt 120582120582 le 3119903119903

0 else

(519)


we have

1198891198891205911205911120573120573ℏ

01198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0) = 1198891198891205911205911120573120573ℏ

0

1120573120573ℏ

119890119890minus119894119894120596120596119899119899(120591120591minus1205911205911)1198661198661199541199541205821205820 (119894119894120596120596119899119899)

119899119899

1120573120573ℏ


119899119899prime

=1

(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591 1198891198891205911205911120573120573ℏ

0119890119890119894119894120596120596119899119899minus120596120596119899119899prime1205911205911 119866119866119954119954120582120582


119899119899119899119899prime=

=1


0 (119894119894120596120596119899119899prime)119899119899119899119899prime

=1120573120573ℏ


0 (119894119894120596120596119899119899)119899119899


21

119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591)

119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =

⎩⎪⎨



0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582 le 31199031199032

lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)


0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582prime le 31199031199032

lt 120582120582 le 3119903119903

0 else

(520)




Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =

⎩⎪⎨


2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)


2lt 120582120582 le 3119903119903

0 else

(521)



119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =

18ℏ2 1198891198891205911205911

120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0lang119879119879120591120591

⎣⎢⎢⎡ 11988111988111989511989511198951198951prime(119948119948120783120783)1198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951prime

dagger (1205911205911)3119903119903

1198951198951prime=31199031199032 +1

31199031199032

1198951198951=1119948119948120783120783


dagger (1205911205911)3119903119903

1198941198941prime=31199031199032 +1

31199031199032

1198941198941=1119948119948120783120783 ⎦⎥⎥⎤

⎣⎢⎢⎡ 11988111988111989511989521198951198952prime(119948119948120784120784)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime

dagger (1205911205912)3119903119903

1198951198952prime=31199031199032 +1

31199031199032

1198951198952=1119948119948120784120784


dagger (1205911205912)3119903119903

1198941198942prime=31199031199032 +1

31199031199032

1198941198942=1119948119948120784120784 ⎦⎥⎥⎤119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime

dagger (0)rang0119888119888 = 1198921198921 + 1198921198922 + 1198921198923 + 1198921198924

where

1198921198921 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


31199031199032 lt1198951198952

primele311990311990311994811994812078412078411989511989521198951198952prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime



(522)

1198921198922 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


1le1198941198942le31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime



(523)

1198921198923 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


1le1198951198952le31199031199032 lt1198951198952

primele311990311990311994811994812078412078411989511989521198951198952prime

1le1198941198941le31199031199032 lt1198941198941

primele311990311990311994811994812078312078311989411989411198941198941prime



(524)

22

1198921198924 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


lowast (119948119948120784120784)1le1198941198942le

31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198941198941le31199031199032 lt1198941198941

primele311990311990311994811994812078312078311989411989411198941198941prime



(525)



dagger we can


1198921198922 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)119881119881

11989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime

times 119868119868 (526)

Where










dagger (0)rang0c










⎩⎪⎨

⎪⎧ 1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951prime


dagger (0)rang0119888119888







(527)


23












0 (1205911205912 1205911205912)12057512057511989411989421198941198942prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582

+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime



0 (1205911205911 1205911205911)12057512057511989511989511198951198951prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582


0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582

= 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime



0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582




























24






























0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 11986611986611994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime




0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime




11989211989221 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198681198681 =

132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951


1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 + 1

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951prime

0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙


0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 =

⎩⎪⎨

⎪⎧

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912) ∙ 119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205912 1205911205911) ∙ 119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582

0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

0 else

and

25

11989211989224 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198681198684 =

132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903




32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903




0 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 =

⎩⎪⎨

⎪⎧120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime0 (1205911205912 0)119866119866119954119954120582120582

0 (120591120591 1205911205911) 120582120582 120582120582prime isin 1 31199031199032

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

0 else


(314)]



1198921198922 =

⎩⎪⎨

⎪⎧

120575120575119954119954119954119954prime

16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

120575120575119954119954119954119954prime

16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582

0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

0 else

(528)



calculated as

119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) = 41198921198922 =

⎩⎪⎨

⎪⎧120575120575119954119954119954119954prime

4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1


0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

120575120575119954119954119954119954prime

4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 31199031199032

+ 13119903119903

0 else

(529)



119866119866119954119954120582120582119954119954prime120582120582prime


(2) (119894119894120596120596119899119899)119899119899


(2) (120591120591)120573120573ℏ0

(530)


26

119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =

120575120575119954119954119954119954prime4

1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954120582120582prime

0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912)

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0

1120573120573ℏ


1198991198992

∙1120573120573ℏ

119890119890minus11989411989412059612059611989911989911205911205911119866119866119954119954120582120582prime0 1198941198941205961205961198991198991

1198991198991

∙1120573120573ℏ

119890119890minus119894119894120596120596119899119899(120591120591minus1205911205912)1198661198661199541199541205821205820 (119894119894120596120596119899119899)

119899119899

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1120573120573ℏ

119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954120582120582prime

0 11989411989412059612059611989911989911198661198661199541199541198951198951prime0 1198941198941205961205961198991198992

11989911989911989911989911198991198992⎩⎪⎨

⎪⎧ 1120573120573ℏ

1198891198891205911205911120573120573ℏ

0119890119890minus1198941198941205961205961198991198991minus12059612059611989911989921205911205911

times1120573120573ℏ

1198891198891205911205912120573120573ℏ

0119890119890minus1198941198941205961205961198991198992minus1205961205961198991198991205911205912

⎭⎪⎬

⎪⎫

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1120573120573ℏ

119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)

11989911989911989911989911198991198992

119866119866119954119954120582120582prime0 11989411989412059612059611989911989911198661198661199541199541198951198951prime

0 1198941198941205961205961198991198992120575120575119899119899111989911989921205751205751198991198991198991198992

=120575120575119954119954119954119954prime120573120573ℏ

119890119890minus119894119894120596120596119899119899120591120591

119899119899

1198661198661199541199541205821205820 (119894119894120596120596119899119899)

14

119881119881120582120582prime1198951198951prime(119954119954



3119903119903

1198951198951prime=31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899)119866119866119954119954120582120582prime

0 (119894119894120596120596119899119899)


119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582

0 (119894119894120596120596119899119899) 120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 119866119866119954119954120582120582prime

0 (119894119894120596120596119899119899) (531)



120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) (532)

and 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582

0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954120582120582prime(2) (119894119894120596120596119899119899) ∙ 119866119866119954119954120582120582prime


+ 13119903119903 we

have

Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) (533)


Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =

⎩⎪⎪⎨

⎪⎪⎧120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903

2

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

0 else

(534)

27



(1) (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) + ⋯

= 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582


0 (119894119894120596120596119899119899) + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime

(2) (119894119894120596120596119899119899)

∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) + ⋯ = 119866119866119954119954120582120582


0 (119894119894120596120596119899119899)

or





(2) (119894119894120596120596119899119899) + ⋯



Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) =

⎩⎪⎪⎪⎪⎨


120575120575119954119954119954119954prime

2

119881119881120582120582120582120582primelowast (119954119954)

120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903

minus120575120575119954119954119954119954prime

2

119881119881120582120582prime120582120582(119954119954)

120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582prime le 3119903119903

2lt 120582120582 le 3119903119903

120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903

2

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

(536)









Σ119954119954120582120582119903119903 (120596120596) via [13]

Γ119954119954120582120582(120596120596) = minus2ImΣ119954119954120582120582119954119954120582120582119903119903 (120596120596) (539)


we have

28

Σ119954119954120582120582119954119954120582120582119903119903 (119894119894120596120596119899119899) =

⎩⎪⎨

⎪⎧14sum

1198811198811205821205821198951198951prime(119954119954)

2

1205961205961199541199541198951198951prime120596120596119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

1

120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894

120582120582 isin 1 31199031199032

14sum 1198811198811198951198951120582120582(119954119954)2

1205961205961199541199541198951198951120596120596119954119954120582120582

311990311990321198951198951=1

1120596120596minus1205961205961199541199541198951198951+119894119894119894119894

120582120582 isin 31199031199032

+ 13119903119903

(540)

Using the relation

Im 1

120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894



Γ119954119954120582120582(120596120596) =

⎩⎪⎨

⎪⎧1205871205872sum

1198811198811205821205821198951198951prime(119954119954)

2

1205961205961199541199541198951198951prime120596120596119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

120575120575120596120596 minus 1205961205961199541199541198951198951prime 120582120582 isin 1 31199031199032

1205871205872sum 1198811198811198951198951120582120582(119954119954)2

1205961205961199541199541198951198951120596120596119954119954120582120582

311990311990321198951198951=1

120575120575120596120596 minus 1205961205961199541199541198951198951 120582120582 isin 31199031199032

+ 13119903119903

(542)

Or equivalently

Γ119954119954120582120582(119864119864 = ℏ120596120596) =

⎩⎪⎨

⎪⎧120587120587ℏ3

2sum

1198811198811205821205821198951198951prime(119954119954)

2

1198641198641199541199541198951198951prime119864119864119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

120575120575119864119864 minus 1198641198641199541199541198951198951prime 120582120582 isin 1 31199031199032

120587120587ℏ3

2sum 1198811198811198951198951120582120582(119954119954)2

1198641198641199541199541198951198951119864119864119954119954120582120582

311990311990321198951198951=1

120575120575119864119864 minus 1198641198641199541199541198951198951 120582120582 isin 31199031199032

+ 13119903119903

(543)








int(⋯ )120588120588119864119864119954119954d119864119864119954119954 giving

Γ119954119954120582120582(119864119864) = 120587120587ℏ3


2119864119864119954119954119895119895prime=119864119864119954119954prime


2120588120588(119864119864)

119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864

119864119864119864119864119954119954120582120582 (544)







29


Γtot = 1119885119885119890119890minus120573120573119864119864119954119954120582120582Γ119954119954120582120582119864119864119954119954120582120582

119954119954120582120582

=120587120587ℏ3

2

119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582 119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864119954119954120582120582

1198641198641199541199541205821205822119954119954120582120582

=120587120587ℏ3

2

119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582 119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582

Finally we get

Γtot = Γ119954119954120582120582(119864119864) =120587120587ℏ3

2sum 119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582 (545)





to make



References


MainText


SI

1 Methodology

11 Model system












11 Basic notations

12 Dynamical matrix


















3


























4



















5



120625120625(119954119954) = 12062512062511(119954119954) 12062512062512(119954119954)12062512062521(119954119954) 12062512062522(119954119954) (1)




1199321199321dagger(119954119954)12062512062511(119954119954)1199321199321(119954119954) and 12062512062522(119954119954) = 1199321199322




matrix of the form

119932119932(119954119954) = 1199321199321(119954119954) 120782120782120782120782 1199321199322(119954119954) (2)


119932119932dagger(119954119954)120625120625(119954119954)119932119932(119954119954) = 12062512062511(119954119954) 12062512062512(119954119954)12062512062521(119954119954) 12062512062522(119954119954)

(3)

where 12062512062512(119954119954) = 1199321199321dagger(119954119954)12062512062512(119954119954)1199321199322(119954119954) and 12062512062521(119954119954) = 12062512062512






derivation)

Γtot = 120587120587ℏ3

2sum 119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582 (4)



2 is the coupling







6
























7














bulk value












8



























ASSOCIATED CONTENT




9


conductivity

AUTHOR INFORMATION



Notes


ACKNOWLEDGMENTS










MP1303

10

References


11


S1










1 Methodology





S2

1 Methodology

11 Model system












S3



















S4




120581120581CP = 119876119860119860Δ119879119879Δ119911119911

(S1)

















S1(b)]







S5















part119864119864t120597120597120597120597

= 119860119860119877119877119879119879bot(119905119905) minus 119879119879top(119905119905) (S3)






S6














119871119871 = |11989911989911199381199381 + 11989911989921199381199382| = 11988611988611989911989912 + 11989911989922 + 11989911989911198991198992 = |1198991198991minus1198991198992|1198861198862 sin(1205791205792) (S4)




2120587120587120575120575119894119894119895119895 such that

1199181199181 = 4120587120587radic3119871119871

radic32

minus12 1199181199182 = 4120587120587

radic3119871119871(01) (S5)

S7





120579120579 (deg) 119860119860 (nm2) 1198991198991 1198991198992 1198981198981 1198981198982

0 60383688 24 0 24 0

0696407 709613169 48 47 47 48

1121311 273718420 30 29 29 30

2000628 85648391 17 16 16 17

3006558 152322047 23 21 21 23

4048894 189013524 26 23 23 26

5085849 53255058 14 12 12 14

7926470 49376245 14 11 11 14

9998709 86277388 19 14 14 19

15178179 31554670 15 4 4 15

19932013 42876612 15 8 8 15

25039660 13942761 9 4 4 9

30158276 18974735 11 4 4 11

32204228 21805221 12 4 4 12









119868119868 = 1119881119881119887119887int 119891119891(119954119954)BZ d119954119954 asymp 1

119873119873sum 119908119908119894119894119891119891(119954119954119894119894)119894119894 (S6)

S8






⎩⎪⎨

⎪⎧ 1199541199541 = 1

9 1205971205973 1199541199542 = 2

9 1205971205973 1199541199543 = 1

18 512059712059718

1199541199544 = 518

512059712059718 1199541199545 = 1

9 21205971205979 1199541199546 = 2

9 21205971205979

1199541199547 = 118

312059712059718 1199541199548 = 1

9 1205971205979 1199541199549 = 1

18 12059712059718

(S7)


119908119908124689 = 112

119908119908357 = 16 (S8)


Γtot = 120587120587ℏ3

2sum 119908119908119896119896 sum

119890119890minus120573120573119864119864119954119954119948119948120582120582

119885119885

120588120588119864119864119954119954119948119948120582120582119881119881120582120582120582120582+31199031199032(119954119954119948119948)

2

1198641198641199541199541199481199481205821205822120582120582

9119896119896=1 (S9)





S9


















contribution [10]

S10



S11


S12


1









119955119955119899119899119899119899(119905119905) = 119929119929119899119899 + 119929119929s0 + 119958119958119899119899119899119899(119905119905) (11)



ℒ = sum sum 119872119872119904119904|119955119899119899119904119904|2(119905119905)2

119903119903119899119899=1

119873119873119899119899=1 minus 119881119881 (12)




ℒ = sum sum 119872119872119904119904|119958119899119899119904119904|2(119905119905)2

119903119903119899119899=1

119873119873119899119899=1 minus 1

2sum sum sum 120601120601120572120572120572120572prime












12 Dynamical matrix


equation as follows

2








form

119906119906119899119899119899119899120572120572(119905119905) = 1119872119872119904119904

119890119890119899119899120572120572119890119890119894119894(119954119954∙119929119929119899119899minus120596120596119905119905) (17)




where

119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = sum 1

119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime



119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime 119890119890

minus119894119894119954119954∙119929119929119897119897119897119897 (19)















120596120596120582120582(minus119954119954) = 120596120596120582120582(119954119954) (112)


sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582

prime119899119899120572120572 = 120575120575120582120582120582120582prime (113)





3





119890119890119899119899120572120572120582120582 (minus119954119954)lowast

= 119890119890119899119899120572120572120582120582 (119954119954) (116)


119906119906119899119899119899119899120572120572(119905119905) = sum 119862119862120582120582(119954119954)119873119873119872119872119904119904

119890119890119899119899120572120572120582120582 (119954119954)119890119890minus119894119894120596120596119954119954120582120582119905119905119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 = sum 119876119876120582120582(119954119954119905119905)119873119873119872119872119904119904

119890119890119899119899120572120572120582120582 (119954119954)119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 (21)




119876119876120582120582(119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)lowast

= 119876119876120582120582(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (minus119954119954) (22)



[119876119876120582120582(119954119954 119905119905)]lowast = 119876119876120582120582(minus119954119954 119905119905) (23)



119879119879 = 12sum 1198721198721198991198991199061198991198991198991198991205721205722 (119905119905)119899119899119899119899120572120572 = 1

2sum 119872119872119904119904





12sum 119876120582120582(119954119954 119905119905)119876120582120582prime

lowast (119954119954 119905119905) sum 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime(119954119954)

lowast119899119899120572120572 =119954119954120582120582120582120582prime

12sum 119876120582120582(119954119954 119905119905)119876120582120582lowast(119954119954 119905119905)119954119954120582120582 =

12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582

ie

119879119879 = 12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582 (24)


1119873119873sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899 = 120575120575119954119954+119954119954prime120782120782 (25)

4





12120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899+119954119954prime∙119929119929119899119899prime

119873119873119872119872119899119899119872119872119899119899prime



=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime


119873119873119872119872119899119899119872119872119899119899prime



=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954prime∙119929119929119897119897


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899

119899119899=120575120575119902119902+119902119902prime=0


=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954119929119929119897119897



=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)

119899119899120572120572



=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)

119899119899120572120572


119954119954120582120582120582120582prime


121205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)

119954119954120582120582

ie

119880119880 = 12sum 1205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)119954119954120582120582 (26)




2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)119954119954120582120582 (27)


119867119867 = 119879119879 + 119881119881 = 12sum [119875119875120582120582(minus119954119954 119905119905)119875119875120582120582(119954119954 119905119905) + 120596120596120582120582

2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)]119954119954120582120582 (28)


119867119867 = 12sum 119875119875minus119954119954120582120582119875119875119954119954120582120582 + 120596120596119954119954120582120582

2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582 (29)






119876119876119954119954120582120582 = ℏ

2120596120596119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582

dagger 119875119875119954119954120582120582 = 119894119894ℏ120596120596119954119954120582120582

2119887119887119954119954120582120582




5

119887119887119954119954120582120582 119887119887119954119954prime120582120582prime



dagger = 0 (212)


119867119867 =12119875119875minus119954119954120582120582119875119875119954119954120582120582 +120596120596119954119954120582120582

2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582

=12minus

ℏ120596120596119954119954120582120582

2119887119887minus119954119954120582120582


2 ℏ2120596120596119954119954120582120582


dagger 119954119954120582120582

=ℏ4120596120596119954119954120582120582minus119887119887minus119954119954120582120582



119954119954120582120582



dagger

=ℏ4120596120596119954119954120582120582119887119887minus119954119954120582120582


dagger + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582

119954119954120582120582

=ℏ41205961205961199541199541205821205822119887119887minus119954119954120582120582


119954119954120582120582

= ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 +

12

119954119954120582120582


119867119867 = sum ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1

2119954119954120582120582 (213)


119906119906119899119899119899119899120572120572 = sum ℏ

2119873119873119872119872119904119904120596120596119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119887119887119954119954120582120582 + 119887119887minus119954119954120582120582

dagger 119954119954120582120582 (214)








119867119867 = 1198671198671 + 1198671198672 + 11986711986712 (31)



119867119867 = 119879119879 + 119881119881 = sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2119899119899119899119899120572120572 + 1

2sum sum sum 120601120601120572120572120572120572prime


6


119904119904 = 1199031199032

+ 1 1199031199032


119879119879 = sum sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2

1199031199032119899119899=1 + sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)

2119903119903119899119899=1199031199032+1

119899119899120572120572 (33)


119881119881 =12 120601120601120572120572120572120572prime


1199031199032

119899119899119899119899prime=1


119903119903

119899119899119899119899prime=1199031199032+1


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime119899119899119899119899prime


1199031199032

119899119899prime=1

119903119903

119899119899=1199031199032+1

= 11988111988111 + 11988111988122 + 11988111988112 + 11988111988121

Or equivalently

⎩⎪⎪⎨

⎪⎪⎧ 11988111988111 = 1

2sum sum sum 120601120601120572120572120572120572prime



11988111988122 = 12sum sum sum 120601120601120572120572120572120572prime





119903119903119899119899prime=1199031199032+1

1199031199032119899119899=1120572120572120572120572prime119899119899119899119899prime



1199031199032119899119899prime=1

119903119903119899119899=1199031199032+1120572120572120572120572prime119899119899119899119899prime

(34)




the following form

119906119906119899119899119899119899120572120572 =

⎩⎨

⎧ sum ℏ

2119873119873119872119872119904119904ω119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger 119954119954120582120582 119904119904 isin 1 1199031199032

sum ℏ

2119873119873119872119872119904119904120596120596119954119954prime120582120582prime119890119899119899120572120572120582120582


2+ 1 119903119903

(35)


dagger )




prime119899119899120572120572 = 120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582

lowast119890119899119899120572120572120582120582

prime119899119899120572120572 =

120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582

prime119899119899120572120572 = sum 119890119890119899119899120572120572120582120582

lowast119890119899119899120572120572120582120582


119876119876119954119954120582120582 equiv ℏ

2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger 119876119876119954119954120582120582 equiv ℏ

2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger (36)


7

119875119875119954119954120582120582 = 119894119894ℏω119954119954120582120582

2119886119886119954119954120582120582



Eq (35) reads

119906119906119899119899119899119899120572120572 = sum 119890119890119904119904119899119899120582120582 (119954119954)

119873119873119872119872119904119904119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1 119903119903

2

sum 119890119904119904119899119899120582120582 (119954119954)119873119873119872119872119904119904

119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1199031199032

+ 1 119903119903 (38)


119879119879 = 1198791198791 + 1198791198792 = 12sum 119875119875119954119954120582120582119875119875minus119954119954120582120582119954119954120582120582 + 1



11988111988111 =12 120601120601120572120572120572120572prime


119899119899119899119899prime


119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime

119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954) 119863119863119899119899120572120572119899119899


120572120572prime

1199031199032

119899119899prime=1

1199031199032

119899119899=1120572120572

=12 ω119954119954120582120582prime


119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime (119954119954)

lowast

1199031199032

119899119899=1

=120572120572

12ω119954119954120582120582

2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582

ie

11988111988111 = 12sum ω119954119954120582120582

2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582 (310)






11988111988122 = 12sum 1205961205961199541199541205821205822 119876119876119954119954120582120582119876119876minus119954119954120582120582119954119954120582120582 (311)


8

11988111988112 =12 120601120601120572120572120572120572prime


119899119899119899119899prime

119903119903

119899119899prime=1199031199032+1

119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime

120582120582prime (119954119954prime)

119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954


lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119890119890119899119899120572120572120582120582 (119954119954)119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954



lowast119881119881119899119899120572120572119899119899


1199031199032119899119899=1120572120572120572120572prime (312)

where


119872119872119904119904119872119872119904119904prime119897119897 120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime (313)




9

11988111988121 =12 120601120601120572120572120572120572prime


119899119899119899119899prime

119903119903

119899119899=1199031199032+1

119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime

120582120582prime (119954119954prime)

119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899prime=1120572120572120572120572prime

=12 120601120601120572120572prime120572120572


119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1


119872119872119899119899prime119872119872119899119899

1119873119873 119890119890


1199031199032

119899119899=1120572120572prime120572120572

=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582


1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582


119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime





119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime


dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954prime 119890119890119899119899120572120572120582120582 (119954119954prime)

lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954prime∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582prime119876119876119954119954120582120582

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954

119890119890119899119899120572120572120582120582 (119954119954)lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger dagger

3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954


lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119890119890119899119899120572120572120582120582 (119954119954)119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

lowast

=

⎩⎨

⎧12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954⎭⎬

⎫dagger

= 11988111988112dagger

Define

1198671198671 = 11987911987911 + 119881119881111198671198672 = 11987911987922 + 1198811198812211986711986712 = 11988111988112 + 11988111988121

(315)


⎩⎪⎨

⎪⎧ 1198671198671 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582

dagger 119886119886119954119954120582120582 + 1231199031199032

120582120582=1119954119954


23119903119903

120582120582prime=1+31199031199032119954119954


dagger3119903119903120582120582prime=31199031199032 +1

31199031199032120582120582=1 + h c 119954119954

(316)




10


119867119867 = 1198671198670 + 119867119867119862119862 (317)

where

1198671198670 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582

dagger 119886119886119954119954120582120582 + 123119903119903

120582120582=1119954119954


dagger3119903119903120582120582prime=31199031199032 +1

31199031199032120582120582=1 + ℎ 119888119888 119954119954

(318)




119886119886119901119901(120591120591) = 119890119890119894119894ℏ119867119867119905119905119886119886119901119901119890119890

minus119894119894ℏ119867119867119905119905 = 119890119890

119867119867120591120591ℏ 119886119886119901119901119890119890

minus119867119867120591120591ℏ (41)



ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119867119867119886119886119953119953(120591120591) (42)


ℏpart119886119886119953119953(120591120591)120597120597120591120591

= 1198671198670119886119886119953119953(120591120591) = 1198671198670 1198901198901198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ = 1198671198670119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ minus 119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ 1198671198670

= 1198901198901198671198670120591120591ℏ 1198671198670119886119886119953119953 minus 1198861198861199531199531198671198670119890119890

minus1198671198670120591120591ℏ = 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ

ie

ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ (43)



2119953119953 119886119886119953119953prime = sum ℏ120596120596119953119953119886119886119953119953


dagger119886119886119953119953119953119953 =




ie


Therefore we have

ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119886119886119953119953(120591120591) (45)

11



dagger = ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 +

12

119953119953

119886119886119953119953primedagger = ℏ120596120596119953119953119886119886119953119953


119953119953

= ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953119886119886119953119953prime


dagger119886119886119953119953119953119953



dagger119886119886119953119953119953119953



dagger 119886119886119953119953dagger119886119886119953119953

119953119953


In summary we have

119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591

119886119886119953119953dagger(120591120591) = 119886119886119953119953

dagger(0)119890119890120596120596119953119953120591120591 = 119886119886119953119953dagger119890119890120596120596119953119953120591120591 (46)






dagger (120591120591prime) (47)



120588120588119867119867 = 119890119890minus120573120573119867119867119885119885119867119867 (48)








= minusTr 120588120588119867119867119890119890119867119867120591120591ℏ 119886119886119953119953119890119890

minus119867119867120591120591ℏ 119890119890

119867119867120591120591primeℏ 119886119886119953119953prime


minus119867119867120591120591primeℏ 119890119890

119867119867120591120591ℏ 119886119886119953119953119890119890


ℏ 119886119886119953119953primedagger




dagger (0)





12






119867119867120591120591ℏ 119886119886119953119953119890119890



ℏ 119886119886119953119953119890119890minus119867119867120591120591ℏ 119890119890

119867119867120591120591primeℏ


119867119867120591120591minus120591120591primeℏ 119886119886119953119953119890119890




dagger (0)








dagger (0)rang

= minusTr 120588120588119867119867119890119890119867119867(120591120591+120573120573ℏ)

ℏ 119886119886119953119953119890119890minus119867119867(120591120591+120573120573ℏ)

ℏ 119886119886119953119953prime = minusTr 120588120588119867119867119890119890120573120573119867119867119890119890

119867119867120591120591ℏ 119886119886119953119953119890119890



1119885119885119867119867


= minus1119885119885119867119867


Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119886119886119953119953(120591120591)

= minus1119885119885119867119867






ℏ 119886119886119953119953119890119890minus119867119867(120591120591minus120573120573ℏ)


119867119867120591120591ℏ 119886119886119953119953119890119890

minus119867119867120591120591ℏ 119890119890120573120573119867119867



= minus1119885119885119867119867


Tr119890119890minus120573120573119867119867119886119886119953119953(120591120591)119886119886119953119953prime(0)

= minus1119885119885119867119867


13



where 120596120596119899119899 = 2120587120587119899119899120573120573ℏ



120573120573ℏ (412)









primelang119886119886119953119953(0)119886119886119953119953prime





primelang119886119886119953119953prime





primelang119886119886119953119953prime




ie

119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =






dagger (0) = 120575120575119953119953119953119953prime


119890119890119860119860119861119861119890119890minus119860119860 = sum 1119899119899119860(119899119899)119861119861infin

119899119899=0 (414)



119891119891(119905119905) = 119891119891(0) + 119905119905119891119891prime(0) + 1199051199052

2119891119891primeprime(0) + ⋯ = sum 119905119905119899119899

119899119899119889119889119899119899119891119889119889119905119905119899119899

119905119905=0

infin119899119899=0 (415)


⎩⎪⎨



⋮119891119891(119899119899)(119905119905) = 119890119890119905119905119860119860119860(119899119899)119861119861119890119890minus119905119905119860119860

(416)

14


119890119890119905119905A119861119861119890119890minus119905119905119860119860 = sum 119905119905119899119899

119899119899119860(119899119899)119861119861infin

119899119899=0 (417)


follows


11198851198851198671198670


11198851198851198671198670


=11198851198851198671198670

Tr (minus120573120573)119899119899

1198991198991198671198670

(119899119899)119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)


dagger (0) we have

1198671198670(119899119899)119886119886119953119953prime


such that


11198851198851198671198670

Tr (minus120573120573)119899119899

1198991198991198671198670

(119899119899)119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)

=11198851198851198671198670

Tr minus120573120573ℏ120596120596119953119953prime

119899119899

119899119899119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)

= 119890119890minus120573120573ℏ120596120596119953119953prime11198851198851198671198670


dagger (0)rang





Hence we have


1119890119890120573120573ℏ120596120596119953119953minus1

equiv 120575120575119953119953prime119953119953119899119899119861119861120596120596119953119953 (419)


we obtain

119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =








119890119890119894119894120596120596119899119899minus120596120596119953119953120573120573ℏ minus

15

1 = minus1+ 1

119890119890120573120573ℏ120596120596119953119953minus1

119894119894120596120596119899119899minus120596120596119953119953119890119890119894119894


119894119894120596120596119899119899minus120596120596119953119953

119890119890minus120573120573ℏ120596120596119953119953minus1

119890119890120573120573ℏ120596120596119953119953minus1= minus 1

119894119894120596120596119899119899minus120596120596119953119953

1minus119890119890120573120573ℏ120596120596119953119953

119890119890120573120573ℏ120596120596119953119953minus1= 1

119894119894120596120596119899119899minus120596120596119953119953

ie

1198661198661199531199530(119894119894120596120596119899119899) == 1119894119894120596120596119899119899minus120596120596119953119953

(421)




119867119867119862119862119868119868 (120591120591) = 1198901198901198671198670120591120591ℏ 119867119867119862119862119890119890

minus1198671198670120591120591ℏ (51)


ℏ part119867119867119862119862119868119868 (120591120591)120597120597120591120591

= 1198671198670119867119867119862119862119868119868 (120591120591) (52)




119867119867120591120591ℏ 119886119886119953119953(0)119890119890minus

119867119867120591120591ℏ 119886119886119953119953prime

dagger (0)

= minusTr 120588120588119867119867119879119879120591120591 119890119890119867119867120591120591ℏ 119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119890119890minus

119867119867120591120591ℏ 119886119886119953119953prime

dagger (0)


where

119886119886119953119953119868119868 (120591120591) equiv 1198901198901198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus

1198671198670120591120591ℏ (53)


119880119880(1205911205911 1205911205912) equiv 11989011989011986711986701205911205911ℏ 119890119890minus

119867119867 (1205911205911minus1205911205912)ℏ 119890119890minus

11986711986701205911205912ℏ (54)


119880119880(1205911205911 1205911205912)119880119880(1205911205912 1205911205913) = 119880119880(1205911205911 1205911205913) (55)


ℏpart119880119880(120591120591 120591120591prime)

120597120597120591120591= 1198671198670119890119890

1198671198670120591120591ℏ 119890119890minus


1198671198670120591120591ℏ 119867119867119890119890minus


1198671198670120591120591primeℏ

= 1198901198901198671198670120591120591ℏ 1198671198670 minus 119867119867119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119890119890minus


1198671198670120591120591primeℏ = 119890119890

1198671198670120591120591ℏ minus119867119867119862119862119890119890

minus1198671198670120591120591ℏ 119880119880(120591120591 120591120591prime)

= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime)

16

ie

ℏ part119880119880120591120591120591120591prime120597120597120591120591

= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime) (56)


119880119880(120591120591 120591120591prime) = summinus1ℏ

119899119899

119899119899 int 1198891198891205911205911120591120591120591120591prime ⋯int 119889119889120591120591119899119899

120591120591120591120591prime 119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119899119899)infin

119899119899=0 (57)


119866119866119953119953119953119953prime(120591120591 0) = minus1119885119885119867119867



dagger (0)

Tr119890119890minus120573120573119867119867


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)=


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)





119866119866119953119953119953119953prime(120591120591 0) = minussum 1

119898119898minus1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)119886119886119953119953119868119868 (120591120591)119886119886


119898119898=0

sum 1119898119898minus

1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0infin

119898119898=0 (58)




120573120573ℏ0 lang119879119879120591120591119867119867119862119862


12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0+⋯

1minus1ℏint 1198891198891205911205911120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)rang0+12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0+⋯

(59)


119863119863119898119898 equiv 1119898119898minus 1

ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0 (510)




17







1198681198682 = minus1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0

= minus1ℏ 1198891198891205911205911120573120573ℏ



0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

= 119866119866119953119953119953119953prime0 (120591120591 0) minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888





(1) (120591120591) equiv

minus1ℏ int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime


1198681198683 = minus1ℏint 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime


(1) (120591120591) (513)


1198681198683 = 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0 = 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)119886119886119953119953(120591120591)rang0 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime


2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 +

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888



follows

18

12ℏ2

1198891198891205911205912120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 +1

2ℏ2 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

=12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

+12 minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0

minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

=12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

+12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

= minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime



1198681198683 = 119866119866119953119953119953119953prime(2) (120591120591) + 119866119866119953119953119953119953prime

(1) (120591120591)1198631198631 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198632 (514)


1198681198684 = minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0 =

minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913



minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913



minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913


dagger (0)rang0119888119888 +

minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 =

119866119866119953119953119953119953prime0 (120591120591 0) minus1

6ℏ3 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0+

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0

minus1ℏ int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888+

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime


120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0+ 119866119866119953119953119953119953prime

(3) (120591120591) = 119866119866119953119953119953119953prime(3) (120591120591) +

119866119866119953119953119953119953prime(2) (120591120591)1198631198631 + 119866119866119953119953119953119953prime

(1) (120591120591)1198631198632 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198633



follows

119866119866119953119953119953119953prime0 (120591120591) minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0

+1

2ℏ2 1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime


= 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime

(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (1 + 1198631198631 + 1198631198632 + ⋯ )



19


(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (515)

where

119866119866119953119953119953119953prime

(1) (120591120591) = minus1ℏ int 1198891198891205911205911


dagger (0)rang0119888119888

119866119866119953119953119953119953prime(2) (120591120591) = 1

2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 (516)




119866119866119953119953119953119953prime(1) (120591120591) = minus

1ℏ

12 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591 119881119881119895119895119895119895prime(119948119948)119876119876119948119948119895119895(1205911205911)119876119876119948119948119895119895prime

dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


= minus12ℏ

1198891198891205911205911120573120573ℏ

0lang119879119879120591120591

ℏ119881119881119895119895119895119895prime(119948119948)2120596120596119948119948119895119895120596120596119948119948119895119895prime


dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948

+ ℏ119881119881119895119895119895119895prime

lowast (119948119948)



3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948





11989211989211 = minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886minus119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)rang0119888119888

= minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886minus119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)119886119886119948119948119895119895(1205911205911)rang0

= minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ



119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


20


11989211989211 =

⎩⎪⎨


4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582


2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)

4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582


2lt 120582120582 le 3119903119903

0 else

(517)


11989211989212 = minus14


4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886119948119948119895119895prime(1205911205911)119886119886119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)rang0119888119888

= minus14


4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886119954119954prime120582120582prime


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0

ie

11989211989212 =

⎩⎪⎨

⎪⎧



0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903



0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903

2lt 120582120582 le 3119903119903

0 else

(518)


can be written as

119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591) =

⎩⎪⎨

⎪⎧minus119881119881

120582120582120582120582prime(minus119954119954)

2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ

0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le3119903119903

2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582

(119954119954)

2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ

0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le3119903119903

2lt 120582120582 le 3119903119903

0 else

(519)


we have

1198891198891205911205911120573120573ℏ

01198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0) = 1198891198891205911205911120573120573ℏ

0

1120573120573ℏ

119890119890minus119894119894120596120596119899119899(120591120591minus1205911205911)1198661198661199541199541205821205820 (119894119894120596120596119899119899)

119899119899

1120573120573ℏ


119899119899prime

=1

(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591 1198891198891205911205911120573120573ℏ

0119890119890119894119894120596120596119899119899minus120596120596119899119899prime1205911205911 119866119866119954119954120582120582


119899119899119899119899prime=

=1


0 (119894119894120596120596119899119899prime)119899119899119899119899prime

=1120573120573ℏ


0 (119894119894120596120596119899119899)119899119899


21

119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591)

119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =

⎩⎪⎨



0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582 le 31199031199032

lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)


0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582prime le 31199031199032

lt 120582120582 le 3119903119903

0 else

(520)




Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =

⎩⎪⎨


2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)


2lt 120582120582 le 3119903119903

0 else

(521)



119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =

18ℏ2 1198891198891205911205911

120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0lang119879119879120591120591

⎣⎢⎢⎡ 11988111988111989511989511198951198951prime(119948119948120783120783)1198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951prime

dagger (1205911205911)3119903119903

1198951198951prime=31199031199032 +1

31199031199032

1198951198951=1119948119948120783120783


dagger (1205911205911)3119903119903

1198941198941prime=31199031199032 +1

31199031199032

1198941198941=1119948119948120783120783 ⎦⎥⎥⎤

⎣⎢⎢⎡ 11988111988111989511989521198951198952prime(119948119948120784120784)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime

dagger (1205911205912)3119903119903

1198951198952prime=31199031199032 +1

31199031199032

1198951198952=1119948119948120784120784


dagger (1205911205912)3119903119903

1198941198942prime=31199031199032 +1

31199031199032

1198941198942=1119948119948120784120784 ⎦⎥⎥⎤119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime

dagger (0)rang0119888119888 = 1198921198921 + 1198921198922 + 1198921198923 + 1198921198924

where

1198921198921 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


31199031199032 lt1198951198952

primele311990311990311994811994812078412078411989511989521198951198952prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime



(522)

1198921198922 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


1le1198941198942le31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime



(523)

1198921198923 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


1le1198951198952le31199031199032 lt1198951198952

primele311990311990311994811994812078412078411989511989521198951198952prime

1le1198941198941le31199031199032 lt1198941198941

primele311990311990311994811994812078312078311989411989411198941198941prime



(524)

22

1198921198924 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


lowast (119948119948120784120784)1le1198941198942le

31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198941198941le31199031199032 lt1198941198941

primele311990311990311994811994812078312078311989411989411198941198941prime



(525)



dagger we can


1198921198922 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)119881119881

11989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime

times 119868119868 (526)

Where










dagger (0)rang0c










⎩⎪⎨

⎪⎧ 1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951prime


dagger (0)rang0119888119888







(527)


23












0 (1205911205912 1205911205912)12057512057511989411989421198941198942prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582

+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime



0 (1205911205911 1205911205911)12057512057511989511989511198951198951prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582


0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582

= 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime



0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582




























24






























0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 11986611986611994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime




0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime




11989211989221 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198681198681 =

132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951


1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 + 1

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951prime

0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙


0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 =

⎩⎪⎨

⎪⎧

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912) ∙ 119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205912 1205911205911) ∙ 119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582

0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

0 else

and

25

11989211989224 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198681198684 =

132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903




32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903




0 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 =

⎩⎪⎨

⎪⎧120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime0 (1205911205912 0)119866119866119954119954120582120582

0 (120591120591 1205911205911) 120582120582 120582120582prime isin 1 31199031199032

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

0 else


(314)]



1198921198922 =

⎩⎪⎨

⎪⎧

120575120575119954119954119954119954prime

16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

120575120575119954119954119954119954prime

16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582

0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

0 else

(528)



calculated as

119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) = 41198921198922 =

⎩⎪⎨

⎪⎧120575120575119954119954119954119954prime

4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1


0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

120575120575119954119954119954119954prime

4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 31199031199032

+ 13119903119903

0 else

(529)



119866119866119954119954120582120582119954119954prime120582120582prime


(2) (119894119894120596120596119899119899)119899119899


(2) (120591120591)120573120573ℏ0

(530)


26

119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =

120575120575119954119954119954119954prime4

1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954120582120582prime

0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912)

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0

1120573120573ℏ


1198991198992

∙1120573120573ℏ

119890119890minus11989411989412059612059611989911989911205911205911119866119866119954119954120582120582prime0 1198941198941205961205961198991198991

1198991198991

∙1120573120573ℏ

119890119890minus119894119894120596120596119899119899(120591120591minus1205911205912)1198661198661199541199541205821205820 (119894119894120596120596119899119899)

119899119899

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1120573120573ℏ

119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954120582120582prime

0 11989411989412059612059611989911989911198661198661199541199541198951198951prime0 1198941198941205961205961198991198992

11989911989911989911989911198991198992⎩⎪⎨

⎪⎧ 1120573120573ℏ

1198891198891205911205911120573120573ℏ

0119890119890minus1198941198941205961205961198991198991minus12059612059611989911989921205911205911

times1120573120573ℏ

1198891198891205911205912120573120573ℏ

0119890119890minus1198941198941205961205961198991198992minus1205961205961198991198991205911205912

⎭⎪⎬

⎪⎫

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1120573120573ℏ

119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)

11989911989911989911989911198991198992

119866119866119954119954120582120582prime0 11989411989412059612059611989911989911198661198661199541199541198951198951prime

0 1198941198941205961205961198991198992120575120575119899119899111989911989921205751205751198991198991198991198992

=120575120575119954119954119954119954prime120573120573ℏ

119890119890minus119894119894120596120596119899119899120591120591

119899119899

1198661198661199541199541205821205820 (119894119894120596120596119899119899)

14

119881119881120582120582prime1198951198951prime(119954119954



3119903119903

1198951198951prime=31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899)119866119866119954119954120582120582prime

0 (119894119894120596120596119899119899)


119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582

0 (119894119894120596120596119899119899) 120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 119866119866119954119954120582120582prime

0 (119894119894120596120596119899119899) (531)



120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) (532)

and 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582

0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954120582120582prime(2) (119894119894120596120596119899119899) ∙ 119866119866119954119954120582120582prime


+ 13119903119903 we

have

Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) (533)


Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =

⎩⎪⎪⎨

⎪⎪⎧120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903

2

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

0 else

(534)

27



(1) (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) + ⋯

= 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582


0 (119894119894120596120596119899119899) + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime

(2) (119894119894120596120596119899119899)

∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) + ⋯ = 119866119866119954119954120582120582


0 (119894119894120596120596119899119899)

or





(2) (119894119894120596120596119899119899) + ⋯



Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) =

⎩⎪⎪⎪⎪⎨


120575120575119954119954119954119954prime

2

119881119881120582120582120582120582primelowast (119954119954)

120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903

minus120575120575119954119954119954119954prime

2

119881119881120582120582prime120582120582(119954119954)

120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582prime le 3119903119903

2lt 120582120582 le 3119903119903

120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903

2

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

(536)









Σ119954119954120582120582119903119903 (120596120596) via [13]

Γ119954119954120582120582(120596120596) = minus2ImΣ119954119954120582120582119954119954120582120582119903119903 (120596120596) (539)


we have

28

Σ119954119954120582120582119954119954120582120582119903119903 (119894119894120596120596119899119899) =

⎩⎪⎨

⎪⎧14sum

1198811198811205821205821198951198951prime(119954119954)

2

1205961205961199541199541198951198951prime120596120596119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

1

120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894

120582120582 isin 1 31199031199032

14sum 1198811198811198951198951120582120582(119954119954)2

1205961205961199541199541198951198951120596120596119954119954120582120582

311990311990321198951198951=1

1120596120596minus1205961205961199541199541198951198951+119894119894119894119894

120582120582 isin 31199031199032

+ 13119903119903

(540)

Using the relation

Im 1

120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894



Γ119954119954120582120582(120596120596) =

⎩⎪⎨

⎪⎧1205871205872sum

1198811198811205821205821198951198951prime(119954119954)

2

1205961205961199541199541198951198951prime120596120596119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

120575120575120596120596 minus 1205961205961199541199541198951198951prime 120582120582 isin 1 31199031199032

1205871205872sum 1198811198811198951198951120582120582(119954119954)2

1205961205961199541199541198951198951120596120596119954119954120582120582

311990311990321198951198951=1

120575120575120596120596 minus 1205961205961199541199541198951198951 120582120582 isin 31199031199032

+ 13119903119903

(542)

Or equivalently

Γ119954119954120582120582(119864119864 = ℏ120596120596) =

⎩⎪⎨

⎪⎧120587120587ℏ3

2sum

1198811198811205821205821198951198951prime(119954119954)

2

1198641198641199541199541198951198951prime119864119864119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

120575120575119864119864 minus 1198641198641199541199541198951198951prime 120582120582 isin 1 31199031199032

120587120587ℏ3

2sum 1198811198811198951198951120582120582(119954119954)2

1198641198641199541199541198951198951119864119864119954119954120582120582

311990311990321198951198951=1

120575120575119864119864 minus 1198641198641199541199541198951198951 120582120582 isin 31199031199032

+ 13119903119903

(543)








int(⋯ )120588120588119864119864119954119954d119864119864119954119954 giving

Γ119954119954120582120582(119864119864) = 120587120587ℏ3


2119864119864119954119954119895119895prime=119864119864119954119954prime


2120588120588(119864119864)

119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864

119864119864119864119864119954119954120582120582 (544)







29


Γtot = 1119885119885119890119890minus120573120573119864119864119954119954120582120582Γ119954119954120582120582119864119864119954119954120582120582

119954119954120582120582

=120587120587ℏ3

2

119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582 119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864119954119954120582120582

1198641198641199541199541205821205822119954119954120582120582

=120587120587ℏ3

2

119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582 119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582

Finally we get

Γtot = Γ119954119954120582120582(119864119864) =120587120587ℏ3

2sum 119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582 (545)





to make



References


MainText


SI

1 Methodology

11 Model system












11 Basic notations

12 Dynamical matrix


















4



















5



120625120625(119954119954) = 12062512062511(119954119954) 12062512062512(119954119954)12062512062521(119954119954) 12062512062522(119954119954) (1)




1199321199321dagger(119954119954)12062512062511(119954119954)1199321199321(119954119954) and 12062512062522(119954119954) = 1199321199322




matrix of the form

119932119932(119954119954) = 1199321199321(119954119954) 120782120782120782120782 1199321199322(119954119954) (2)


119932119932dagger(119954119954)120625120625(119954119954)119932119932(119954119954) = 12062512062511(119954119954) 12062512062512(119954119954)12062512062521(119954119954) 12062512062522(119954119954)

(3)

where 12062512062512(119954119954) = 1199321199321dagger(119954119954)12062512062512(119954119954)1199321199322(119954119954) and 12062512062521(119954119954) = 12062512062512






derivation)

Γtot = 120587120587ℏ3

2sum 119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582 (4)



2 is the coupling







6
























7














bulk value












8



























ASSOCIATED CONTENT




9


conductivity

AUTHOR INFORMATION



Notes


ACKNOWLEDGMENTS










MP1303

10

References


11


S1










1 Methodology





S2

1 Methodology

11 Model system












S3



















S4




120581120581CP = 119876119860119860Δ119879119879Δ119911119911

(S1)

















S1(b)]







S5















part119864119864t120597120597120597120597

= 119860119860119877119877119879119879bot(119905119905) minus 119879119879top(119905119905) (S3)






S6














119871119871 = |11989911989911199381199381 + 11989911989921199381199382| = 11988611988611989911989912 + 11989911989922 + 11989911989911198991198992 = |1198991198991minus1198991198992|1198861198862 sin(1205791205792) (S4)




2120587120587120575120575119894119894119895119895 such that

1199181199181 = 4120587120587radic3119871119871

radic32

minus12 1199181199182 = 4120587120587

radic3119871119871(01) (S5)

S7





120579120579 (deg) 119860119860 (nm2) 1198991198991 1198991198992 1198981198981 1198981198982

0 60383688 24 0 24 0

0696407 709613169 48 47 47 48

1121311 273718420 30 29 29 30

2000628 85648391 17 16 16 17

3006558 152322047 23 21 21 23

4048894 189013524 26 23 23 26

5085849 53255058 14 12 12 14

7926470 49376245 14 11 11 14

9998709 86277388 19 14 14 19

15178179 31554670 15 4 4 15

19932013 42876612 15 8 8 15

25039660 13942761 9 4 4 9

30158276 18974735 11 4 4 11

32204228 21805221 12 4 4 12









119868119868 = 1119881119881119887119887int 119891119891(119954119954)BZ d119954119954 asymp 1

119873119873sum 119908119908119894119894119891119891(119954119954119894119894)119894119894 (S6)

S8






⎩⎪⎨

⎪⎧ 1199541199541 = 1

9 1205971205973 1199541199542 = 2

9 1205971205973 1199541199543 = 1

18 512059712059718

1199541199544 = 518

512059712059718 1199541199545 = 1

9 21205971205979 1199541199546 = 2

9 21205971205979

1199541199547 = 118

312059712059718 1199541199548 = 1

9 1205971205979 1199541199549 = 1

18 12059712059718

(S7)


119908119908124689 = 112

119908119908357 = 16 (S8)


Γtot = 120587120587ℏ3

2sum 119908119908119896119896 sum

119890119890minus120573120573119864119864119954119954119948119948120582120582

119885119885

120588120588119864119864119954119954119948119948120582120582119881119881120582120582120582120582+31199031199032(119954119954119948119948)

2

1198641198641199541199541199481199481205821205822120582120582

9119896119896=1 (S9)





S9


















contribution [10]

S10



S11


S12


1









119955119955119899119899119899119899(119905119905) = 119929119929119899119899 + 119929119929s0 + 119958119958119899119899119899119899(119905119905) (11)



ℒ = sum sum 119872119872119904119904|119955119899119899119904119904|2(119905119905)2

119903119903119899119899=1

119873119873119899119899=1 minus 119881119881 (12)




ℒ = sum sum 119872119872119904119904|119958119899119899119904119904|2(119905119905)2

119903119903119899119899=1

119873119873119899119899=1 minus 1

2sum sum sum 120601120601120572120572120572120572prime












12 Dynamical matrix


equation as follows

2








form

119906119906119899119899119899119899120572120572(119905119905) = 1119872119872119904119904

119890119890119899119899120572120572119890119890119894119894(119954119954∙119929119929119899119899minus120596120596119905119905) (17)




where

119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = sum 1

119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime



119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime 119890119890

minus119894119894119954119954∙119929119929119897119897119897119897 (19)















120596120596120582120582(minus119954119954) = 120596120596120582120582(119954119954) (112)


sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582

prime119899119899120572120572 = 120575120575120582120582120582120582prime (113)





3





119890119890119899119899120572120572120582120582 (minus119954119954)lowast

= 119890119890119899119899120572120572120582120582 (119954119954) (116)


119906119906119899119899119899119899120572120572(119905119905) = sum 119862119862120582120582(119954119954)119873119873119872119872119904119904

119890119890119899119899120572120572120582120582 (119954119954)119890119890minus119894119894120596120596119954119954120582120582119905119905119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 = sum 119876119876120582120582(119954119954119905119905)119873119873119872119872119904119904

119890119890119899119899120572120572120582120582 (119954119954)119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 (21)




119876119876120582120582(119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)lowast

= 119876119876120582120582(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (minus119954119954) (22)



[119876119876120582120582(119954119954 119905119905)]lowast = 119876119876120582120582(minus119954119954 119905119905) (23)



119879119879 = 12sum 1198721198721198991198991199061198991198991198991198991205721205722 (119905119905)119899119899119899119899120572120572 = 1

2sum 119872119872119904119904





12sum 119876120582120582(119954119954 119905119905)119876120582120582prime

lowast (119954119954 119905119905) sum 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime(119954119954)

lowast119899119899120572120572 =119954119954120582120582120582120582prime

12sum 119876120582120582(119954119954 119905119905)119876120582120582lowast(119954119954 119905119905)119954119954120582120582 =

12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582

ie

119879119879 = 12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582 (24)


1119873119873sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899 = 120575120575119954119954+119954119954prime120782120782 (25)

4





12120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899+119954119954prime∙119929119929119899119899prime

119873119873119872119872119899119899119872119872119899119899prime



=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime


119873119873119872119872119899119899119872119872119899119899prime



=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954prime∙119929119929119897119897


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899

119899119899=120575120575119902119902+119902119902prime=0


=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954119929119929119897119897



=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)

119899119899120572120572



=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)

119899119899120572120572


119954119954120582120582120582120582prime


121205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)

119954119954120582120582

ie

119880119880 = 12sum 1205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)119954119954120582120582 (26)




2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)119954119954120582120582 (27)


119867119867 = 119879119879 + 119881119881 = 12sum [119875119875120582120582(minus119954119954 119905119905)119875119875120582120582(119954119954 119905119905) + 120596120596120582120582

2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)]119954119954120582120582 (28)


119867119867 = 12sum 119875119875minus119954119954120582120582119875119875119954119954120582120582 + 120596120596119954119954120582120582

2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582 (29)






119876119876119954119954120582120582 = ℏ

2120596120596119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582

dagger 119875119875119954119954120582120582 = 119894119894ℏ120596120596119954119954120582120582

2119887119887119954119954120582120582




5

119887119887119954119954120582120582 119887119887119954119954prime120582120582prime



dagger = 0 (212)


119867119867 =12119875119875minus119954119954120582120582119875119875119954119954120582120582 +120596120596119954119954120582120582

2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582

=12minus

ℏ120596120596119954119954120582120582

2119887119887minus119954119954120582120582


2 ℏ2120596120596119954119954120582120582


dagger 119954119954120582120582

=ℏ4120596120596119954119954120582120582minus119887119887minus119954119954120582120582



119954119954120582120582



dagger

=ℏ4120596120596119954119954120582120582119887119887minus119954119954120582120582


dagger + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582

119954119954120582120582

=ℏ41205961205961199541199541205821205822119887119887minus119954119954120582120582


119954119954120582120582

= ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 +

12

119954119954120582120582


119867119867 = sum ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1

2119954119954120582120582 (213)


119906119906119899119899119899119899120572120572 = sum ℏ

2119873119873119872119872119904119904120596120596119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119887119887119954119954120582120582 + 119887119887minus119954119954120582120582

dagger 119954119954120582120582 (214)








119867119867 = 1198671198671 + 1198671198672 + 11986711986712 (31)



119867119867 = 119879119879 + 119881119881 = sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2119899119899119899119899120572120572 + 1

2sum sum sum 120601120601120572120572120572120572prime


6


119904119904 = 1199031199032

+ 1 1199031199032


119879119879 = sum sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2

1199031199032119899119899=1 + sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)

2119903119903119899119899=1199031199032+1

119899119899120572120572 (33)


119881119881 =12 120601120601120572120572120572120572prime


1199031199032

119899119899119899119899prime=1


119903119903

119899119899119899119899prime=1199031199032+1


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime119899119899119899119899prime


1199031199032

119899119899prime=1

119903119903

119899119899=1199031199032+1

= 11988111988111 + 11988111988122 + 11988111988112 + 11988111988121

Or equivalently

⎩⎪⎪⎨

⎪⎪⎧ 11988111988111 = 1

2sum sum sum 120601120601120572120572120572120572prime



11988111988122 = 12sum sum sum 120601120601120572120572120572120572prime





119903119903119899119899prime=1199031199032+1

1199031199032119899119899=1120572120572120572120572prime119899119899119899119899prime



1199031199032119899119899prime=1

119903119903119899119899=1199031199032+1120572120572120572120572prime119899119899119899119899prime

(34)




the following form

119906119906119899119899119899119899120572120572 =

⎩⎨

⎧ sum ℏ

2119873119873119872119872119904119904ω119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger 119954119954120582120582 119904119904 isin 1 1199031199032

sum ℏ

2119873119873119872119872119904119904120596120596119954119954prime120582120582prime119890119899119899120572120572120582120582


2+ 1 119903119903

(35)


dagger )




prime119899119899120572120572 = 120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582

lowast119890119899119899120572120572120582120582

prime119899119899120572120572 =

120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582

prime119899119899120572120572 = sum 119890119890119899119899120572120572120582120582

lowast119890119899119899120572120572120582120582


119876119876119954119954120582120582 equiv ℏ

2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger 119876119876119954119954120582120582 equiv ℏ

2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger (36)


7

119875119875119954119954120582120582 = 119894119894ℏω119954119954120582120582

2119886119886119954119954120582120582



Eq (35) reads

119906119906119899119899119899119899120572120572 = sum 119890119890119904119904119899119899120582120582 (119954119954)

119873119873119872119872119904119904119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1 119903119903

2

sum 119890119904119904119899119899120582120582 (119954119954)119873119873119872119872119904119904

119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1199031199032

+ 1 119903119903 (38)


119879119879 = 1198791198791 + 1198791198792 = 12sum 119875119875119954119954120582120582119875119875minus119954119954120582120582119954119954120582120582 + 1



11988111988111 =12 120601120601120572120572120572120572prime


119899119899119899119899prime


119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime

119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954) 119863119863119899119899120572120572119899119899


120572120572prime

1199031199032

119899119899prime=1

1199031199032

119899119899=1120572120572

=12 ω119954119954120582120582prime


119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime (119954119954)

lowast

1199031199032

119899119899=1

=120572120572

12ω119954119954120582120582

2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582

ie

11988111988111 = 12sum ω119954119954120582120582

2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582 (310)






11988111988122 = 12sum 1205961205961199541199541205821205822 119876119876119954119954120582120582119876119876minus119954119954120582120582119954119954120582120582 (311)


8

11988111988112 =12 120601120601120572120572120572120572prime


119899119899119899119899prime

119903119903

119899119899prime=1199031199032+1

119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime

120582120582prime (119954119954prime)

119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954


lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119890119890119899119899120572120572120582120582 (119954119954)119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954



lowast119881119881119899119899120572120572119899119899


1199031199032119899119899=1120572120572120572120572prime (312)

where


119872119872119904119904119872119872119904119904prime119897119897 120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime (313)




9

11988111988121 =12 120601120601120572120572120572120572prime


119899119899119899119899prime

119903119903

119899119899=1199031199032+1

119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime

120582120582prime (119954119954prime)

119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899prime=1120572120572120572120572prime

=12 120601120601120572120572prime120572120572


119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1


119872119872119899119899prime119872119872119899119899

1119873119873 119890119890


1199031199032

119899119899=1120572120572prime120572120572

=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582


1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582


119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime





119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime


dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954prime 119890119890119899119899120572120572120582120582 (119954119954prime)

lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954prime∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582prime119876119876119954119954120582120582

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954

119890119890119899119899120572120572120582120582 (119954119954)lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger dagger

3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954


lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119890119890119899119899120572120572120582120582 (119954119954)119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

lowast

=

⎩⎨

⎧12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954⎭⎬

⎫dagger

= 11988111988112dagger

Define

1198671198671 = 11987911987911 + 119881119881111198671198672 = 11987911987922 + 1198811198812211986711986712 = 11988111988112 + 11988111988121

(315)


⎩⎪⎨

⎪⎧ 1198671198671 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582

dagger 119886119886119954119954120582120582 + 1231199031199032

120582120582=1119954119954


23119903119903

120582120582prime=1+31199031199032119954119954


dagger3119903119903120582120582prime=31199031199032 +1

31199031199032120582120582=1 + h c 119954119954

(316)




10


119867119867 = 1198671198670 + 119867119867119862119862 (317)

where

1198671198670 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582

dagger 119886119886119954119954120582120582 + 123119903119903

120582120582=1119954119954


dagger3119903119903120582120582prime=31199031199032 +1

31199031199032120582120582=1 + ℎ 119888119888 119954119954

(318)




119886119886119901119901(120591120591) = 119890119890119894119894ℏ119867119867119905119905119886119886119901119901119890119890

minus119894119894ℏ119867119867119905119905 = 119890119890

119867119867120591120591ℏ 119886119886119901119901119890119890

minus119867119867120591120591ℏ (41)



ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119867119867119886119886119953119953(120591120591) (42)


ℏpart119886119886119953119953(120591120591)120597120597120591120591

= 1198671198670119886119886119953119953(120591120591) = 1198671198670 1198901198901198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ = 1198671198670119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ minus 119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ 1198671198670

= 1198901198901198671198670120591120591ℏ 1198671198670119886119886119953119953 minus 1198861198861199531199531198671198670119890119890

minus1198671198670120591120591ℏ = 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ

ie

ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ (43)



2119953119953 119886119886119953119953prime = sum ℏ120596120596119953119953119886119886119953119953


dagger119886119886119953119953119953119953 =




ie


Therefore we have

ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119886119886119953119953(120591120591) (45)

11



dagger = ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 +

12

119953119953

119886119886119953119953primedagger = ℏ120596120596119953119953119886119886119953119953


119953119953

= ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953119886119886119953119953prime


dagger119886119886119953119953119953119953



dagger119886119886119953119953119953119953



dagger 119886119886119953119953dagger119886119886119953119953

119953119953


In summary we have

119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591

119886119886119953119953dagger(120591120591) = 119886119886119953119953

dagger(0)119890119890120596120596119953119953120591120591 = 119886119886119953119953dagger119890119890120596120596119953119953120591120591 (46)






dagger (120591120591prime) (47)



120588120588119867119867 = 119890119890minus120573120573119867119867119885119885119867119867 (48)








= minusTr 120588120588119867119867119890119890119867119867120591120591ℏ 119886119886119953119953119890119890

minus119867119867120591120591ℏ 119890119890

119867119867120591120591primeℏ 119886119886119953119953prime


minus119867119867120591120591primeℏ 119890119890

119867119867120591120591ℏ 119886119886119953119953119890119890


ℏ 119886119886119953119953primedagger




dagger (0)





12






119867119867120591120591ℏ 119886119886119953119953119890119890



ℏ 119886119886119953119953119890119890minus119867119867120591120591ℏ 119890119890

119867119867120591120591primeℏ


119867119867120591120591minus120591120591primeℏ 119886119886119953119953119890119890




dagger (0)








dagger (0)rang

= minusTr 120588120588119867119867119890119890119867119867(120591120591+120573120573ℏ)

ℏ 119886119886119953119953119890119890minus119867119867(120591120591+120573120573ℏ)

ℏ 119886119886119953119953prime = minusTr 120588120588119867119867119890119890120573120573119867119867119890119890

119867119867120591120591ℏ 119886119886119953119953119890119890



1119885119885119867119867


= minus1119885119885119867119867


Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119886119886119953119953(120591120591)

= minus1119885119885119867119867






ℏ 119886119886119953119953119890119890minus119867119867(120591120591minus120573120573ℏ)


119867119867120591120591ℏ 119886119886119953119953119890119890

minus119867119867120591120591ℏ 119890119890120573120573119867119867



= minus1119885119885119867119867


Tr119890119890minus120573120573119867119867119886119886119953119953(120591120591)119886119886119953119953prime(0)

= minus1119885119885119867119867


13



where 120596120596119899119899 = 2120587120587119899119899120573120573ℏ



120573120573ℏ (412)









primelang119886119886119953119953(0)119886119886119953119953prime





primelang119886119886119953119953prime





primelang119886119886119953119953prime




ie

119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =






dagger (0) = 120575120575119953119953119953119953prime


119890119890119860119860119861119861119890119890minus119860119860 = sum 1119899119899119860(119899119899)119861119861infin

119899119899=0 (414)



119891119891(119905119905) = 119891119891(0) + 119905119905119891119891prime(0) + 1199051199052

2119891119891primeprime(0) + ⋯ = sum 119905119905119899119899

119899119899119889119889119899119899119891119889119889119905119905119899119899

119905119905=0

infin119899119899=0 (415)


⎩⎪⎨



⋮119891119891(119899119899)(119905119905) = 119890119890119905119905119860119860119860(119899119899)119861119861119890119890minus119905119905119860119860

(416)

14


119890119890119905119905A119861119861119890119890minus119905119905119860119860 = sum 119905119905119899119899

119899119899119860(119899119899)119861119861infin

119899119899=0 (417)


follows


11198851198851198671198670


11198851198851198671198670


=11198851198851198671198670

Tr (minus120573120573)119899119899

1198991198991198671198670

(119899119899)119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)


dagger (0) we have

1198671198670(119899119899)119886119886119953119953prime


such that


11198851198851198671198670

Tr (minus120573120573)119899119899

1198991198991198671198670

(119899119899)119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)

=11198851198851198671198670

Tr minus120573120573ℏ120596120596119953119953prime

119899119899

119899119899119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)

= 119890119890minus120573120573ℏ120596120596119953119953prime11198851198851198671198670


dagger (0)rang





Hence we have


1119890119890120573120573ℏ120596120596119953119953minus1

equiv 120575120575119953119953prime119953119953119899119899119861119861120596120596119953119953 (419)


we obtain

119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =








119890119890119894119894120596120596119899119899minus120596120596119953119953120573120573ℏ minus

15

1 = minus1+ 1

119890119890120573120573ℏ120596120596119953119953minus1

119894119894120596120596119899119899minus120596120596119953119953119890119890119894119894


119894119894120596120596119899119899minus120596120596119953119953

119890119890minus120573120573ℏ120596120596119953119953minus1

119890119890120573120573ℏ120596120596119953119953minus1= minus 1

119894119894120596120596119899119899minus120596120596119953119953

1minus119890119890120573120573ℏ120596120596119953119953

119890119890120573120573ℏ120596120596119953119953minus1= 1

119894119894120596120596119899119899minus120596120596119953119953

ie

1198661198661199531199530(119894119894120596120596119899119899) == 1119894119894120596120596119899119899minus120596120596119953119953

(421)




119867119867119862119862119868119868 (120591120591) = 1198901198901198671198670120591120591ℏ 119867119867119862119862119890119890

minus1198671198670120591120591ℏ (51)


ℏ part119867119867119862119862119868119868 (120591120591)120597120597120591120591

= 1198671198670119867119867119862119862119868119868 (120591120591) (52)




119867119867120591120591ℏ 119886119886119953119953(0)119890119890minus

119867119867120591120591ℏ 119886119886119953119953prime

dagger (0)

= minusTr 120588120588119867119867119879119879120591120591 119890119890119867119867120591120591ℏ 119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119890119890minus

119867119867120591120591ℏ 119886119886119953119953prime

dagger (0)


where

119886119886119953119953119868119868 (120591120591) equiv 1198901198901198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus

1198671198670120591120591ℏ (53)


119880119880(1205911205911 1205911205912) equiv 11989011989011986711986701205911205911ℏ 119890119890minus

119867119867 (1205911205911minus1205911205912)ℏ 119890119890minus

11986711986701205911205912ℏ (54)


119880119880(1205911205911 1205911205912)119880119880(1205911205912 1205911205913) = 119880119880(1205911205911 1205911205913) (55)


ℏpart119880119880(120591120591 120591120591prime)

120597120597120591120591= 1198671198670119890119890

1198671198670120591120591ℏ 119890119890minus


1198671198670120591120591ℏ 119867119867119890119890minus


1198671198670120591120591primeℏ

= 1198901198901198671198670120591120591ℏ 1198671198670 minus 119867119867119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119890119890minus


1198671198670120591120591primeℏ = 119890119890

1198671198670120591120591ℏ minus119867119867119862119862119890119890

minus1198671198670120591120591ℏ 119880119880(120591120591 120591120591prime)

= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime)

16

ie

ℏ part119880119880120591120591120591120591prime120597120597120591120591

= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime) (56)


119880119880(120591120591 120591120591prime) = summinus1ℏ

119899119899

119899119899 int 1198891198891205911205911120591120591120591120591prime ⋯int 119889119889120591120591119899119899

120591120591120591120591prime 119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119899119899)infin

119899119899=0 (57)


119866119866119953119953119953119953prime(120591120591 0) = minus1119885119885119867119867



dagger (0)

Tr119890119890minus120573120573119867119867


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)=


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)





119866119866119953119953119953119953prime(120591120591 0) = minussum 1

119898119898minus1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)119886119886119953119953119868119868 (120591120591)119886119886


119898119898=0

sum 1119898119898minus

1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0infin

119898119898=0 (58)




120573120573ℏ0 lang119879119879120591120591119867119867119862119862


12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0+⋯

1minus1ℏint 1198891198891205911205911120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)rang0+12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0+⋯

(59)


119863119863119898119898 equiv 1119898119898minus 1

ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0 (510)




17







1198681198682 = minus1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0

= minus1ℏ 1198891198891205911205911120573120573ℏ



0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

= 119866119866119953119953119953119953prime0 (120591120591 0) minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888





(1) (120591120591) equiv

minus1ℏ int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime


1198681198683 = minus1ℏint 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime


(1) (120591120591) (513)


1198681198683 = 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0 = 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)119886119886119953119953(120591120591)rang0 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime


2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 +

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888



follows

18

12ℏ2

1198891198891205911205912120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 +1

2ℏ2 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

=12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

+12 minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0

minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

=12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

+12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

= minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime



1198681198683 = 119866119866119953119953119953119953prime(2) (120591120591) + 119866119866119953119953119953119953prime

(1) (120591120591)1198631198631 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198632 (514)


1198681198684 = minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0 =

minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913



minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913



minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913


dagger (0)rang0119888119888 +

minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 =

119866119866119953119953119953119953prime0 (120591120591 0) minus1

6ℏ3 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0+

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0

minus1ℏ int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888+

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime


120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0+ 119866119866119953119953119953119953prime

(3) (120591120591) = 119866119866119953119953119953119953prime(3) (120591120591) +

119866119866119953119953119953119953prime(2) (120591120591)1198631198631 + 119866119866119953119953119953119953prime

(1) (120591120591)1198631198632 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198633



follows

119866119866119953119953119953119953prime0 (120591120591) minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0

+1

2ℏ2 1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime


= 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime

(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (1 + 1198631198631 + 1198631198632 + ⋯ )



19


(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (515)

where

119866119866119953119953119953119953prime

(1) (120591120591) = minus1ℏ int 1198891198891205911205911


dagger (0)rang0119888119888

119866119866119953119953119953119953prime(2) (120591120591) = 1

2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 (516)




119866119866119953119953119953119953prime(1) (120591120591) = minus

1ℏ

12 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591 119881119881119895119895119895119895prime(119948119948)119876119876119948119948119895119895(1205911205911)119876119876119948119948119895119895prime

dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


= minus12ℏ

1198891198891205911205911120573120573ℏ

0lang119879119879120591120591

ℏ119881119881119895119895119895119895prime(119948119948)2120596120596119948119948119895119895120596120596119948119948119895119895prime


dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948

+ ℏ119881119881119895119895119895119895prime

lowast (119948119948)



3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948





11989211989211 = minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886minus119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)rang0119888119888

= minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886minus119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)119886119886119948119948119895119895(1205911205911)rang0

= minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ



119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


20


11989211989211 =

⎩⎪⎨


4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582


2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)

4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582


2lt 120582120582 le 3119903119903

0 else

(517)


11989211989212 = minus14


4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886119948119948119895119895prime(1205911205911)119886119886119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)rang0119888119888

= minus14


4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886119954119954prime120582120582prime


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0

ie

11989211989212 =

⎩⎪⎨

⎪⎧



0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903



0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903

2lt 120582120582 le 3119903119903

0 else

(518)


can be written as

119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591) =

⎩⎪⎨

⎪⎧minus119881119881

120582120582120582120582prime(minus119954119954)

2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ

0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le3119903119903

2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582

(119954119954)

2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ

0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le3119903119903

2lt 120582120582 le 3119903119903

0 else

(519)


we have

1198891198891205911205911120573120573ℏ

01198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0) = 1198891198891205911205911120573120573ℏ

0

1120573120573ℏ

119890119890minus119894119894120596120596119899119899(120591120591minus1205911205911)1198661198661199541199541205821205820 (119894119894120596120596119899119899)

119899119899

1120573120573ℏ


119899119899prime

=1

(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591 1198891198891205911205911120573120573ℏ

0119890119890119894119894120596120596119899119899minus120596120596119899119899prime1205911205911 119866119866119954119954120582120582


119899119899119899119899prime=

=1


0 (119894119894120596120596119899119899prime)119899119899119899119899prime

=1120573120573ℏ


0 (119894119894120596120596119899119899)119899119899


21

119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591)

119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =

⎩⎪⎨



0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582 le 31199031199032

lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)


0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582prime le 31199031199032

lt 120582120582 le 3119903119903

0 else

(520)




Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =

⎩⎪⎨


2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)


2lt 120582120582 le 3119903119903

0 else

(521)



119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =

18ℏ2 1198891198891205911205911

120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0lang119879119879120591120591

⎣⎢⎢⎡ 11988111988111989511989511198951198951prime(119948119948120783120783)1198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951prime

dagger (1205911205911)3119903119903

1198951198951prime=31199031199032 +1

31199031199032

1198951198951=1119948119948120783120783


dagger (1205911205911)3119903119903

1198941198941prime=31199031199032 +1

31199031199032

1198941198941=1119948119948120783120783 ⎦⎥⎥⎤

⎣⎢⎢⎡ 11988111988111989511989521198951198952prime(119948119948120784120784)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime

dagger (1205911205912)3119903119903

1198951198952prime=31199031199032 +1

31199031199032

1198951198952=1119948119948120784120784


dagger (1205911205912)3119903119903

1198941198942prime=31199031199032 +1

31199031199032

1198941198942=1119948119948120784120784 ⎦⎥⎥⎤119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime

dagger (0)rang0119888119888 = 1198921198921 + 1198921198922 + 1198921198923 + 1198921198924

where

1198921198921 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


31199031199032 lt1198951198952

primele311990311990311994811994812078412078411989511989521198951198952prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime



(522)

1198921198922 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


1le1198941198942le31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime



(523)

1198921198923 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


1le1198951198952le31199031199032 lt1198951198952

primele311990311990311994811994812078412078411989511989521198951198952prime

1le1198941198941le31199031199032 lt1198941198941

primele311990311990311994811994812078312078311989411989411198941198941prime



(524)

22

1198921198924 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


lowast (119948119948120784120784)1le1198941198942le

31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198941198941le31199031199032 lt1198941198941

primele311990311990311994811994812078312078311989411989411198941198941prime



(525)



dagger we can


1198921198922 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)119881119881

11989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime

times 119868119868 (526)

Where










dagger (0)rang0c










⎩⎪⎨

⎪⎧ 1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951prime


dagger (0)rang0119888119888







(527)


23












0 (1205911205912 1205911205912)12057512057511989411989421198941198942prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582

+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime



0 (1205911205911 1205911205911)12057512057511989511989511198951198951prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582


0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582

= 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime



0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582




























24






























0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 11986611986611994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime




0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime




11989211989221 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198681198681 =

132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951


1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 + 1

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951prime

0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙


0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 =

⎩⎪⎨

⎪⎧

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912) ∙ 119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205912 1205911205911) ∙ 119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582

0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

0 else

and

25

11989211989224 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198681198684 =

132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903




32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903




0 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 =

⎩⎪⎨

⎪⎧120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime0 (1205911205912 0)119866119866119954119954120582120582

0 (120591120591 1205911205911) 120582120582 120582120582prime isin 1 31199031199032

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

0 else


(314)]



1198921198922 =

⎩⎪⎨

⎪⎧

120575120575119954119954119954119954prime

16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

120575120575119954119954119954119954prime

16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582

0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

0 else

(528)



calculated as

119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) = 41198921198922 =

⎩⎪⎨

⎪⎧120575120575119954119954119954119954prime

4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1


0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

120575120575119954119954119954119954prime

4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 31199031199032

+ 13119903119903

0 else

(529)



119866119866119954119954120582120582119954119954prime120582120582prime


(2) (119894119894120596120596119899119899)119899119899


(2) (120591120591)120573120573ℏ0

(530)


26

119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =

120575120575119954119954119954119954prime4

1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954120582120582prime

0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912)

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0

1120573120573ℏ


1198991198992

∙1120573120573ℏ

119890119890minus11989411989412059612059611989911989911205911205911119866119866119954119954120582120582prime0 1198941198941205961205961198991198991

1198991198991

∙1120573120573ℏ

119890119890minus119894119894120596120596119899119899(120591120591minus1205911205912)1198661198661199541199541205821205820 (119894119894120596120596119899119899)

119899119899

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1120573120573ℏ

119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954120582120582prime

0 11989411989412059612059611989911989911198661198661199541199541198951198951prime0 1198941198941205961205961198991198992

11989911989911989911989911198991198992⎩⎪⎨

⎪⎧ 1120573120573ℏ

1198891198891205911205911120573120573ℏ

0119890119890minus1198941198941205961205961198991198991minus12059612059611989911989921205911205911

times1120573120573ℏ

1198891198891205911205912120573120573ℏ

0119890119890minus1198941198941205961205961198991198992minus1205961205961198991198991205911205912

⎭⎪⎬

⎪⎫

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1120573120573ℏ

119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)

11989911989911989911989911198991198992

119866119866119954119954120582120582prime0 11989411989412059612059611989911989911198661198661199541199541198951198951prime

0 1198941198941205961205961198991198992120575120575119899119899111989911989921205751205751198991198991198991198992

=120575120575119954119954119954119954prime120573120573ℏ

119890119890minus119894119894120596120596119899119899120591120591

119899119899

1198661198661199541199541205821205820 (119894119894120596120596119899119899)

14

119881119881120582120582prime1198951198951prime(119954119954



3119903119903

1198951198951prime=31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899)119866119866119954119954120582120582prime

0 (119894119894120596120596119899119899)


119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582

0 (119894119894120596120596119899119899) 120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 119866119866119954119954120582120582prime

0 (119894119894120596120596119899119899) (531)



120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) (532)

and 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582

0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954120582120582prime(2) (119894119894120596120596119899119899) ∙ 119866119866119954119954120582120582prime


+ 13119903119903 we

have

Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) (533)


Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =

⎩⎪⎪⎨

⎪⎪⎧120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903

2

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

0 else

(534)

27



(1) (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) + ⋯

= 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582


0 (119894119894120596120596119899119899) + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime

(2) (119894119894120596120596119899119899)

∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) + ⋯ = 119866119866119954119954120582120582


0 (119894119894120596120596119899119899)

or





(2) (119894119894120596120596119899119899) + ⋯



Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) =

⎩⎪⎪⎪⎪⎨


120575120575119954119954119954119954prime

2

119881119881120582120582120582120582primelowast (119954119954)

120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903

minus120575120575119954119954119954119954prime

2

119881119881120582120582prime120582120582(119954119954)

120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582prime le 3119903119903

2lt 120582120582 le 3119903119903

120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903

2

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

(536)









Σ119954119954120582120582119903119903 (120596120596) via [13]

Γ119954119954120582120582(120596120596) = minus2ImΣ119954119954120582120582119954119954120582120582119903119903 (120596120596) (539)


we have

28

Σ119954119954120582120582119954119954120582120582119903119903 (119894119894120596120596119899119899) =

⎩⎪⎨

⎪⎧14sum

1198811198811205821205821198951198951prime(119954119954)

2

1205961205961199541199541198951198951prime120596120596119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

1

120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894

120582120582 isin 1 31199031199032

14sum 1198811198811198951198951120582120582(119954119954)2

1205961205961199541199541198951198951120596120596119954119954120582120582

311990311990321198951198951=1

1120596120596minus1205961205961199541199541198951198951+119894119894119894119894

120582120582 isin 31199031199032

+ 13119903119903

(540)

Using the relation

Im 1

120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894



Γ119954119954120582120582(120596120596) =

⎩⎪⎨

⎪⎧1205871205872sum

1198811198811205821205821198951198951prime(119954119954)

2

1205961205961199541199541198951198951prime120596120596119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

120575120575120596120596 minus 1205961205961199541199541198951198951prime 120582120582 isin 1 31199031199032

1205871205872sum 1198811198811198951198951120582120582(119954119954)2

1205961205961199541199541198951198951120596120596119954119954120582120582

311990311990321198951198951=1

120575120575120596120596 minus 1205961205961199541199541198951198951 120582120582 isin 31199031199032

+ 13119903119903

(542)

Or equivalently

Γ119954119954120582120582(119864119864 = ℏ120596120596) =

⎩⎪⎨

⎪⎧120587120587ℏ3

2sum

1198811198811205821205821198951198951prime(119954119954)

2

1198641198641199541199541198951198951prime119864119864119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

120575120575119864119864 minus 1198641198641199541199541198951198951prime 120582120582 isin 1 31199031199032

120587120587ℏ3

2sum 1198811198811198951198951120582120582(119954119954)2

1198641198641199541199541198951198951119864119864119954119954120582120582

311990311990321198951198951=1

120575120575119864119864 minus 1198641198641199541199541198951198951 120582120582 isin 31199031199032

+ 13119903119903

(543)








int(⋯ )120588120588119864119864119954119954d119864119864119954119954 giving

Γ119954119954120582120582(119864119864) = 120587120587ℏ3


2119864119864119954119954119895119895prime=119864119864119954119954prime


2120588120588(119864119864)

119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864

119864119864119864119864119954119954120582120582 (544)







29


Γtot = 1119885119885119890119890minus120573120573119864119864119954119954120582120582Γ119954119954120582120582119864119864119954119954120582120582

119954119954120582120582

=120587120587ℏ3

2

119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582 119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864119954119954120582120582

1198641198641199541199541205821205822119954119954120582120582

=120587120587ℏ3

2

119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582 119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582

Finally we get

Γtot = Γ119954119954120582120582(119864119864) =120587120587ℏ3

2sum 119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582 (545)





to make



References


MainText


SI

1 Methodology

11 Model system












11 Basic notations

12 Dynamical matrix


















5



120625120625(119954119954) = 12062512062511(119954119954) 12062512062512(119954119954)12062512062521(119954119954) 12062512062522(119954119954) (1)




1199321199321dagger(119954119954)12062512062511(119954119954)1199321199321(119954119954) and 12062512062522(119954119954) = 1199321199322




matrix of the form

119932119932(119954119954) = 1199321199321(119954119954) 120782120782120782120782 1199321199322(119954119954) (2)


119932119932dagger(119954119954)120625120625(119954119954)119932119932(119954119954) = 12062512062511(119954119954) 12062512062512(119954119954)12062512062521(119954119954) 12062512062522(119954119954)

(3)

where 12062512062512(119954119954) = 1199321199321dagger(119954119954)12062512062512(119954119954)1199321199322(119954119954) and 12062512062521(119954119954) = 12062512062512






derivation)

Γtot = 120587120587ℏ3

2sum 119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582 (4)



2 is the coupling







6
























7














bulk value












8



























ASSOCIATED CONTENT




9


conductivity

AUTHOR INFORMATION



Notes


ACKNOWLEDGMENTS










MP1303

10

References


11


S1










1 Methodology





S2

1 Methodology

11 Model system












S3



















S4




120581120581CP = 119876119860119860Δ119879119879Δ119911119911

(S1)

















S1(b)]







S5















part119864119864t120597120597120597120597

= 119860119860119877119877119879119879bot(119905119905) minus 119879119879top(119905119905) (S3)






S6














119871119871 = |11989911989911199381199381 + 11989911989921199381199382| = 11988611988611989911989912 + 11989911989922 + 11989911989911198991198992 = |1198991198991minus1198991198992|1198861198862 sin(1205791205792) (S4)




2120587120587120575120575119894119894119895119895 such that

1199181199181 = 4120587120587radic3119871119871

radic32

minus12 1199181199182 = 4120587120587

radic3119871119871(01) (S5)

S7





120579120579 (deg) 119860119860 (nm2) 1198991198991 1198991198992 1198981198981 1198981198982

0 60383688 24 0 24 0

0696407 709613169 48 47 47 48

1121311 273718420 30 29 29 30

2000628 85648391 17 16 16 17

3006558 152322047 23 21 21 23

4048894 189013524 26 23 23 26

5085849 53255058 14 12 12 14

7926470 49376245 14 11 11 14

9998709 86277388 19 14 14 19

15178179 31554670 15 4 4 15

19932013 42876612 15 8 8 15

25039660 13942761 9 4 4 9

30158276 18974735 11 4 4 11

32204228 21805221 12 4 4 12









119868119868 = 1119881119881119887119887int 119891119891(119954119954)BZ d119954119954 asymp 1

119873119873sum 119908119908119894119894119891119891(119954119954119894119894)119894119894 (S6)

S8






⎩⎪⎨

⎪⎧ 1199541199541 = 1

9 1205971205973 1199541199542 = 2

9 1205971205973 1199541199543 = 1

18 512059712059718

1199541199544 = 518

512059712059718 1199541199545 = 1

9 21205971205979 1199541199546 = 2

9 21205971205979

1199541199547 = 118

312059712059718 1199541199548 = 1

9 1205971205979 1199541199549 = 1

18 12059712059718

(S7)


119908119908124689 = 112

119908119908357 = 16 (S8)


Γtot = 120587120587ℏ3

2sum 119908119908119896119896 sum

119890119890minus120573120573119864119864119954119954119948119948120582120582

119885119885

120588120588119864119864119954119954119948119948120582120582119881119881120582120582120582120582+31199031199032(119954119954119948119948)

2

1198641198641199541199541199481199481205821205822120582120582

9119896119896=1 (S9)





S9


















contribution [10]

S10



S11


S12


1









119955119955119899119899119899119899(119905119905) = 119929119929119899119899 + 119929119929s0 + 119958119958119899119899119899119899(119905119905) (11)



ℒ = sum sum 119872119872119904119904|119955119899119899119904119904|2(119905119905)2

119903119903119899119899=1

119873119873119899119899=1 minus 119881119881 (12)




ℒ = sum sum 119872119872119904119904|119958119899119899119904119904|2(119905119905)2

119903119903119899119899=1

119873119873119899119899=1 minus 1

2sum sum sum 120601120601120572120572120572120572prime












12 Dynamical matrix


equation as follows

2








form

119906119906119899119899119899119899120572120572(119905119905) = 1119872119872119904119904

119890119890119899119899120572120572119890119890119894119894(119954119954∙119929119929119899119899minus120596120596119905119905) (17)




where

119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = sum 1

119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime



119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime 119890119890

minus119894119894119954119954∙119929119929119897119897119897119897 (19)















120596120596120582120582(minus119954119954) = 120596120596120582120582(119954119954) (112)


sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582

prime119899119899120572120572 = 120575120575120582120582120582120582prime (113)





3





119890119890119899119899120572120572120582120582 (minus119954119954)lowast

= 119890119890119899119899120572120572120582120582 (119954119954) (116)


119906119906119899119899119899119899120572120572(119905119905) = sum 119862119862120582120582(119954119954)119873119873119872119872119904119904

119890119890119899119899120572120572120582120582 (119954119954)119890119890minus119894119894120596120596119954119954120582120582119905119905119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 = sum 119876119876120582120582(119954119954119905119905)119873119873119872119872119904119904

119890119890119899119899120572120572120582120582 (119954119954)119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 (21)




119876119876120582120582(119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)lowast

= 119876119876120582120582(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (minus119954119954) (22)



[119876119876120582120582(119954119954 119905119905)]lowast = 119876119876120582120582(minus119954119954 119905119905) (23)



119879119879 = 12sum 1198721198721198991198991199061198991198991198991198991205721205722 (119905119905)119899119899119899119899120572120572 = 1

2sum 119872119872119904119904





12sum 119876120582120582(119954119954 119905119905)119876120582120582prime

lowast (119954119954 119905119905) sum 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime(119954119954)

lowast119899119899120572120572 =119954119954120582120582120582120582prime

12sum 119876120582120582(119954119954 119905119905)119876120582120582lowast(119954119954 119905119905)119954119954120582120582 =

12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582

ie

119879119879 = 12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582 (24)


1119873119873sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899 = 120575120575119954119954+119954119954prime120782120782 (25)

4





12120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899+119954119954prime∙119929119929119899119899prime

119873119873119872119872119899119899119872119872119899119899prime



=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime


119873119873119872119872119899119899119872119872119899119899prime



=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954prime∙119929119929119897119897


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899

119899119899=120575120575119902119902+119902119902prime=0


=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954119929119929119897119897



=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)

119899119899120572120572



=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)

119899119899120572120572


119954119954120582120582120582120582prime


121205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)

119954119954120582120582

ie

119880119880 = 12sum 1205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)119954119954120582120582 (26)




2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)119954119954120582120582 (27)


119867119867 = 119879119879 + 119881119881 = 12sum [119875119875120582120582(minus119954119954 119905119905)119875119875120582120582(119954119954 119905119905) + 120596120596120582120582

2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)]119954119954120582120582 (28)


119867119867 = 12sum 119875119875minus119954119954120582120582119875119875119954119954120582120582 + 120596120596119954119954120582120582

2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582 (29)






119876119876119954119954120582120582 = ℏ

2120596120596119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582

dagger 119875119875119954119954120582120582 = 119894119894ℏ120596120596119954119954120582120582

2119887119887119954119954120582120582




5

119887119887119954119954120582120582 119887119887119954119954prime120582120582prime



dagger = 0 (212)


119867119867 =12119875119875minus119954119954120582120582119875119875119954119954120582120582 +120596120596119954119954120582120582

2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582

=12minus

ℏ120596120596119954119954120582120582

2119887119887minus119954119954120582120582


2 ℏ2120596120596119954119954120582120582


dagger 119954119954120582120582

=ℏ4120596120596119954119954120582120582minus119887119887minus119954119954120582120582



119954119954120582120582



dagger

=ℏ4120596120596119954119954120582120582119887119887minus119954119954120582120582


dagger + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582

119954119954120582120582

=ℏ41205961205961199541199541205821205822119887119887minus119954119954120582120582


119954119954120582120582

= ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 +

12

119954119954120582120582


119867119867 = sum ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1

2119954119954120582120582 (213)


119906119906119899119899119899119899120572120572 = sum ℏ

2119873119873119872119872119904119904120596120596119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119887119887119954119954120582120582 + 119887119887minus119954119954120582120582

dagger 119954119954120582120582 (214)








119867119867 = 1198671198671 + 1198671198672 + 11986711986712 (31)



119867119867 = 119879119879 + 119881119881 = sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2119899119899119899119899120572120572 + 1

2sum sum sum 120601120601120572120572120572120572prime


6


119904119904 = 1199031199032

+ 1 1199031199032


119879119879 = sum sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2

1199031199032119899119899=1 + sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)

2119903119903119899119899=1199031199032+1

119899119899120572120572 (33)


119881119881 =12 120601120601120572120572120572120572prime


1199031199032

119899119899119899119899prime=1


119903119903

119899119899119899119899prime=1199031199032+1


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime119899119899119899119899prime


1199031199032

119899119899prime=1

119903119903

119899119899=1199031199032+1

= 11988111988111 + 11988111988122 + 11988111988112 + 11988111988121

Or equivalently

⎩⎪⎪⎨

⎪⎪⎧ 11988111988111 = 1

2sum sum sum 120601120601120572120572120572120572prime



11988111988122 = 12sum sum sum 120601120601120572120572120572120572prime





119903119903119899119899prime=1199031199032+1

1199031199032119899119899=1120572120572120572120572prime119899119899119899119899prime



1199031199032119899119899prime=1

119903119903119899119899=1199031199032+1120572120572120572120572prime119899119899119899119899prime

(34)




the following form

119906119906119899119899119899119899120572120572 =

⎩⎨

⎧ sum ℏ

2119873119873119872119872119904119904ω119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger 119954119954120582120582 119904119904 isin 1 1199031199032

sum ℏ

2119873119873119872119872119904119904120596120596119954119954prime120582120582prime119890119899119899120572120572120582120582


2+ 1 119903119903

(35)


dagger )




prime119899119899120572120572 = 120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582

lowast119890119899119899120572120572120582120582

prime119899119899120572120572 =

120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582

prime119899119899120572120572 = sum 119890119890119899119899120572120572120582120582

lowast119890119899119899120572120572120582120582


119876119876119954119954120582120582 equiv ℏ

2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger 119876119876119954119954120582120582 equiv ℏ

2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger (36)


7

119875119875119954119954120582120582 = 119894119894ℏω119954119954120582120582

2119886119886119954119954120582120582



Eq (35) reads

119906119906119899119899119899119899120572120572 = sum 119890119890119904119904119899119899120582120582 (119954119954)

119873119873119872119872119904119904119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1 119903119903

2

sum 119890119904119904119899119899120582120582 (119954119954)119873119873119872119872119904119904

119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1199031199032

+ 1 119903119903 (38)


119879119879 = 1198791198791 + 1198791198792 = 12sum 119875119875119954119954120582120582119875119875minus119954119954120582120582119954119954120582120582 + 1



11988111988111 =12 120601120601120572120572120572120572prime


119899119899119899119899prime


119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime

119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954) 119863119863119899119899120572120572119899119899


120572120572prime

1199031199032

119899119899prime=1

1199031199032

119899119899=1120572120572

=12 ω119954119954120582120582prime


119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime (119954119954)

lowast

1199031199032

119899119899=1

=120572120572

12ω119954119954120582120582

2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582

ie

11988111988111 = 12sum ω119954119954120582120582

2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582 (310)






11988111988122 = 12sum 1205961205961199541199541205821205822 119876119876119954119954120582120582119876119876minus119954119954120582120582119954119954120582120582 (311)


8

11988111988112 =12 120601120601120572120572120572120572prime


119899119899119899119899prime

119903119903

119899119899prime=1199031199032+1

119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime

120582120582prime (119954119954prime)

119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954


lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119890119890119899119899120572120572120582120582 (119954119954)119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954



lowast119881119881119899119899120572120572119899119899


1199031199032119899119899=1120572120572120572120572prime (312)

where


119872119872119904119904119872119872119904119904prime119897119897 120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime (313)




9

11988111988121 =12 120601120601120572120572120572120572prime


119899119899119899119899prime

119903119903

119899119899=1199031199032+1

119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime

120582120582prime (119954119954prime)

119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899prime=1120572120572120572120572prime

=12 120601120601120572120572prime120572120572


119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1


119872119872119899119899prime119872119872119899119899

1119873119873 119890119890


1199031199032

119899119899=1120572120572prime120572120572

=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582


1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582


119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime





119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime


dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954prime 119890119890119899119899120572120572120582120582 (119954119954prime)

lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954prime∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582prime119876119876119954119954120582120582

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954

119890119890119899119899120572120572120582120582 (119954119954)lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger dagger

3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954


lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119890119890119899119899120572120572120582120582 (119954119954)119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

lowast

=

⎩⎨

⎧12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954⎭⎬

⎫dagger

= 11988111988112dagger

Define

1198671198671 = 11987911987911 + 119881119881111198671198672 = 11987911987922 + 1198811198812211986711986712 = 11988111988112 + 11988111988121

(315)


⎩⎪⎨

⎪⎧ 1198671198671 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582

dagger 119886119886119954119954120582120582 + 1231199031199032

120582120582=1119954119954


23119903119903

120582120582prime=1+31199031199032119954119954


dagger3119903119903120582120582prime=31199031199032 +1

31199031199032120582120582=1 + h c 119954119954

(316)




10


119867119867 = 1198671198670 + 119867119867119862119862 (317)

where

1198671198670 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582

dagger 119886119886119954119954120582120582 + 123119903119903

120582120582=1119954119954


dagger3119903119903120582120582prime=31199031199032 +1

31199031199032120582120582=1 + ℎ 119888119888 119954119954

(318)




119886119886119901119901(120591120591) = 119890119890119894119894ℏ119867119867119905119905119886119886119901119901119890119890

minus119894119894ℏ119867119867119905119905 = 119890119890

119867119867120591120591ℏ 119886119886119901119901119890119890

minus119867119867120591120591ℏ (41)



ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119867119867119886119886119953119953(120591120591) (42)


ℏpart119886119886119953119953(120591120591)120597120597120591120591

= 1198671198670119886119886119953119953(120591120591) = 1198671198670 1198901198901198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ = 1198671198670119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ minus 119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ 1198671198670

= 1198901198901198671198670120591120591ℏ 1198671198670119886119886119953119953 minus 1198861198861199531199531198671198670119890119890

minus1198671198670120591120591ℏ = 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ

ie

ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ (43)



2119953119953 119886119886119953119953prime = sum ℏ120596120596119953119953119886119886119953119953


dagger119886119886119953119953119953119953 =




ie


Therefore we have

ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119886119886119953119953(120591120591) (45)

11



dagger = ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 +

12

119953119953

119886119886119953119953primedagger = ℏ120596120596119953119953119886119886119953119953


119953119953

= ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953119886119886119953119953prime


dagger119886119886119953119953119953119953



dagger119886119886119953119953119953119953



dagger 119886119886119953119953dagger119886119886119953119953

119953119953


In summary we have

119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591

119886119886119953119953dagger(120591120591) = 119886119886119953119953

dagger(0)119890119890120596120596119953119953120591120591 = 119886119886119953119953dagger119890119890120596120596119953119953120591120591 (46)






dagger (120591120591prime) (47)



120588120588119867119867 = 119890119890minus120573120573119867119867119885119885119867119867 (48)








= minusTr 120588120588119867119867119890119890119867119867120591120591ℏ 119886119886119953119953119890119890

minus119867119867120591120591ℏ 119890119890

119867119867120591120591primeℏ 119886119886119953119953prime


minus119867119867120591120591primeℏ 119890119890

119867119867120591120591ℏ 119886119886119953119953119890119890


ℏ 119886119886119953119953primedagger




dagger (0)





12






119867119867120591120591ℏ 119886119886119953119953119890119890



ℏ 119886119886119953119953119890119890minus119867119867120591120591ℏ 119890119890

119867119867120591120591primeℏ


119867119867120591120591minus120591120591primeℏ 119886119886119953119953119890119890




dagger (0)








dagger (0)rang

= minusTr 120588120588119867119867119890119890119867119867(120591120591+120573120573ℏ)

ℏ 119886119886119953119953119890119890minus119867119867(120591120591+120573120573ℏ)

ℏ 119886119886119953119953prime = minusTr 120588120588119867119867119890119890120573120573119867119867119890119890

119867119867120591120591ℏ 119886119886119953119953119890119890



1119885119885119867119867


= minus1119885119885119867119867


Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119886119886119953119953(120591120591)

= minus1119885119885119867119867






ℏ 119886119886119953119953119890119890minus119867119867(120591120591minus120573120573ℏ)


119867119867120591120591ℏ 119886119886119953119953119890119890

minus119867119867120591120591ℏ 119890119890120573120573119867119867



= minus1119885119885119867119867


Tr119890119890minus120573120573119867119867119886119886119953119953(120591120591)119886119886119953119953prime(0)

= minus1119885119885119867119867


13



where 120596120596119899119899 = 2120587120587119899119899120573120573ℏ



120573120573ℏ (412)









primelang119886119886119953119953(0)119886119886119953119953prime





primelang119886119886119953119953prime





primelang119886119886119953119953prime




ie

119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =






dagger (0) = 120575120575119953119953119953119953prime


119890119890119860119860119861119861119890119890minus119860119860 = sum 1119899119899119860(119899119899)119861119861infin

119899119899=0 (414)



119891119891(119905119905) = 119891119891(0) + 119905119905119891119891prime(0) + 1199051199052

2119891119891primeprime(0) + ⋯ = sum 119905119905119899119899

119899119899119889119889119899119899119891119889119889119905119905119899119899

119905119905=0

infin119899119899=0 (415)


⎩⎪⎨



⋮119891119891(119899119899)(119905119905) = 119890119890119905119905119860119860119860(119899119899)119861119861119890119890minus119905119905119860119860

(416)

14


119890119890119905119905A119861119861119890119890minus119905119905119860119860 = sum 119905119905119899119899

119899119899119860(119899119899)119861119861infin

119899119899=0 (417)


follows


11198851198851198671198670


11198851198851198671198670


=11198851198851198671198670

Tr (minus120573120573)119899119899

1198991198991198671198670

(119899119899)119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)


dagger (0) we have

1198671198670(119899119899)119886119886119953119953prime


such that


11198851198851198671198670

Tr (minus120573120573)119899119899

1198991198991198671198670

(119899119899)119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)

=11198851198851198671198670

Tr minus120573120573ℏ120596120596119953119953prime

119899119899

119899119899119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)

= 119890119890minus120573120573ℏ120596120596119953119953prime11198851198851198671198670


dagger (0)rang





Hence we have


1119890119890120573120573ℏ120596120596119953119953minus1

equiv 120575120575119953119953prime119953119953119899119899119861119861120596120596119953119953 (419)


we obtain

119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =








119890119890119894119894120596120596119899119899minus120596120596119953119953120573120573ℏ minus

15

1 = minus1+ 1

119890119890120573120573ℏ120596120596119953119953minus1

119894119894120596120596119899119899minus120596120596119953119953119890119890119894119894


119894119894120596120596119899119899minus120596120596119953119953

119890119890minus120573120573ℏ120596120596119953119953minus1

119890119890120573120573ℏ120596120596119953119953minus1= minus 1

119894119894120596120596119899119899minus120596120596119953119953

1minus119890119890120573120573ℏ120596120596119953119953

119890119890120573120573ℏ120596120596119953119953minus1= 1

119894119894120596120596119899119899minus120596120596119953119953

ie

1198661198661199531199530(119894119894120596120596119899119899) == 1119894119894120596120596119899119899minus120596120596119953119953

(421)




119867119867119862119862119868119868 (120591120591) = 1198901198901198671198670120591120591ℏ 119867119867119862119862119890119890

minus1198671198670120591120591ℏ (51)


ℏ part119867119867119862119862119868119868 (120591120591)120597120597120591120591

= 1198671198670119867119867119862119862119868119868 (120591120591) (52)




119867119867120591120591ℏ 119886119886119953119953(0)119890119890minus

119867119867120591120591ℏ 119886119886119953119953prime

dagger (0)

= minusTr 120588120588119867119867119879119879120591120591 119890119890119867119867120591120591ℏ 119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119890119890minus

119867119867120591120591ℏ 119886119886119953119953prime

dagger (0)


where

119886119886119953119953119868119868 (120591120591) equiv 1198901198901198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus

1198671198670120591120591ℏ (53)


119880119880(1205911205911 1205911205912) equiv 11989011989011986711986701205911205911ℏ 119890119890minus

119867119867 (1205911205911minus1205911205912)ℏ 119890119890minus

11986711986701205911205912ℏ (54)


119880119880(1205911205911 1205911205912)119880119880(1205911205912 1205911205913) = 119880119880(1205911205911 1205911205913) (55)


ℏpart119880119880(120591120591 120591120591prime)

120597120597120591120591= 1198671198670119890119890

1198671198670120591120591ℏ 119890119890minus


1198671198670120591120591ℏ 119867119867119890119890minus


1198671198670120591120591primeℏ

= 1198901198901198671198670120591120591ℏ 1198671198670 minus 119867119867119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119890119890minus


1198671198670120591120591primeℏ = 119890119890

1198671198670120591120591ℏ minus119867119867119862119862119890119890

minus1198671198670120591120591ℏ 119880119880(120591120591 120591120591prime)

= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime)

16

ie

ℏ part119880119880120591120591120591120591prime120597120597120591120591

= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime) (56)


119880119880(120591120591 120591120591prime) = summinus1ℏ

119899119899

119899119899 int 1198891198891205911205911120591120591120591120591prime ⋯int 119889119889120591120591119899119899

120591120591120591120591prime 119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119899119899)infin

119899119899=0 (57)


119866119866119953119953119953119953prime(120591120591 0) = minus1119885119885119867119867



dagger (0)

Tr119890119890minus120573120573119867119867


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)=


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)





119866119866119953119953119953119953prime(120591120591 0) = minussum 1

119898119898minus1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)119886119886119953119953119868119868 (120591120591)119886119886


119898119898=0

sum 1119898119898minus

1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0infin

119898119898=0 (58)




120573120573ℏ0 lang119879119879120591120591119867119867119862119862


12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0+⋯

1minus1ℏint 1198891198891205911205911120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)rang0+12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0+⋯

(59)


119863119863119898119898 equiv 1119898119898minus 1

ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0 (510)




17







1198681198682 = minus1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0

= minus1ℏ 1198891198891205911205911120573120573ℏ



0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

= 119866119866119953119953119953119953prime0 (120591120591 0) minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888





(1) (120591120591) equiv

minus1ℏ int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime


1198681198683 = minus1ℏint 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime


(1) (120591120591) (513)


1198681198683 = 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0 = 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)119886119886119953119953(120591120591)rang0 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime


2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 +

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888



follows

18

12ℏ2

1198891198891205911205912120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 +1

2ℏ2 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

=12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

+12 minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0

minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

=12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

+12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

= minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime



1198681198683 = 119866119866119953119953119953119953prime(2) (120591120591) + 119866119866119953119953119953119953prime

(1) (120591120591)1198631198631 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198632 (514)


1198681198684 = minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0 =

minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913



minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913



minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913


dagger (0)rang0119888119888 +

minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 =

119866119866119953119953119953119953prime0 (120591120591 0) minus1

6ℏ3 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0+

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0

minus1ℏ int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888+

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime


120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0+ 119866119866119953119953119953119953prime

(3) (120591120591) = 119866119866119953119953119953119953prime(3) (120591120591) +

119866119866119953119953119953119953prime(2) (120591120591)1198631198631 + 119866119866119953119953119953119953prime

(1) (120591120591)1198631198632 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198633



follows

119866119866119953119953119953119953prime0 (120591120591) minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0

+1

2ℏ2 1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime


= 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime

(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (1 + 1198631198631 + 1198631198632 + ⋯ )



19


(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (515)

where

119866119866119953119953119953119953prime

(1) (120591120591) = minus1ℏ int 1198891198891205911205911


dagger (0)rang0119888119888

119866119866119953119953119953119953prime(2) (120591120591) = 1

2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 (516)




119866119866119953119953119953119953prime(1) (120591120591) = minus

1ℏ

12 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591 119881119881119895119895119895119895prime(119948119948)119876119876119948119948119895119895(1205911205911)119876119876119948119948119895119895prime

dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


= minus12ℏ

1198891198891205911205911120573120573ℏ

0lang119879119879120591120591

ℏ119881119881119895119895119895119895prime(119948119948)2120596120596119948119948119895119895120596120596119948119948119895119895prime


dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948

+ ℏ119881119881119895119895119895119895prime

lowast (119948119948)



3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948





11989211989211 = minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886minus119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)rang0119888119888

= minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886minus119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)119886119886119948119948119895119895(1205911205911)rang0

= minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ



119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


20


11989211989211 =

⎩⎪⎨


4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582


2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)

4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582


2lt 120582120582 le 3119903119903

0 else

(517)


11989211989212 = minus14


4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886119948119948119895119895prime(1205911205911)119886119886119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)rang0119888119888

= minus14


4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886119954119954prime120582120582prime


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0

ie

11989211989212 =

⎩⎪⎨

⎪⎧



0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903



0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903

2lt 120582120582 le 3119903119903

0 else

(518)


can be written as

119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591) =

⎩⎪⎨

⎪⎧minus119881119881

120582120582120582120582prime(minus119954119954)

2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ

0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le3119903119903

2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582

(119954119954)

2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ

0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le3119903119903

2lt 120582120582 le 3119903119903

0 else

(519)


we have

1198891198891205911205911120573120573ℏ

01198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0) = 1198891198891205911205911120573120573ℏ

0

1120573120573ℏ

119890119890minus119894119894120596120596119899119899(120591120591minus1205911205911)1198661198661199541199541205821205820 (119894119894120596120596119899119899)

119899119899

1120573120573ℏ


119899119899prime

=1

(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591 1198891198891205911205911120573120573ℏ

0119890119890119894119894120596120596119899119899minus120596120596119899119899prime1205911205911 119866119866119954119954120582120582


119899119899119899119899prime=

=1


0 (119894119894120596120596119899119899prime)119899119899119899119899prime

=1120573120573ℏ


0 (119894119894120596120596119899119899)119899119899


21

119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591)

119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =

⎩⎪⎨



0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582 le 31199031199032

lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)


0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582prime le 31199031199032

lt 120582120582 le 3119903119903

0 else

(520)




Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =

⎩⎪⎨


2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)


2lt 120582120582 le 3119903119903

0 else

(521)



119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =

18ℏ2 1198891198891205911205911

120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0lang119879119879120591120591

⎣⎢⎢⎡ 11988111988111989511989511198951198951prime(119948119948120783120783)1198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951prime

dagger (1205911205911)3119903119903

1198951198951prime=31199031199032 +1

31199031199032

1198951198951=1119948119948120783120783


dagger (1205911205911)3119903119903

1198941198941prime=31199031199032 +1

31199031199032

1198941198941=1119948119948120783120783 ⎦⎥⎥⎤

⎣⎢⎢⎡ 11988111988111989511989521198951198952prime(119948119948120784120784)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime

dagger (1205911205912)3119903119903

1198951198952prime=31199031199032 +1

31199031199032

1198951198952=1119948119948120784120784


dagger (1205911205912)3119903119903

1198941198942prime=31199031199032 +1

31199031199032

1198941198942=1119948119948120784120784 ⎦⎥⎥⎤119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime

dagger (0)rang0119888119888 = 1198921198921 + 1198921198922 + 1198921198923 + 1198921198924

where

1198921198921 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


31199031199032 lt1198951198952

primele311990311990311994811994812078412078411989511989521198951198952prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime



(522)

1198921198922 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


1le1198941198942le31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime



(523)

1198921198923 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


1le1198951198952le31199031199032 lt1198951198952

primele311990311990311994811994812078412078411989511989521198951198952prime

1le1198941198941le31199031199032 lt1198941198941

primele311990311990311994811994812078312078311989411989411198941198941prime



(524)

22

1198921198924 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


lowast (119948119948120784120784)1le1198941198942le

31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198941198941le31199031199032 lt1198941198941

primele311990311990311994811994812078312078311989411989411198941198941prime



(525)



dagger we can


1198921198922 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)119881119881

11989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime

times 119868119868 (526)

Where










dagger (0)rang0c










⎩⎪⎨

⎪⎧ 1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951prime


dagger (0)rang0119888119888







(527)


23












0 (1205911205912 1205911205912)12057512057511989411989421198941198942prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582

+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime



0 (1205911205911 1205911205911)12057512057511989511989511198951198951prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582


0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582

= 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime



0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582




























24






























0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 11986611986611994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime




0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime




11989211989221 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198681198681 =

132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951


1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 + 1

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951prime

0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙


0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 =

⎩⎪⎨

⎪⎧

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912) ∙ 119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205912 1205911205911) ∙ 119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582

0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

0 else

and

25

11989211989224 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198681198684 =

132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903




32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903




0 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 =

⎩⎪⎨

⎪⎧120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime0 (1205911205912 0)119866119866119954119954120582120582

0 (120591120591 1205911205911) 120582120582 120582120582prime isin 1 31199031199032

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

0 else


(314)]



1198921198922 =

⎩⎪⎨

⎪⎧

120575120575119954119954119954119954prime

16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

120575120575119954119954119954119954prime

16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582

0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

0 else

(528)



calculated as

119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) = 41198921198922 =

⎩⎪⎨

⎪⎧120575120575119954119954119954119954prime

4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1


0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

120575120575119954119954119954119954prime

4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 31199031199032

+ 13119903119903

0 else

(529)



119866119866119954119954120582120582119954119954prime120582120582prime


(2) (119894119894120596120596119899119899)119899119899


(2) (120591120591)120573120573ℏ0

(530)


26

119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =

120575120575119954119954119954119954prime4

1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954120582120582prime

0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912)

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0

1120573120573ℏ


1198991198992

∙1120573120573ℏ

119890119890minus11989411989412059612059611989911989911205911205911119866119866119954119954120582120582prime0 1198941198941205961205961198991198991

1198991198991

∙1120573120573ℏ

119890119890minus119894119894120596120596119899119899(120591120591minus1205911205912)1198661198661199541199541205821205820 (119894119894120596120596119899119899)

119899119899

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1120573120573ℏ

119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954120582120582prime

0 11989411989412059612059611989911989911198661198661199541199541198951198951prime0 1198941198941205961205961198991198992

11989911989911989911989911198991198992⎩⎪⎨

⎪⎧ 1120573120573ℏ

1198891198891205911205911120573120573ℏ

0119890119890minus1198941198941205961205961198991198991minus12059612059611989911989921205911205911

times1120573120573ℏ

1198891198891205911205912120573120573ℏ

0119890119890minus1198941198941205961205961198991198992minus1205961205961198991198991205911205912

⎭⎪⎬

⎪⎫

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1120573120573ℏ

119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)

11989911989911989911989911198991198992

119866119866119954119954120582120582prime0 11989411989412059612059611989911989911198661198661199541199541198951198951prime

0 1198941198941205961205961198991198992120575120575119899119899111989911989921205751205751198991198991198991198992

=120575120575119954119954119954119954prime120573120573ℏ

119890119890minus119894119894120596120596119899119899120591120591

119899119899

1198661198661199541199541205821205820 (119894119894120596120596119899119899)

14

119881119881120582120582prime1198951198951prime(119954119954



3119903119903

1198951198951prime=31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899)119866119866119954119954120582120582prime

0 (119894119894120596120596119899119899)


119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582

0 (119894119894120596120596119899119899) 120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 119866119866119954119954120582120582prime

0 (119894119894120596120596119899119899) (531)



120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) (532)

and 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582

0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954120582120582prime(2) (119894119894120596120596119899119899) ∙ 119866119866119954119954120582120582prime


+ 13119903119903 we

have

Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) (533)


Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =

⎩⎪⎪⎨

⎪⎪⎧120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903

2

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

0 else

(534)

27



(1) (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) + ⋯

= 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582


0 (119894119894120596120596119899119899) + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime

(2) (119894119894120596120596119899119899)

∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) + ⋯ = 119866119866119954119954120582120582


0 (119894119894120596120596119899119899)

or





(2) (119894119894120596120596119899119899) + ⋯



Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) =

⎩⎪⎪⎪⎪⎨


120575120575119954119954119954119954prime

2

119881119881120582120582120582120582primelowast (119954119954)

120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903

minus120575120575119954119954119954119954prime

2

119881119881120582120582prime120582120582(119954119954)

120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582prime le 3119903119903

2lt 120582120582 le 3119903119903

120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903

2

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

(536)









Σ119954119954120582120582119903119903 (120596120596) via [13]

Γ119954119954120582120582(120596120596) = minus2ImΣ119954119954120582120582119954119954120582120582119903119903 (120596120596) (539)


we have

28

Σ119954119954120582120582119954119954120582120582119903119903 (119894119894120596120596119899119899) =

⎩⎪⎨

⎪⎧14sum

1198811198811205821205821198951198951prime(119954119954)

2

1205961205961199541199541198951198951prime120596120596119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

1

120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894

120582120582 isin 1 31199031199032

14sum 1198811198811198951198951120582120582(119954119954)2

1205961205961199541199541198951198951120596120596119954119954120582120582

311990311990321198951198951=1

1120596120596minus1205961205961199541199541198951198951+119894119894119894119894

120582120582 isin 31199031199032

+ 13119903119903

(540)

Using the relation

Im 1

120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894



Γ119954119954120582120582(120596120596) =

⎩⎪⎨

⎪⎧1205871205872sum

1198811198811205821205821198951198951prime(119954119954)

2

1205961205961199541199541198951198951prime120596120596119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

120575120575120596120596 minus 1205961205961199541199541198951198951prime 120582120582 isin 1 31199031199032

1205871205872sum 1198811198811198951198951120582120582(119954119954)2

1205961205961199541199541198951198951120596120596119954119954120582120582

311990311990321198951198951=1

120575120575120596120596 minus 1205961205961199541199541198951198951 120582120582 isin 31199031199032

+ 13119903119903

(542)

Or equivalently

Γ119954119954120582120582(119864119864 = ℏ120596120596) =

⎩⎪⎨

⎪⎧120587120587ℏ3

2sum

1198811198811205821205821198951198951prime(119954119954)

2

1198641198641199541199541198951198951prime119864119864119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

120575120575119864119864 minus 1198641198641199541199541198951198951prime 120582120582 isin 1 31199031199032

120587120587ℏ3

2sum 1198811198811198951198951120582120582(119954119954)2

1198641198641199541199541198951198951119864119864119954119954120582120582

311990311990321198951198951=1

120575120575119864119864 minus 1198641198641199541199541198951198951 120582120582 isin 31199031199032

+ 13119903119903

(543)








int(⋯ )120588120588119864119864119954119954d119864119864119954119954 giving

Γ119954119954120582120582(119864119864) = 120587120587ℏ3


2119864119864119954119954119895119895prime=119864119864119954119954prime


2120588120588(119864119864)

119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864

119864119864119864119864119954119954120582120582 (544)







29


Γtot = 1119885119885119890119890minus120573120573119864119864119954119954120582120582Γ119954119954120582120582119864119864119954119954120582120582

119954119954120582120582

=120587120587ℏ3

2

119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582 119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864119954119954120582120582

1198641198641199541199541205821205822119954119954120582120582

=120587120587ℏ3

2

119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582 119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582

Finally we get

Γtot = Γ119954119954120582120582(119864119864) =120587120587ℏ3

2sum 119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582 (545)





to make



References


MainText


SI

1 Methodology

11 Model system












11 Basic notations

12 Dynamical matrix


















6
























7














bulk value












8



























ASSOCIATED CONTENT




9


conductivity

AUTHOR INFORMATION



Notes


ACKNOWLEDGMENTS










MP1303

10

References


11


S1










1 Methodology





S2

1 Methodology

11 Model system












S3



















S4




120581120581CP = 119876119860119860Δ119879119879Δ119911119911

(S1)

















S1(b)]







S5















part119864119864t120597120597120597120597

= 119860119860119877119877119879119879bot(119905119905) minus 119879119879top(119905119905) (S3)






S6














119871119871 = |11989911989911199381199381 + 11989911989921199381199382| = 11988611988611989911989912 + 11989911989922 + 11989911989911198991198992 = |1198991198991minus1198991198992|1198861198862 sin(1205791205792) (S4)




2120587120587120575120575119894119894119895119895 such that

1199181199181 = 4120587120587radic3119871119871

radic32

minus12 1199181199182 = 4120587120587

radic3119871119871(01) (S5)

S7





120579120579 (deg) 119860119860 (nm2) 1198991198991 1198991198992 1198981198981 1198981198982

0 60383688 24 0 24 0

0696407 709613169 48 47 47 48

1121311 273718420 30 29 29 30

2000628 85648391 17 16 16 17

3006558 152322047 23 21 21 23

4048894 189013524 26 23 23 26

5085849 53255058 14 12 12 14

7926470 49376245 14 11 11 14

9998709 86277388 19 14 14 19

15178179 31554670 15 4 4 15

19932013 42876612 15 8 8 15

25039660 13942761 9 4 4 9

30158276 18974735 11 4 4 11

32204228 21805221 12 4 4 12









119868119868 = 1119881119881119887119887int 119891119891(119954119954)BZ d119954119954 asymp 1

119873119873sum 119908119908119894119894119891119891(119954119954119894119894)119894119894 (S6)

S8






⎩⎪⎨

⎪⎧ 1199541199541 = 1

9 1205971205973 1199541199542 = 2

9 1205971205973 1199541199543 = 1

18 512059712059718

1199541199544 = 518

512059712059718 1199541199545 = 1

9 21205971205979 1199541199546 = 2

9 21205971205979

1199541199547 = 118

312059712059718 1199541199548 = 1

9 1205971205979 1199541199549 = 1

18 12059712059718

(S7)


119908119908124689 = 112

119908119908357 = 16 (S8)


Γtot = 120587120587ℏ3

2sum 119908119908119896119896 sum

119890119890minus120573120573119864119864119954119954119948119948120582120582

119885119885

120588120588119864119864119954119954119948119948120582120582119881119881120582120582120582120582+31199031199032(119954119954119948119948)

2

1198641198641199541199541199481199481205821205822120582120582

9119896119896=1 (S9)





S9


















contribution [10]

S10



S11


S12


1









119955119955119899119899119899119899(119905119905) = 119929119929119899119899 + 119929119929s0 + 119958119958119899119899119899119899(119905119905) (11)



ℒ = sum sum 119872119872119904119904|119955119899119899119904119904|2(119905119905)2

119903119903119899119899=1

119873119873119899119899=1 minus 119881119881 (12)




ℒ = sum sum 119872119872119904119904|119958119899119899119904119904|2(119905119905)2

119903119903119899119899=1

119873119873119899119899=1 minus 1

2sum sum sum 120601120601120572120572120572120572prime












12 Dynamical matrix


equation as follows

2








form

119906119906119899119899119899119899120572120572(119905119905) = 1119872119872119904119904

119890119890119899119899120572120572119890119890119894119894(119954119954∙119929119929119899119899minus120596120596119905119905) (17)




where

119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = sum 1

119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime



119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime 119890119890

minus119894119894119954119954∙119929119929119897119897119897119897 (19)















120596120596120582120582(minus119954119954) = 120596120596120582120582(119954119954) (112)


sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582

prime119899119899120572120572 = 120575120575120582120582120582120582prime (113)





3





119890119890119899119899120572120572120582120582 (minus119954119954)lowast

= 119890119890119899119899120572120572120582120582 (119954119954) (116)


119906119906119899119899119899119899120572120572(119905119905) = sum 119862119862120582120582(119954119954)119873119873119872119872119904119904

119890119890119899119899120572120572120582120582 (119954119954)119890119890minus119894119894120596120596119954119954120582120582119905119905119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 = sum 119876119876120582120582(119954119954119905119905)119873119873119872119872119904119904

119890119890119899119899120572120572120582120582 (119954119954)119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 (21)




119876119876120582120582(119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)lowast

= 119876119876120582120582(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (minus119954119954) (22)



[119876119876120582120582(119954119954 119905119905)]lowast = 119876119876120582120582(minus119954119954 119905119905) (23)



119879119879 = 12sum 1198721198721198991198991199061198991198991198991198991205721205722 (119905119905)119899119899119899119899120572120572 = 1

2sum 119872119872119904119904





12sum 119876120582120582(119954119954 119905119905)119876120582120582prime

lowast (119954119954 119905119905) sum 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime(119954119954)

lowast119899119899120572120572 =119954119954120582120582120582120582prime

12sum 119876120582120582(119954119954 119905119905)119876120582120582lowast(119954119954 119905119905)119954119954120582120582 =

12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582

ie

119879119879 = 12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582 (24)


1119873119873sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899 = 120575120575119954119954+119954119954prime120782120782 (25)

4





12120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899+119954119954prime∙119929119929119899119899prime

119873119873119872119872119899119899119872119872119899119899prime



=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime


119873119873119872119872119899119899119872119872119899119899prime



=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954prime∙119929119929119897119897


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899

119899119899=120575120575119902119902+119902119902prime=0


=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954119929119929119897119897



=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)

119899119899120572120572



=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)

119899119899120572120572


119954119954120582120582120582120582prime


121205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)

119954119954120582120582

ie

119880119880 = 12sum 1205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)119954119954120582120582 (26)




2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)119954119954120582120582 (27)


119867119867 = 119879119879 + 119881119881 = 12sum [119875119875120582120582(minus119954119954 119905119905)119875119875120582120582(119954119954 119905119905) + 120596120596120582120582

2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)]119954119954120582120582 (28)


119867119867 = 12sum 119875119875minus119954119954120582120582119875119875119954119954120582120582 + 120596120596119954119954120582120582

2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582 (29)






119876119876119954119954120582120582 = ℏ

2120596120596119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582

dagger 119875119875119954119954120582120582 = 119894119894ℏ120596120596119954119954120582120582

2119887119887119954119954120582120582




5

119887119887119954119954120582120582 119887119887119954119954prime120582120582prime



dagger = 0 (212)


119867119867 =12119875119875minus119954119954120582120582119875119875119954119954120582120582 +120596120596119954119954120582120582

2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582

=12minus

ℏ120596120596119954119954120582120582

2119887119887minus119954119954120582120582


2 ℏ2120596120596119954119954120582120582


dagger 119954119954120582120582

=ℏ4120596120596119954119954120582120582minus119887119887minus119954119954120582120582



119954119954120582120582



dagger

=ℏ4120596120596119954119954120582120582119887119887minus119954119954120582120582


dagger + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582

119954119954120582120582

=ℏ41205961205961199541199541205821205822119887119887minus119954119954120582120582


119954119954120582120582

= ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 +

12

119954119954120582120582


119867119867 = sum ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1

2119954119954120582120582 (213)


119906119906119899119899119899119899120572120572 = sum ℏ

2119873119873119872119872119904119904120596120596119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119887119887119954119954120582120582 + 119887119887minus119954119954120582120582

dagger 119954119954120582120582 (214)








119867119867 = 1198671198671 + 1198671198672 + 11986711986712 (31)



119867119867 = 119879119879 + 119881119881 = sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2119899119899119899119899120572120572 + 1

2sum sum sum 120601120601120572120572120572120572prime


6


119904119904 = 1199031199032

+ 1 1199031199032


119879119879 = sum sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2

1199031199032119899119899=1 + sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)

2119903119903119899119899=1199031199032+1

119899119899120572120572 (33)


119881119881 =12 120601120601120572120572120572120572prime


1199031199032

119899119899119899119899prime=1


119903119903

119899119899119899119899prime=1199031199032+1


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime119899119899119899119899prime


1199031199032

119899119899prime=1

119903119903

119899119899=1199031199032+1

= 11988111988111 + 11988111988122 + 11988111988112 + 11988111988121

Or equivalently

⎩⎪⎪⎨

⎪⎪⎧ 11988111988111 = 1

2sum sum sum 120601120601120572120572120572120572prime



11988111988122 = 12sum sum sum 120601120601120572120572120572120572prime





119903119903119899119899prime=1199031199032+1

1199031199032119899119899=1120572120572120572120572prime119899119899119899119899prime



1199031199032119899119899prime=1

119903119903119899119899=1199031199032+1120572120572120572120572prime119899119899119899119899prime

(34)




the following form

119906119906119899119899119899119899120572120572 =

⎩⎨

⎧ sum ℏ

2119873119873119872119872119904119904ω119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger 119954119954120582120582 119904119904 isin 1 1199031199032

sum ℏ

2119873119873119872119872119904119904120596120596119954119954prime120582120582prime119890119899119899120572120572120582120582


2+ 1 119903119903

(35)


dagger )




prime119899119899120572120572 = 120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582

lowast119890119899119899120572120572120582120582

prime119899119899120572120572 =

120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582

prime119899119899120572120572 = sum 119890119890119899119899120572120572120582120582

lowast119890119899119899120572120572120582120582


119876119876119954119954120582120582 equiv ℏ

2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger 119876119876119954119954120582120582 equiv ℏ

2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger (36)


7

119875119875119954119954120582120582 = 119894119894ℏω119954119954120582120582

2119886119886119954119954120582120582



Eq (35) reads

119906119906119899119899119899119899120572120572 = sum 119890119890119904119904119899119899120582120582 (119954119954)

119873119873119872119872119904119904119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1 119903119903

2

sum 119890119904119904119899119899120582120582 (119954119954)119873119873119872119872119904119904

119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1199031199032

+ 1 119903119903 (38)


119879119879 = 1198791198791 + 1198791198792 = 12sum 119875119875119954119954120582120582119875119875minus119954119954120582120582119954119954120582120582 + 1



11988111988111 =12 120601120601120572120572120572120572prime


119899119899119899119899prime


119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime

119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954) 119863119863119899119899120572120572119899119899


120572120572prime

1199031199032

119899119899prime=1

1199031199032

119899119899=1120572120572

=12 ω119954119954120582120582prime


119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime (119954119954)

lowast

1199031199032

119899119899=1

=120572120572

12ω119954119954120582120582

2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582

ie

11988111988111 = 12sum ω119954119954120582120582

2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582 (310)






11988111988122 = 12sum 1205961205961199541199541205821205822 119876119876119954119954120582120582119876119876minus119954119954120582120582119954119954120582120582 (311)


8

11988111988112 =12 120601120601120572120572120572120572prime


119899119899119899119899prime

119903119903

119899119899prime=1199031199032+1

119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime

120582120582prime (119954119954prime)

119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954


lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119890119890119899119899120572120572120582120582 (119954119954)119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954



lowast119881119881119899119899120572120572119899119899


1199031199032119899119899=1120572120572120572120572prime (312)

where


119872119872119904119904119872119872119904119904prime119897119897 120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime (313)




9

11988111988121 =12 120601120601120572120572120572120572prime


119899119899119899119899prime

119903119903

119899119899=1199031199032+1

119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime

120582120582prime (119954119954prime)

119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899prime=1120572120572120572120572prime

=12 120601120601120572120572prime120572120572


119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1


119872119872119899119899prime119872119872119899119899

1119873119873 119890119890


1199031199032

119899119899=1120572120572prime120572120572

=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582


1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582


119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime





119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime


dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954prime 119890119890119899119899120572120572120582120582 (119954119954prime)

lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954prime∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582prime119876119876119954119954120582120582

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954

119890119890119899119899120572120572120582120582 (119954119954)lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger dagger

3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954


lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119890119890119899119899120572120572120582120582 (119954119954)119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

lowast

=

⎩⎨

⎧12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954⎭⎬

⎫dagger

= 11988111988112dagger

Define

1198671198671 = 11987911987911 + 119881119881111198671198672 = 11987911987922 + 1198811198812211986711986712 = 11988111988112 + 11988111988121

(315)


⎩⎪⎨

⎪⎧ 1198671198671 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582

dagger 119886119886119954119954120582120582 + 1231199031199032

120582120582=1119954119954


23119903119903

120582120582prime=1+31199031199032119954119954


dagger3119903119903120582120582prime=31199031199032 +1

31199031199032120582120582=1 + h c 119954119954

(316)




10


119867119867 = 1198671198670 + 119867119867119862119862 (317)

where

1198671198670 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582

dagger 119886119886119954119954120582120582 + 123119903119903

120582120582=1119954119954


dagger3119903119903120582120582prime=31199031199032 +1

31199031199032120582120582=1 + ℎ 119888119888 119954119954

(318)




119886119886119901119901(120591120591) = 119890119890119894119894ℏ119867119867119905119905119886119886119901119901119890119890

minus119894119894ℏ119867119867119905119905 = 119890119890

119867119867120591120591ℏ 119886119886119901119901119890119890

minus119867119867120591120591ℏ (41)



ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119867119867119886119886119953119953(120591120591) (42)


ℏpart119886119886119953119953(120591120591)120597120597120591120591

= 1198671198670119886119886119953119953(120591120591) = 1198671198670 1198901198901198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ = 1198671198670119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ minus 119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ 1198671198670

= 1198901198901198671198670120591120591ℏ 1198671198670119886119886119953119953 minus 1198861198861199531199531198671198670119890119890

minus1198671198670120591120591ℏ = 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ

ie

ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ (43)



2119953119953 119886119886119953119953prime = sum ℏ120596120596119953119953119886119886119953119953


dagger119886119886119953119953119953119953 =




ie


Therefore we have

ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119886119886119953119953(120591120591) (45)

11



dagger = ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 +

12

119953119953

119886119886119953119953primedagger = ℏ120596120596119953119953119886119886119953119953


119953119953

= ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953119886119886119953119953prime


dagger119886119886119953119953119953119953



dagger119886119886119953119953119953119953



dagger 119886119886119953119953dagger119886119886119953119953

119953119953


In summary we have

119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591

119886119886119953119953dagger(120591120591) = 119886119886119953119953

dagger(0)119890119890120596120596119953119953120591120591 = 119886119886119953119953dagger119890119890120596120596119953119953120591120591 (46)






dagger (120591120591prime) (47)



120588120588119867119867 = 119890119890minus120573120573119867119867119885119885119867119867 (48)








= minusTr 120588120588119867119867119890119890119867119867120591120591ℏ 119886119886119953119953119890119890

minus119867119867120591120591ℏ 119890119890

119867119867120591120591primeℏ 119886119886119953119953prime


minus119867119867120591120591primeℏ 119890119890

119867119867120591120591ℏ 119886119886119953119953119890119890


ℏ 119886119886119953119953primedagger




dagger (0)





12






119867119867120591120591ℏ 119886119886119953119953119890119890



ℏ 119886119886119953119953119890119890minus119867119867120591120591ℏ 119890119890

119867119867120591120591primeℏ


119867119867120591120591minus120591120591primeℏ 119886119886119953119953119890119890




dagger (0)








dagger (0)rang

= minusTr 120588120588119867119867119890119890119867119867(120591120591+120573120573ℏ)

ℏ 119886119886119953119953119890119890minus119867119867(120591120591+120573120573ℏ)

ℏ 119886119886119953119953prime = minusTr 120588120588119867119867119890119890120573120573119867119867119890119890

119867119867120591120591ℏ 119886119886119953119953119890119890



1119885119885119867119867


= minus1119885119885119867119867


Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119886119886119953119953(120591120591)

= minus1119885119885119867119867






ℏ 119886119886119953119953119890119890minus119867119867(120591120591minus120573120573ℏ)


119867119867120591120591ℏ 119886119886119953119953119890119890

minus119867119867120591120591ℏ 119890119890120573120573119867119867



= minus1119885119885119867119867


Tr119890119890minus120573120573119867119867119886119886119953119953(120591120591)119886119886119953119953prime(0)

= minus1119885119885119867119867


13



where 120596120596119899119899 = 2120587120587119899119899120573120573ℏ



120573120573ℏ (412)









primelang119886119886119953119953(0)119886119886119953119953prime





primelang119886119886119953119953prime





primelang119886119886119953119953prime




ie

119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =






dagger (0) = 120575120575119953119953119953119953prime


119890119890119860119860119861119861119890119890minus119860119860 = sum 1119899119899119860(119899119899)119861119861infin

119899119899=0 (414)



119891119891(119905119905) = 119891119891(0) + 119905119905119891119891prime(0) + 1199051199052

2119891119891primeprime(0) + ⋯ = sum 119905119905119899119899

119899119899119889119889119899119899119891119889119889119905119905119899119899

119905119905=0

infin119899119899=0 (415)


⎩⎪⎨



⋮119891119891(119899119899)(119905119905) = 119890119890119905119905119860119860119860(119899119899)119861119861119890119890minus119905119905119860119860

(416)

14


119890119890119905119905A119861119861119890119890minus119905119905119860119860 = sum 119905119905119899119899

119899119899119860(119899119899)119861119861infin

119899119899=0 (417)


follows


11198851198851198671198670


11198851198851198671198670


=11198851198851198671198670

Tr (minus120573120573)119899119899

1198991198991198671198670

(119899119899)119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)


dagger (0) we have

1198671198670(119899119899)119886119886119953119953prime


such that


11198851198851198671198670

Tr (minus120573120573)119899119899

1198991198991198671198670

(119899119899)119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)

=11198851198851198671198670

Tr minus120573120573ℏ120596120596119953119953prime

119899119899

119899119899119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)

= 119890119890minus120573120573ℏ120596120596119953119953prime11198851198851198671198670


dagger (0)rang





Hence we have


1119890119890120573120573ℏ120596120596119953119953minus1

equiv 120575120575119953119953prime119953119953119899119899119861119861120596120596119953119953 (419)


we obtain

119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =








119890119890119894119894120596120596119899119899minus120596120596119953119953120573120573ℏ minus

15

1 = minus1+ 1

119890119890120573120573ℏ120596120596119953119953minus1

119894119894120596120596119899119899minus120596120596119953119953119890119890119894119894


119894119894120596120596119899119899minus120596120596119953119953

119890119890minus120573120573ℏ120596120596119953119953minus1

119890119890120573120573ℏ120596120596119953119953minus1= minus 1

119894119894120596120596119899119899minus120596120596119953119953

1minus119890119890120573120573ℏ120596120596119953119953

119890119890120573120573ℏ120596120596119953119953minus1= 1

119894119894120596120596119899119899minus120596120596119953119953

ie

1198661198661199531199530(119894119894120596120596119899119899) == 1119894119894120596120596119899119899minus120596120596119953119953

(421)




119867119867119862119862119868119868 (120591120591) = 1198901198901198671198670120591120591ℏ 119867119867119862119862119890119890

minus1198671198670120591120591ℏ (51)


ℏ part119867119867119862119862119868119868 (120591120591)120597120597120591120591

= 1198671198670119867119867119862119862119868119868 (120591120591) (52)




119867119867120591120591ℏ 119886119886119953119953(0)119890119890minus

119867119867120591120591ℏ 119886119886119953119953prime

dagger (0)

= minusTr 120588120588119867119867119879119879120591120591 119890119890119867119867120591120591ℏ 119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119890119890minus

119867119867120591120591ℏ 119886119886119953119953prime

dagger (0)


where

119886119886119953119953119868119868 (120591120591) equiv 1198901198901198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus

1198671198670120591120591ℏ (53)


119880119880(1205911205911 1205911205912) equiv 11989011989011986711986701205911205911ℏ 119890119890minus

119867119867 (1205911205911minus1205911205912)ℏ 119890119890minus

11986711986701205911205912ℏ (54)


119880119880(1205911205911 1205911205912)119880119880(1205911205912 1205911205913) = 119880119880(1205911205911 1205911205913) (55)


ℏpart119880119880(120591120591 120591120591prime)

120597120597120591120591= 1198671198670119890119890

1198671198670120591120591ℏ 119890119890minus


1198671198670120591120591ℏ 119867119867119890119890minus


1198671198670120591120591primeℏ

= 1198901198901198671198670120591120591ℏ 1198671198670 minus 119867119867119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119890119890minus


1198671198670120591120591primeℏ = 119890119890

1198671198670120591120591ℏ minus119867119867119862119862119890119890

minus1198671198670120591120591ℏ 119880119880(120591120591 120591120591prime)

= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime)

16

ie

ℏ part119880119880120591120591120591120591prime120597120597120591120591

= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime) (56)


119880119880(120591120591 120591120591prime) = summinus1ℏ

119899119899

119899119899 int 1198891198891205911205911120591120591120591120591prime ⋯int 119889119889120591120591119899119899

120591120591120591120591prime 119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119899119899)infin

119899119899=0 (57)


119866119866119953119953119953119953prime(120591120591 0) = minus1119885119885119867119867



dagger (0)

Tr119890119890minus120573120573119867119867


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)=


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)





119866119866119953119953119953119953prime(120591120591 0) = minussum 1

119898119898minus1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)119886119886119953119953119868119868 (120591120591)119886119886


119898119898=0

sum 1119898119898minus

1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0infin

119898119898=0 (58)




120573120573ℏ0 lang119879119879120591120591119867119867119862119862


12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0+⋯

1minus1ℏint 1198891198891205911205911120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)rang0+12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0+⋯

(59)


119863119863119898119898 equiv 1119898119898minus 1

ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0 (510)




17







1198681198682 = minus1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0

= minus1ℏ 1198891198891205911205911120573120573ℏ



0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

= 119866119866119953119953119953119953prime0 (120591120591 0) minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888





(1) (120591120591) equiv

minus1ℏ int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime


1198681198683 = minus1ℏint 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime


(1) (120591120591) (513)


1198681198683 = 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0 = 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)119886119886119953119953(120591120591)rang0 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime


2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 +

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888



follows

18

12ℏ2

1198891198891205911205912120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 +1

2ℏ2 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

=12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

+12 minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0

minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

=12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

+12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

= minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime



1198681198683 = 119866119866119953119953119953119953prime(2) (120591120591) + 119866119866119953119953119953119953prime

(1) (120591120591)1198631198631 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198632 (514)


1198681198684 = minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0 =

minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913



minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913



minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913


dagger (0)rang0119888119888 +

minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 =

119866119866119953119953119953119953prime0 (120591120591 0) minus1

6ℏ3 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0+

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0

minus1ℏ int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888+

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime


120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0+ 119866119866119953119953119953119953prime

(3) (120591120591) = 119866119866119953119953119953119953prime(3) (120591120591) +

119866119866119953119953119953119953prime(2) (120591120591)1198631198631 + 119866119866119953119953119953119953prime

(1) (120591120591)1198631198632 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198633



follows

119866119866119953119953119953119953prime0 (120591120591) minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0

+1

2ℏ2 1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime


= 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime

(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (1 + 1198631198631 + 1198631198632 + ⋯ )



19


(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (515)

where

119866119866119953119953119953119953prime

(1) (120591120591) = minus1ℏ int 1198891198891205911205911


dagger (0)rang0119888119888

119866119866119953119953119953119953prime(2) (120591120591) = 1

2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 (516)




119866119866119953119953119953119953prime(1) (120591120591) = minus

1ℏ

12 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591 119881119881119895119895119895119895prime(119948119948)119876119876119948119948119895119895(1205911205911)119876119876119948119948119895119895prime

dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


= minus12ℏ

1198891198891205911205911120573120573ℏ

0lang119879119879120591120591

ℏ119881119881119895119895119895119895prime(119948119948)2120596120596119948119948119895119895120596120596119948119948119895119895prime


dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948

+ ℏ119881119881119895119895119895119895prime

lowast (119948119948)



3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948





11989211989211 = minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886minus119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)rang0119888119888

= minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886minus119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)119886119886119948119948119895119895(1205911205911)rang0

= minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ



119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


20


11989211989211 =

⎩⎪⎨


4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582


2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)

4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582


2lt 120582120582 le 3119903119903

0 else

(517)


11989211989212 = minus14


4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886119948119948119895119895prime(1205911205911)119886119886119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)rang0119888119888

= minus14


4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886119954119954prime120582120582prime


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0

ie

11989211989212 =

⎩⎪⎨

⎪⎧



0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903



0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903

2lt 120582120582 le 3119903119903

0 else

(518)


can be written as

119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591) =

⎩⎪⎨

⎪⎧minus119881119881

120582120582120582120582prime(minus119954119954)

2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ

0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le3119903119903

2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582

(119954119954)

2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ

0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le3119903119903

2lt 120582120582 le 3119903119903

0 else

(519)


we have

1198891198891205911205911120573120573ℏ

01198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0) = 1198891198891205911205911120573120573ℏ

0

1120573120573ℏ

119890119890minus119894119894120596120596119899119899(120591120591minus1205911205911)1198661198661199541199541205821205820 (119894119894120596120596119899119899)

119899119899

1120573120573ℏ


119899119899prime

=1

(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591 1198891198891205911205911120573120573ℏ

0119890119890119894119894120596120596119899119899minus120596120596119899119899prime1205911205911 119866119866119954119954120582120582


119899119899119899119899prime=

=1


0 (119894119894120596120596119899119899prime)119899119899119899119899prime

=1120573120573ℏ


0 (119894119894120596120596119899119899)119899119899


21

119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591)

119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =

⎩⎪⎨



0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582 le 31199031199032

lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)


0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582prime le 31199031199032

lt 120582120582 le 3119903119903

0 else

(520)




Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =

⎩⎪⎨


2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)


2lt 120582120582 le 3119903119903

0 else

(521)



119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =

18ℏ2 1198891198891205911205911

120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0lang119879119879120591120591

⎣⎢⎢⎡ 11988111988111989511989511198951198951prime(119948119948120783120783)1198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951prime

dagger (1205911205911)3119903119903

1198951198951prime=31199031199032 +1

31199031199032

1198951198951=1119948119948120783120783


dagger (1205911205911)3119903119903

1198941198941prime=31199031199032 +1

31199031199032

1198941198941=1119948119948120783120783 ⎦⎥⎥⎤

⎣⎢⎢⎡ 11988111988111989511989521198951198952prime(119948119948120784120784)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime

dagger (1205911205912)3119903119903

1198951198952prime=31199031199032 +1

31199031199032

1198951198952=1119948119948120784120784


dagger (1205911205912)3119903119903

1198941198942prime=31199031199032 +1

31199031199032

1198941198942=1119948119948120784120784 ⎦⎥⎥⎤119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime

dagger (0)rang0119888119888 = 1198921198921 + 1198921198922 + 1198921198923 + 1198921198924

where

1198921198921 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


31199031199032 lt1198951198952

primele311990311990311994811994812078412078411989511989521198951198952prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime



(522)

1198921198922 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


1le1198941198942le31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime



(523)

1198921198923 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


1le1198951198952le31199031199032 lt1198951198952

primele311990311990311994811994812078412078411989511989521198951198952prime

1le1198941198941le31199031199032 lt1198941198941

primele311990311990311994811994812078312078311989411989411198941198941prime



(524)

22

1198921198924 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


lowast (119948119948120784120784)1le1198941198942le

31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198941198941le31199031199032 lt1198941198941

primele311990311990311994811994812078312078311989411989411198941198941prime



(525)



dagger we can


1198921198922 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)119881119881

11989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime

times 119868119868 (526)

Where










dagger (0)rang0c










⎩⎪⎨

⎪⎧ 1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951prime


dagger (0)rang0119888119888







(527)


23












0 (1205911205912 1205911205912)12057512057511989411989421198941198942prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582

+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime



0 (1205911205911 1205911205911)12057512057511989511989511198951198951prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582


0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582

= 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime



0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582




























24






























0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 11986611986611994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime




0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime




11989211989221 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198681198681 =

132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951


1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 + 1

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951prime

0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙


0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 =

⎩⎪⎨

⎪⎧

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912) ∙ 119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205912 1205911205911) ∙ 119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582

0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

0 else

and

25

11989211989224 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198681198684 =

132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903




32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903




0 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 =

⎩⎪⎨

⎪⎧120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime0 (1205911205912 0)119866119866119954119954120582120582

0 (120591120591 1205911205911) 120582120582 120582120582prime isin 1 31199031199032

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

0 else


(314)]



1198921198922 =

⎩⎪⎨

⎪⎧

120575120575119954119954119954119954prime

16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

120575120575119954119954119954119954prime

16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582

0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

0 else

(528)



calculated as

119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) = 41198921198922 =

⎩⎪⎨

⎪⎧120575120575119954119954119954119954prime

4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1


0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

120575120575119954119954119954119954prime

4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 31199031199032

+ 13119903119903

0 else

(529)



119866119866119954119954120582120582119954119954prime120582120582prime


(2) (119894119894120596120596119899119899)119899119899


(2) (120591120591)120573120573ℏ0

(530)


26

119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =

120575120575119954119954119954119954prime4

1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954120582120582prime

0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912)

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0

1120573120573ℏ


1198991198992

∙1120573120573ℏ

119890119890minus11989411989412059612059611989911989911205911205911119866119866119954119954120582120582prime0 1198941198941205961205961198991198991

1198991198991

∙1120573120573ℏ

119890119890minus119894119894120596120596119899119899(120591120591minus1205911205912)1198661198661199541199541205821205820 (119894119894120596120596119899119899)

119899119899

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1120573120573ℏ

119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954120582120582prime

0 11989411989412059612059611989911989911198661198661199541199541198951198951prime0 1198941198941205961205961198991198992

11989911989911989911989911198991198992⎩⎪⎨

⎪⎧ 1120573120573ℏ

1198891198891205911205911120573120573ℏ

0119890119890minus1198941198941205961205961198991198991minus12059612059611989911989921205911205911

times1120573120573ℏ

1198891198891205911205912120573120573ℏ

0119890119890minus1198941198941205961205961198991198992minus1205961205961198991198991205911205912

⎭⎪⎬

⎪⎫

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1120573120573ℏ

119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)

11989911989911989911989911198991198992

119866119866119954119954120582120582prime0 11989411989412059612059611989911989911198661198661199541199541198951198951prime

0 1198941198941205961205961198991198992120575120575119899119899111989911989921205751205751198991198991198991198992

=120575120575119954119954119954119954prime120573120573ℏ

119890119890minus119894119894120596120596119899119899120591120591

119899119899

1198661198661199541199541205821205820 (119894119894120596120596119899119899)

14

119881119881120582120582prime1198951198951prime(119954119954



3119903119903

1198951198951prime=31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899)119866119866119954119954120582120582prime

0 (119894119894120596120596119899119899)


119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582

0 (119894119894120596120596119899119899) 120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 119866119866119954119954120582120582prime

0 (119894119894120596120596119899119899) (531)



120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) (532)

and 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582

0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954120582120582prime(2) (119894119894120596120596119899119899) ∙ 119866119866119954119954120582120582prime


+ 13119903119903 we

have

Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) (533)


Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =

⎩⎪⎪⎨

⎪⎪⎧120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903

2

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

0 else

(534)

27



(1) (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) + ⋯

= 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582


0 (119894119894120596120596119899119899) + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime

(2) (119894119894120596120596119899119899)

∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) + ⋯ = 119866119866119954119954120582120582


0 (119894119894120596120596119899119899)

or





(2) (119894119894120596120596119899119899) + ⋯



Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) =

⎩⎪⎪⎪⎪⎨


120575120575119954119954119954119954prime

2

119881119881120582120582120582120582primelowast (119954119954)

120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903

minus120575120575119954119954119954119954prime

2

119881119881120582120582prime120582120582(119954119954)

120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582prime le 3119903119903

2lt 120582120582 le 3119903119903

120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903

2

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

(536)









Σ119954119954120582120582119903119903 (120596120596) via [13]

Γ119954119954120582120582(120596120596) = minus2ImΣ119954119954120582120582119954119954120582120582119903119903 (120596120596) (539)


we have

28

Σ119954119954120582120582119954119954120582120582119903119903 (119894119894120596120596119899119899) =

⎩⎪⎨

⎪⎧14sum

1198811198811205821205821198951198951prime(119954119954)

2

1205961205961199541199541198951198951prime120596120596119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

1

120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894

120582120582 isin 1 31199031199032

14sum 1198811198811198951198951120582120582(119954119954)2

1205961205961199541199541198951198951120596120596119954119954120582120582

311990311990321198951198951=1

1120596120596minus1205961205961199541199541198951198951+119894119894119894119894

120582120582 isin 31199031199032

+ 13119903119903

(540)

Using the relation

Im 1

120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894



Γ119954119954120582120582(120596120596) =

⎩⎪⎨

⎪⎧1205871205872sum

1198811198811205821205821198951198951prime(119954119954)

2

1205961205961199541199541198951198951prime120596120596119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

120575120575120596120596 minus 1205961205961199541199541198951198951prime 120582120582 isin 1 31199031199032

1205871205872sum 1198811198811198951198951120582120582(119954119954)2

1205961205961199541199541198951198951120596120596119954119954120582120582

311990311990321198951198951=1

120575120575120596120596 minus 1205961205961199541199541198951198951 120582120582 isin 31199031199032

+ 13119903119903

(542)

Or equivalently

Γ119954119954120582120582(119864119864 = ℏ120596120596) =

⎩⎪⎨

⎪⎧120587120587ℏ3

2sum

1198811198811205821205821198951198951prime(119954119954)

2

1198641198641199541199541198951198951prime119864119864119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

120575120575119864119864 minus 1198641198641199541199541198951198951prime 120582120582 isin 1 31199031199032

120587120587ℏ3

2sum 1198811198811198951198951120582120582(119954119954)2

1198641198641199541199541198951198951119864119864119954119954120582120582

311990311990321198951198951=1

120575120575119864119864 minus 1198641198641199541199541198951198951 120582120582 isin 31199031199032

+ 13119903119903

(543)








int(⋯ )120588120588119864119864119954119954d119864119864119954119954 giving

Γ119954119954120582120582(119864119864) = 120587120587ℏ3


2119864119864119954119954119895119895prime=119864119864119954119954prime


2120588120588(119864119864)

119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864

119864119864119864119864119954119954120582120582 (544)







29


Γtot = 1119885119885119890119890minus120573120573119864119864119954119954120582120582Γ119954119954120582120582119864119864119954119954120582120582

119954119954120582120582

=120587120587ℏ3

2

119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582 119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864119954119954120582120582

1198641198641199541199541205821205822119954119954120582120582

=120587120587ℏ3

2

119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582 119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582

Finally we get

Γtot = Γ119954119954120582120582(119864119864) =120587120587ℏ3

2sum 119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582 (545)





to make



References


MainText


SI

1 Methodology

11 Model system












11 Basic notations

12 Dynamical matrix


















7














bulk value












8



























ASSOCIATED CONTENT




9


conductivity

AUTHOR INFORMATION



Notes


ACKNOWLEDGMENTS










MP1303

10

References


11


S1










1 Methodology





S2

1 Methodology

11 Model system












S3



















S4




120581120581CP = 119876119860119860Δ119879119879Δ119911119911

(S1)

















S1(b)]







S5















part119864119864t120597120597120597120597

= 119860119860119877119877119879119879bot(119905119905) minus 119879119879top(119905119905) (S3)






S6














119871119871 = |11989911989911199381199381 + 11989911989921199381199382| = 11988611988611989911989912 + 11989911989922 + 11989911989911198991198992 = |1198991198991minus1198991198992|1198861198862 sin(1205791205792) (S4)




2120587120587120575120575119894119894119895119895 such that

1199181199181 = 4120587120587radic3119871119871

radic32

minus12 1199181199182 = 4120587120587

radic3119871119871(01) (S5)

S7





120579120579 (deg) 119860119860 (nm2) 1198991198991 1198991198992 1198981198981 1198981198982

0 60383688 24 0 24 0

0696407 709613169 48 47 47 48

1121311 273718420 30 29 29 30

2000628 85648391 17 16 16 17

3006558 152322047 23 21 21 23

4048894 189013524 26 23 23 26

5085849 53255058 14 12 12 14

7926470 49376245 14 11 11 14

9998709 86277388 19 14 14 19

15178179 31554670 15 4 4 15

19932013 42876612 15 8 8 15

25039660 13942761 9 4 4 9

30158276 18974735 11 4 4 11

32204228 21805221 12 4 4 12









119868119868 = 1119881119881119887119887int 119891119891(119954119954)BZ d119954119954 asymp 1

119873119873sum 119908119908119894119894119891119891(119954119954119894119894)119894119894 (S6)

S8






⎩⎪⎨

⎪⎧ 1199541199541 = 1

9 1205971205973 1199541199542 = 2

9 1205971205973 1199541199543 = 1

18 512059712059718

1199541199544 = 518

512059712059718 1199541199545 = 1

9 21205971205979 1199541199546 = 2

9 21205971205979

1199541199547 = 118

312059712059718 1199541199548 = 1

9 1205971205979 1199541199549 = 1

18 12059712059718

(S7)


119908119908124689 = 112

119908119908357 = 16 (S8)


Γtot = 120587120587ℏ3

2sum 119908119908119896119896 sum

119890119890minus120573120573119864119864119954119954119948119948120582120582

119885119885

120588120588119864119864119954119954119948119948120582120582119881119881120582120582120582120582+31199031199032(119954119954119948119948)

2

1198641198641199541199541199481199481205821205822120582120582

9119896119896=1 (S9)





S9


















contribution [10]

S10



S11


S12


1









119955119955119899119899119899119899(119905119905) = 119929119929119899119899 + 119929119929s0 + 119958119958119899119899119899119899(119905119905) (11)



ℒ = sum sum 119872119872119904119904|119955119899119899119904119904|2(119905119905)2

119903119903119899119899=1

119873119873119899119899=1 minus 119881119881 (12)




ℒ = sum sum 119872119872119904119904|119958119899119899119904119904|2(119905119905)2

119903119903119899119899=1

119873119873119899119899=1 minus 1

2sum sum sum 120601120601120572120572120572120572prime












12 Dynamical matrix


equation as follows

2








form

119906119906119899119899119899119899120572120572(119905119905) = 1119872119872119904119904

119890119890119899119899120572120572119890119890119894119894(119954119954∙119929119929119899119899minus120596120596119905119905) (17)




where

119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = sum 1

119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime



119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime 119890119890

minus119894119894119954119954∙119929119929119897119897119897119897 (19)















120596120596120582120582(minus119954119954) = 120596120596120582120582(119954119954) (112)


sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582

prime119899119899120572120572 = 120575120575120582120582120582120582prime (113)





3





119890119890119899119899120572120572120582120582 (minus119954119954)lowast

= 119890119890119899119899120572120572120582120582 (119954119954) (116)


119906119906119899119899119899119899120572120572(119905119905) = sum 119862119862120582120582(119954119954)119873119873119872119872119904119904

119890119890119899119899120572120572120582120582 (119954119954)119890119890minus119894119894120596120596119954119954120582120582119905119905119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 = sum 119876119876120582120582(119954119954119905119905)119873119873119872119872119904119904

119890119890119899119899120572120572120582120582 (119954119954)119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 (21)




119876119876120582120582(119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)lowast

= 119876119876120582120582(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (minus119954119954) (22)



[119876119876120582120582(119954119954 119905119905)]lowast = 119876119876120582120582(minus119954119954 119905119905) (23)



119879119879 = 12sum 1198721198721198991198991199061198991198991198991198991205721205722 (119905119905)119899119899119899119899120572120572 = 1

2sum 119872119872119904119904





12sum 119876120582120582(119954119954 119905119905)119876120582120582prime

lowast (119954119954 119905119905) sum 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime(119954119954)

lowast119899119899120572120572 =119954119954120582120582120582120582prime

12sum 119876120582120582(119954119954 119905119905)119876120582120582lowast(119954119954 119905119905)119954119954120582120582 =

12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582

ie

119879119879 = 12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582 (24)


1119873119873sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899 = 120575120575119954119954+119954119954prime120782120782 (25)

4





12120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899+119954119954prime∙119929119929119899119899prime

119873119873119872119872119899119899119872119872119899119899prime



=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime


119873119873119872119872119899119899119872119872119899119899prime



=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954prime∙119929119929119897119897


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899

119899119899=120575120575119902119902+119902119902prime=0


=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954119929119929119897119897



=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)

119899119899120572120572



=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)

119899119899120572120572


119954119954120582120582120582120582prime


121205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)

119954119954120582120582

ie

119880119880 = 12sum 1205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)119954119954120582120582 (26)




2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)119954119954120582120582 (27)


119867119867 = 119879119879 + 119881119881 = 12sum [119875119875120582120582(minus119954119954 119905119905)119875119875120582120582(119954119954 119905119905) + 120596120596120582120582

2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)]119954119954120582120582 (28)


119867119867 = 12sum 119875119875minus119954119954120582120582119875119875119954119954120582120582 + 120596120596119954119954120582120582

2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582 (29)






119876119876119954119954120582120582 = ℏ

2120596120596119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582

dagger 119875119875119954119954120582120582 = 119894119894ℏ120596120596119954119954120582120582

2119887119887119954119954120582120582




5

119887119887119954119954120582120582 119887119887119954119954prime120582120582prime



dagger = 0 (212)


119867119867 =12119875119875minus119954119954120582120582119875119875119954119954120582120582 +120596120596119954119954120582120582

2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582

=12minus

ℏ120596120596119954119954120582120582

2119887119887minus119954119954120582120582


2 ℏ2120596120596119954119954120582120582


dagger 119954119954120582120582

=ℏ4120596120596119954119954120582120582minus119887119887minus119954119954120582120582



119954119954120582120582



dagger

=ℏ4120596120596119954119954120582120582119887119887minus119954119954120582120582


dagger + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582

119954119954120582120582

=ℏ41205961205961199541199541205821205822119887119887minus119954119954120582120582


119954119954120582120582

= ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 +

12

119954119954120582120582


119867119867 = sum ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1

2119954119954120582120582 (213)


119906119906119899119899119899119899120572120572 = sum ℏ

2119873119873119872119872119904119904120596120596119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119887119887119954119954120582120582 + 119887119887minus119954119954120582120582

dagger 119954119954120582120582 (214)








119867119867 = 1198671198671 + 1198671198672 + 11986711986712 (31)



119867119867 = 119879119879 + 119881119881 = sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2119899119899119899119899120572120572 + 1

2sum sum sum 120601120601120572120572120572120572prime


6


119904119904 = 1199031199032

+ 1 1199031199032


119879119879 = sum sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2

1199031199032119899119899=1 + sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)

2119903119903119899119899=1199031199032+1

119899119899120572120572 (33)


119881119881 =12 120601120601120572120572120572120572prime


1199031199032

119899119899119899119899prime=1


119903119903

119899119899119899119899prime=1199031199032+1


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime119899119899119899119899prime


1199031199032

119899119899prime=1

119903119903

119899119899=1199031199032+1

= 11988111988111 + 11988111988122 + 11988111988112 + 11988111988121

Or equivalently

⎩⎪⎪⎨

⎪⎪⎧ 11988111988111 = 1

2sum sum sum 120601120601120572120572120572120572prime



11988111988122 = 12sum sum sum 120601120601120572120572120572120572prime





119903119903119899119899prime=1199031199032+1

1199031199032119899119899=1120572120572120572120572prime119899119899119899119899prime



1199031199032119899119899prime=1

119903119903119899119899=1199031199032+1120572120572120572120572prime119899119899119899119899prime

(34)




the following form

119906119906119899119899119899119899120572120572 =

⎩⎨

⎧ sum ℏ

2119873119873119872119872119904119904ω119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger 119954119954120582120582 119904119904 isin 1 1199031199032

sum ℏ

2119873119873119872119872119904119904120596120596119954119954prime120582120582prime119890119899119899120572120572120582120582


2+ 1 119903119903

(35)


dagger )




prime119899119899120572120572 = 120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582

lowast119890119899119899120572120572120582120582

prime119899119899120572120572 =

120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582

prime119899119899120572120572 = sum 119890119890119899119899120572120572120582120582

lowast119890119899119899120572120572120582120582


119876119876119954119954120582120582 equiv ℏ

2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger 119876119876119954119954120582120582 equiv ℏ

2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger (36)


7

119875119875119954119954120582120582 = 119894119894ℏω119954119954120582120582

2119886119886119954119954120582120582



Eq (35) reads

119906119906119899119899119899119899120572120572 = sum 119890119890119904119904119899119899120582120582 (119954119954)

119873119873119872119872119904119904119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1 119903119903

2

sum 119890119904119904119899119899120582120582 (119954119954)119873119873119872119872119904119904

119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1199031199032

+ 1 119903119903 (38)


119879119879 = 1198791198791 + 1198791198792 = 12sum 119875119875119954119954120582120582119875119875minus119954119954120582120582119954119954120582120582 + 1



11988111988111 =12 120601120601120572120572120572120572prime


119899119899119899119899prime


119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime

119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954) 119863119863119899119899120572120572119899119899


120572120572prime

1199031199032

119899119899prime=1

1199031199032

119899119899=1120572120572

=12 ω119954119954120582120582prime


119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime (119954119954)

lowast

1199031199032

119899119899=1

=120572120572

12ω119954119954120582120582

2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582

ie

11988111988111 = 12sum ω119954119954120582120582

2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582 (310)






11988111988122 = 12sum 1205961205961199541199541205821205822 119876119876119954119954120582120582119876119876minus119954119954120582120582119954119954120582120582 (311)


8

11988111988112 =12 120601120601120572120572120572120572prime


119899119899119899119899prime

119903119903

119899119899prime=1199031199032+1

119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime

120582120582prime (119954119954prime)

119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954


lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119890119890119899119899120572120572120582120582 (119954119954)119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954



lowast119881119881119899119899120572120572119899119899


1199031199032119899119899=1120572120572120572120572prime (312)

where


119872119872119904119904119872119872119904119904prime119897119897 120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime (313)




9

11988111988121 =12 120601120601120572120572120572120572prime


119899119899119899119899prime

119903119903

119899119899=1199031199032+1

119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime

120582120582prime (119954119954prime)

119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899prime=1120572120572120572120572prime

=12 120601120601120572120572prime120572120572


119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1


119872119872119899119899prime119872119872119899119899

1119873119873 119890119890


1199031199032

119899119899=1120572120572prime120572120572

=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582


1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582


119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime





119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime


dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954prime 119890119890119899119899120572120572120582120582 (119954119954prime)

lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954prime∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582prime119876119876119954119954120582120582

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954

119890119890119899119899120572120572120582120582 (119954119954)lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger dagger

3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954


lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119890119890119899119899120572120572120582120582 (119954119954)119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

lowast

=

⎩⎨

⎧12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954⎭⎬

⎫dagger

= 11988111988112dagger

Define

1198671198671 = 11987911987911 + 119881119881111198671198672 = 11987911987922 + 1198811198812211986711986712 = 11988111988112 + 11988111988121

(315)


⎩⎪⎨

⎪⎧ 1198671198671 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582

dagger 119886119886119954119954120582120582 + 1231199031199032

120582120582=1119954119954


23119903119903

120582120582prime=1+31199031199032119954119954


dagger3119903119903120582120582prime=31199031199032 +1

31199031199032120582120582=1 + h c 119954119954

(316)




10


119867119867 = 1198671198670 + 119867119867119862119862 (317)

where

1198671198670 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582

dagger 119886119886119954119954120582120582 + 123119903119903

120582120582=1119954119954


dagger3119903119903120582120582prime=31199031199032 +1

31199031199032120582120582=1 + ℎ 119888119888 119954119954

(318)




119886119886119901119901(120591120591) = 119890119890119894119894ℏ119867119867119905119905119886119886119901119901119890119890

minus119894119894ℏ119867119867119905119905 = 119890119890

119867119867120591120591ℏ 119886119886119901119901119890119890

minus119867119867120591120591ℏ (41)



ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119867119867119886119886119953119953(120591120591) (42)


ℏpart119886119886119953119953(120591120591)120597120597120591120591

= 1198671198670119886119886119953119953(120591120591) = 1198671198670 1198901198901198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ = 1198671198670119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ minus 119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ 1198671198670

= 1198901198901198671198670120591120591ℏ 1198671198670119886119886119953119953 minus 1198861198861199531199531198671198670119890119890

minus1198671198670120591120591ℏ = 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ

ie

ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ (43)



2119953119953 119886119886119953119953prime = sum ℏ120596120596119953119953119886119886119953119953


dagger119886119886119953119953119953119953 =




ie


Therefore we have

ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119886119886119953119953(120591120591) (45)

11



dagger = ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 +

12

119953119953

119886119886119953119953primedagger = ℏ120596120596119953119953119886119886119953119953


119953119953

= ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953119886119886119953119953prime


dagger119886119886119953119953119953119953



dagger119886119886119953119953119953119953



dagger 119886119886119953119953dagger119886119886119953119953

119953119953


In summary we have

119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591

119886119886119953119953dagger(120591120591) = 119886119886119953119953

dagger(0)119890119890120596120596119953119953120591120591 = 119886119886119953119953dagger119890119890120596120596119953119953120591120591 (46)






dagger (120591120591prime) (47)



120588120588119867119867 = 119890119890minus120573120573119867119867119885119885119867119867 (48)








= minusTr 120588120588119867119867119890119890119867119867120591120591ℏ 119886119886119953119953119890119890

minus119867119867120591120591ℏ 119890119890

119867119867120591120591primeℏ 119886119886119953119953prime


minus119867119867120591120591primeℏ 119890119890

119867119867120591120591ℏ 119886119886119953119953119890119890


ℏ 119886119886119953119953primedagger




dagger (0)





12






119867119867120591120591ℏ 119886119886119953119953119890119890



ℏ 119886119886119953119953119890119890minus119867119867120591120591ℏ 119890119890

119867119867120591120591primeℏ


119867119867120591120591minus120591120591primeℏ 119886119886119953119953119890119890




dagger (0)








dagger (0)rang

= minusTr 120588120588119867119867119890119890119867119867(120591120591+120573120573ℏ)

ℏ 119886119886119953119953119890119890minus119867119867(120591120591+120573120573ℏ)

ℏ 119886119886119953119953prime = minusTr 120588120588119867119867119890119890120573120573119867119867119890119890

119867119867120591120591ℏ 119886119886119953119953119890119890



1119885119885119867119867


= minus1119885119885119867119867


Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119886119886119953119953(120591120591)

= minus1119885119885119867119867






ℏ 119886119886119953119953119890119890minus119867119867(120591120591minus120573120573ℏ)


119867119867120591120591ℏ 119886119886119953119953119890119890

minus119867119867120591120591ℏ 119890119890120573120573119867119867



= minus1119885119885119867119867


Tr119890119890minus120573120573119867119867119886119886119953119953(120591120591)119886119886119953119953prime(0)

= minus1119885119885119867119867


13



where 120596120596119899119899 = 2120587120587119899119899120573120573ℏ



120573120573ℏ (412)









primelang119886119886119953119953(0)119886119886119953119953prime





primelang119886119886119953119953prime





primelang119886119886119953119953prime




ie

119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =






dagger (0) = 120575120575119953119953119953119953prime


119890119890119860119860119861119861119890119890minus119860119860 = sum 1119899119899119860(119899119899)119861119861infin

119899119899=0 (414)



119891119891(119905119905) = 119891119891(0) + 119905119905119891119891prime(0) + 1199051199052

2119891119891primeprime(0) + ⋯ = sum 119905119905119899119899

119899119899119889119889119899119899119891119889119889119905119905119899119899

119905119905=0

infin119899119899=0 (415)


⎩⎪⎨



⋮119891119891(119899119899)(119905119905) = 119890119890119905119905119860119860119860(119899119899)119861119861119890119890minus119905119905119860119860

(416)

14


119890119890119905119905A119861119861119890119890minus119905119905119860119860 = sum 119905119905119899119899

119899119899119860(119899119899)119861119861infin

119899119899=0 (417)


follows


11198851198851198671198670


11198851198851198671198670


=11198851198851198671198670

Tr (minus120573120573)119899119899

1198991198991198671198670

(119899119899)119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)


dagger (0) we have

1198671198670(119899119899)119886119886119953119953prime


such that


11198851198851198671198670

Tr (minus120573120573)119899119899

1198991198991198671198670

(119899119899)119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)

=11198851198851198671198670

Tr minus120573120573ℏ120596120596119953119953prime

119899119899

119899119899119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)

= 119890119890minus120573120573ℏ120596120596119953119953prime11198851198851198671198670


dagger (0)rang





Hence we have


1119890119890120573120573ℏ120596120596119953119953minus1

equiv 120575120575119953119953prime119953119953119899119899119861119861120596120596119953119953 (419)


we obtain

119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =








119890119890119894119894120596120596119899119899minus120596120596119953119953120573120573ℏ minus

15

1 = minus1+ 1

119890119890120573120573ℏ120596120596119953119953minus1

119894119894120596120596119899119899minus120596120596119953119953119890119890119894119894


119894119894120596120596119899119899minus120596120596119953119953

119890119890minus120573120573ℏ120596120596119953119953minus1

119890119890120573120573ℏ120596120596119953119953minus1= minus 1

119894119894120596120596119899119899minus120596120596119953119953

1minus119890119890120573120573ℏ120596120596119953119953

119890119890120573120573ℏ120596120596119953119953minus1= 1

119894119894120596120596119899119899minus120596120596119953119953

ie

1198661198661199531199530(119894119894120596120596119899119899) == 1119894119894120596120596119899119899minus120596120596119953119953

(421)




119867119867119862119862119868119868 (120591120591) = 1198901198901198671198670120591120591ℏ 119867119867119862119862119890119890

minus1198671198670120591120591ℏ (51)


ℏ part119867119867119862119862119868119868 (120591120591)120597120597120591120591

= 1198671198670119867119867119862119862119868119868 (120591120591) (52)




119867119867120591120591ℏ 119886119886119953119953(0)119890119890minus

119867119867120591120591ℏ 119886119886119953119953prime

dagger (0)

= minusTr 120588120588119867119867119879119879120591120591 119890119890119867119867120591120591ℏ 119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119890119890minus

119867119867120591120591ℏ 119886119886119953119953prime

dagger (0)


where

119886119886119953119953119868119868 (120591120591) equiv 1198901198901198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus

1198671198670120591120591ℏ (53)


119880119880(1205911205911 1205911205912) equiv 11989011989011986711986701205911205911ℏ 119890119890minus

119867119867 (1205911205911minus1205911205912)ℏ 119890119890minus

11986711986701205911205912ℏ (54)


119880119880(1205911205911 1205911205912)119880119880(1205911205912 1205911205913) = 119880119880(1205911205911 1205911205913) (55)


ℏpart119880119880(120591120591 120591120591prime)

120597120597120591120591= 1198671198670119890119890

1198671198670120591120591ℏ 119890119890minus


1198671198670120591120591ℏ 119867119867119890119890minus


1198671198670120591120591primeℏ

= 1198901198901198671198670120591120591ℏ 1198671198670 minus 119867119867119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119890119890minus


1198671198670120591120591primeℏ = 119890119890

1198671198670120591120591ℏ minus119867119867119862119862119890119890

minus1198671198670120591120591ℏ 119880119880(120591120591 120591120591prime)

= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime)

16

ie

ℏ part119880119880120591120591120591120591prime120597120597120591120591

= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime) (56)


119880119880(120591120591 120591120591prime) = summinus1ℏ

119899119899

119899119899 int 1198891198891205911205911120591120591120591120591prime ⋯int 119889119889120591120591119899119899

120591120591120591120591prime 119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119899119899)infin

119899119899=0 (57)


119866119866119953119953119953119953prime(120591120591 0) = minus1119885119885119867119867



dagger (0)

Tr119890119890minus120573120573119867119867


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)=


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)





119866119866119953119953119953119953prime(120591120591 0) = minussum 1

119898119898minus1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)119886119886119953119953119868119868 (120591120591)119886119886


119898119898=0

sum 1119898119898minus

1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0infin

119898119898=0 (58)




120573120573ℏ0 lang119879119879120591120591119867119867119862119862


12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0+⋯

1minus1ℏint 1198891198891205911205911120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)rang0+12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0+⋯

(59)


119863119863119898119898 equiv 1119898119898minus 1

ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0 (510)




17







1198681198682 = minus1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0

= minus1ℏ 1198891198891205911205911120573120573ℏ



0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

= 119866119866119953119953119953119953prime0 (120591120591 0) minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888





(1) (120591120591) equiv

minus1ℏ int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime


1198681198683 = minus1ℏint 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime


(1) (120591120591) (513)


1198681198683 = 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0 = 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)119886119886119953119953(120591120591)rang0 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime


2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 +

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888



follows

18

12ℏ2

1198891198891205911205912120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 +1

2ℏ2 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

=12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

+12 minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0

minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

=12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

+12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

= minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime



1198681198683 = 119866119866119953119953119953119953prime(2) (120591120591) + 119866119866119953119953119953119953prime

(1) (120591120591)1198631198631 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198632 (514)


1198681198684 = minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0 =

minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913



minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913



minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913


dagger (0)rang0119888119888 +

minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 =

119866119866119953119953119953119953prime0 (120591120591 0) minus1

6ℏ3 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0+

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0

minus1ℏ int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888+

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime


120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0+ 119866119866119953119953119953119953prime

(3) (120591120591) = 119866119866119953119953119953119953prime(3) (120591120591) +

119866119866119953119953119953119953prime(2) (120591120591)1198631198631 + 119866119866119953119953119953119953prime

(1) (120591120591)1198631198632 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198633



follows

119866119866119953119953119953119953prime0 (120591120591) minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0

+1

2ℏ2 1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime


= 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime

(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (1 + 1198631198631 + 1198631198632 + ⋯ )



19


(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (515)

where

119866119866119953119953119953119953prime

(1) (120591120591) = minus1ℏ int 1198891198891205911205911


dagger (0)rang0119888119888

119866119866119953119953119953119953prime(2) (120591120591) = 1

2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 (516)




119866119866119953119953119953119953prime(1) (120591120591) = minus

1ℏ

12 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591 119881119881119895119895119895119895prime(119948119948)119876119876119948119948119895119895(1205911205911)119876119876119948119948119895119895prime

dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


= minus12ℏ

1198891198891205911205911120573120573ℏ

0lang119879119879120591120591

ℏ119881119881119895119895119895119895prime(119948119948)2120596120596119948119948119895119895120596120596119948119948119895119895prime


dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948

+ ℏ119881119881119895119895119895119895prime

lowast (119948119948)



3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948





11989211989211 = minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886minus119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)rang0119888119888

= minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886minus119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)119886119886119948119948119895119895(1205911205911)rang0

= minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ



119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


20


11989211989211 =

⎩⎪⎨


4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582


2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)

4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582


2lt 120582120582 le 3119903119903

0 else

(517)


11989211989212 = minus14


4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886119948119948119895119895prime(1205911205911)119886119886119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)rang0119888119888

= minus14


4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886119954119954prime120582120582prime


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0

ie

11989211989212 =

⎩⎪⎨

⎪⎧



0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903



0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903

2lt 120582120582 le 3119903119903

0 else

(518)


can be written as

119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591) =

⎩⎪⎨

⎪⎧minus119881119881

120582120582120582120582prime(minus119954119954)

2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ

0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le3119903119903

2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582

(119954119954)

2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ

0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le3119903119903

2lt 120582120582 le 3119903119903

0 else

(519)


we have

1198891198891205911205911120573120573ℏ

01198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0) = 1198891198891205911205911120573120573ℏ

0

1120573120573ℏ

119890119890minus119894119894120596120596119899119899(120591120591minus1205911205911)1198661198661199541199541205821205820 (119894119894120596120596119899119899)

119899119899

1120573120573ℏ


119899119899prime

=1

(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591 1198891198891205911205911120573120573ℏ

0119890119890119894119894120596120596119899119899minus120596120596119899119899prime1205911205911 119866119866119954119954120582120582


119899119899119899119899prime=

=1


0 (119894119894120596120596119899119899prime)119899119899119899119899prime

=1120573120573ℏ


0 (119894119894120596120596119899119899)119899119899


21

119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591)

119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =

⎩⎪⎨



0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582 le 31199031199032

lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)


0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582prime le 31199031199032

lt 120582120582 le 3119903119903

0 else

(520)




Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =

⎩⎪⎨


2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)


2lt 120582120582 le 3119903119903

0 else

(521)



119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =

18ℏ2 1198891198891205911205911

120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0lang119879119879120591120591

⎣⎢⎢⎡ 11988111988111989511989511198951198951prime(119948119948120783120783)1198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951prime

dagger (1205911205911)3119903119903

1198951198951prime=31199031199032 +1

31199031199032

1198951198951=1119948119948120783120783


dagger (1205911205911)3119903119903

1198941198941prime=31199031199032 +1

31199031199032

1198941198941=1119948119948120783120783 ⎦⎥⎥⎤

⎣⎢⎢⎡ 11988111988111989511989521198951198952prime(119948119948120784120784)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime

dagger (1205911205912)3119903119903

1198951198952prime=31199031199032 +1

31199031199032

1198951198952=1119948119948120784120784


dagger (1205911205912)3119903119903

1198941198942prime=31199031199032 +1

31199031199032

1198941198942=1119948119948120784120784 ⎦⎥⎥⎤119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime

dagger (0)rang0119888119888 = 1198921198921 + 1198921198922 + 1198921198923 + 1198921198924

where

1198921198921 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


31199031199032 lt1198951198952

primele311990311990311994811994812078412078411989511989521198951198952prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime



(522)

1198921198922 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


1le1198941198942le31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime



(523)

1198921198923 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


1le1198951198952le31199031199032 lt1198951198952

primele311990311990311994811994812078412078411989511989521198951198952prime

1le1198941198941le31199031199032 lt1198941198941

primele311990311990311994811994812078312078311989411989411198941198941prime



(524)

22

1198921198924 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


lowast (119948119948120784120784)1le1198941198942le

31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198941198941le31199031199032 lt1198941198941

primele311990311990311994811994812078312078311989411989411198941198941prime



(525)



dagger we can


1198921198922 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)119881119881

11989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime

times 119868119868 (526)

Where










dagger (0)rang0c










⎩⎪⎨

⎪⎧ 1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951prime


dagger (0)rang0119888119888







(527)


23












0 (1205911205912 1205911205912)12057512057511989411989421198941198942prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582

+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime



0 (1205911205911 1205911205911)12057512057511989511989511198951198951prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582


0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582

= 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime



0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582




























24






























0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 11986611986611994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime




0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime




11989211989221 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198681198681 =

132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951


1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 + 1

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951prime

0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙


0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 =

⎩⎪⎨

⎪⎧

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912) ∙ 119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205912 1205911205911) ∙ 119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582

0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

0 else

and

25

11989211989224 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198681198684 =

132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903




32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903




0 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 =

⎩⎪⎨

⎪⎧120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime0 (1205911205912 0)119866119866119954119954120582120582

0 (120591120591 1205911205911) 120582120582 120582120582prime isin 1 31199031199032

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

0 else


(314)]



1198921198922 =

⎩⎪⎨

⎪⎧

120575120575119954119954119954119954prime

16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

120575120575119954119954119954119954prime

16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582

0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

0 else

(528)



calculated as

119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) = 41198921198922 =

⎩⎪⎨

⎪⎧120575120575119954119954119954119954prime

4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1


0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

120575120575119954119954119954119954prime

4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 31199031199032

+ 13119903119903

0 else

(529)



119866119866119954119954120582120582119954119954prime120582120582prime


(2) (119894119894120596120596119899119899)119899119899


(2) (120591120591)120573120573ℏ0

(530)


26

119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =

120575120575119954119954119954119954prime4

1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954120582120582prime

0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912)

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0

1120573120573ℏ


1198991198992

∙1120573120573ℏ

119890119890minus11989411989412059612059611989911989911205911205911119866119866119954119954120582120582prime0 1198941198941205961205961198991198991

1198991198991

∙1120573120573ℏ

119890119890minus119894119894120596120596119899119899(120591120591minus1205911205912)1198661198661199541199541205821205820 (119894119894120596120596119899119899)

119899119899

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1120573120573ℏ

119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954120582120582prime

0 11989411989412059612059611989911989911198661198661199541199541198951198951prime0 1198941198941205961205961198991198992

11989911989911989911989911198991198992⎩⎪⎨

⎪⎧ 1120573120573ℏ

1198891198891205911205911120573120573ℏ

0119890119890minus1198941198941205961205961198991198991minus12059612059611989911989921205911205911

times1120573120573ℏ

1198891198891205911205912120573120573ℏ

0119890119890minus1198941198941205961205961198991198992minus1205961205961198991198991205911205912

⎭⎪⎬

⎪⎫

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1120573120573ℏ

119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)

11989911989911989911989911198991198992

119866119866119954119954120582120582prime0 11989411989412059612059611989911989911198661198661199541199541198951198951prime

0 1198941198941205961205961198991198992120575120575119899119899111989911989921205751205751198991198991198991198992

=120575120575119954119954119954119954prime120573120573ℏ

119890119890minus119894119894120596120596119899119899120591120591

119899119899

1198661198661199541199541205821205820 (119894119894120596120596119899119899)

14

119881119881120582120582prime1198951198951prime(119954119954



3119903119903

1198951198951prime=31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899)119866119866119954119954120582120582prime

0 (119894119894120596120596119899119899)


119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582

0 (119894119894120596120596119899119899) 120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 119866119866119954119954120582120582prime

0 (119894119894120596120596119899119899) (531)



120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) (532)

and 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582

0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954120582120582prime(2) (119894119894120596120596119899119899) ∙ 119866119866119954119954120582120582prime


+ 13119903119903 we

have

Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) (533)


Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =

⎩⎪⎪⎨

⎪⎪⎧120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903

2

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

0 else

(534)

27



(1) (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) + ⋯

= 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582


0 (119894119894120596120596119899119899) + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime

(2) (119894119894120596120596119899119899)

∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) + ⋯ = 119866119866119954119954120582120582


0 (119894119894120596120596119899119899)

or





(2) (119894119894120596120596119899119899) + ⋯



Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) =

⎩⎪⎪⎪⎪⎨


120575120575119954119954119954119954prime

2

119881119881120582120582120582120582primelowast (119954119954)

120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903

minus120575120575119954119954119954119954prime

2

119881119881120582120582prime120582120582(119954119954)

120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582prime le 3119903119903

2lt 120582120582 le 3119903119903

120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903

2

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

(536)









Σ119954119954120582120582119903119903 (120596120596) via [13]

Γ119954119954120582120582(120596120596) = minus2ImΣ119954119954120582120582119954119954120582120582119903119903 (120596120596) (539)


we have

28

Σ119954119954120582120582119954119954120582120582119903119903 (119894119894120596120596119899119899) =

⎩⎪⎨

⎪⎧14sum

1198811198811205821205821198951198951prime(119954119954)

2

1205961205961199541199541198951198951prime120596120596119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

1

120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894

120582120582 isin 1 31199031199032

14sum 1198811198811198951198951120582120582(119954119954)2

1205961205961199541199541198951198951120596120596119954119954120582120582

311990311990321198951198951=1

1120596120596minus1205961205961199541199541198951198951+119894119894119894119894

120582120582 isin 31199031199032

+ 13119903119903

(540)

Using the relation

Im 1

120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894



Γ119954119954120582120582(120596120596) =

⎩⎪⎨

⎪⎧1205871205872sum

1198811198811205821205821198951198951prime(119954119954)

2

1205961205961199541199541198951198951prime120596120596119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

120575120575120596120596 minus 1205961205961199541199541198951198951prime 120582120582 isin 1 31199031199032

1205871205872sum 1198811198811198951198951120582120582(119954119954)2

1205961205961199541199541198951198951120596120596119954119954120582120582

311990311990321198951198951=1

120575120575120596120596 minus 1205961205961199541199541198951198951 120582120582 isin 31199031199032

+ 13119903119903

(542)

Or equivalently

Γ119954119954120582120582(119864119864 = ℏ120596120596) =

⎩⎪⎨

⎪⎧120587120587ℏ3

2sum

1198811198811205821205821198951198951prime(119954119954)

2

1198641198641199541199541198951198951prime119864119864119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

120575120575119864119864 minus 1198641198641199541199541198951198951prime 120582120582 isin 1 31199031199032

120587120587ℏ3

2sum 1198811198811198951198951120582120582(119954119954)2

1198641198641199541199541198951198951119864119864119954119954120582120582

311990311990321198951198951=1

120575120575119864119864 minus 1198641198641199541199541198951198951 120582120582 isin 31199031199032

+ 13119903119903

(543)








int(⋯ )120588120588119864119864119954119954d119864119864119954119954 giving

Γ119954119954120582120582(119864119864) = 120587120587ℏ3


2119864119864119954119954119895119895prime=119864119864119954119954prime


2120588120588(119864119864)

119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864

119864119864119864119864119954119954120582120582 (544)







29


Γtot = 1119885119885119890119890minus120573120573119864119864119954119954120582120582Γ119954119954120582120582119864119864119954119954120582120582

119954119954120582120582

=120587120587ℏ3

2

119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582 119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864119954119954120582120582

1198641198641199541199541205821205822119954119954120582120582

=120587120587ℏ3

2

119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582 119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582

Finally we get

Γtot = Γ119954119954120582120582(119864119864) =120587120587ℏ3

2sum 119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582 (545)





to make



References


MainText


SI

1 Methodology

11 Model system












11 Basic notations

12 Dynamical matrix


















8



























ASSOCIATED CONTENT




9


conductivity

AUTHOR INFORMATION



Notes


ACKNOWLEDGMENTS










MP1303

10

References


11


S1










1 Methodology





S2

1 Methodology

11 Model system












S3



















S4




120581120581CP = 119876119860119860Δ119879119879Δ119911119911

(S1)

















S1(b)]







S5















part119864119864t120597120597120597120597

= 119860119860119877119877119879119879bot(119905119905) minus 119879119879top(119905119905) (S3)






S6














119871119871 = |11989911989911199381199381 + 11989911989921199381199382| = 11988611988611989911989912 + 11989911989922 + 11989911989911198991198992 = |1198991198991minus1198991198992|1198861198862 sin(1205791205792) (S4)




2120587120587120575120575119894119894119895119895 such that

1199181199181 = 4120587120587radic3119871119871

radic32

minus12 1199181199182 = 4120587120587

radic3119871119871(01) (S5)

S7





120579120579 (deg) 119860119860 (nm2) 1198991198991 1198991198992 1198981198981 1198981198982

0 60383688 24 0 24 0

0696407 709613169 48 47 47 48

1121311 273718420 30 29 29 30

2000628 85648391 17 16 16 17

3006558 152322047 23 21 21 23

4048894 189013524 26 23 23 26

5085849 53255058 14 12 12 14

7926470 49376245 14 11 11 14

9998709 86277388 19 14 14 19

15178179 31554670 15 4 4 15

19932013 42876612 15 8 8 15

25039660 13942761 9 4 4 9

30158276 18974735 11 4 4 11

32204228 21805221 12 4 4 12









119868119868 = 1119881119881119887119887int 119891119891(119954119954)BZ d119954119954 asymp 1

119873119873sum 119908119908119894119894119891119891(119954119954119894119894)119894119894 (S6)

S8






⎩⎪⎨

⎪⎧ 1199541199541 = 1

9 1205971205973 1199541199542 = 2

9 1205971205973 1199541199543 = 1

18 512059712059718

1199541199544 = 518

512059712059718 1199541199545 = 1

9 21205971205979 1199541199546 = 2

9 21205971205979

1199541199547 = 118

312059712059718 1199541199548 = 1

9 1205971205979 1199541199549 = 1

18 12059712059718

(S7)


119908119908124689 = 112

119908119908357 = 16 (S8)


Γtot = 120587120587ℏ3

2sum 119908119908119896119896 sum

119890119890minus120573120573119864119864119954119954119948119948120582120582

119885119885

120588120588119864119864119954119954119948119948120582120582119881119881120582120582120582120582+31199031199032(119954119954119948119948)

2

1198641198641199541199541199481199481205821205822120582120582

9119896119896=1 (S9)





S9


















contribution [10]

S10



S11


S12


1









119955119955119899119899119899119899(119905119905) = 119929119929119899119899 + 119929119929s0 + 119958119958119899119899119899119899(119905119905) (11)



ℒ = sum sum 119872119872119904119904|119955119899119899119904119904|2(119905119905)2

119903119903119899119899=1

119873119873119899119899=1 minus 119881119881 (12)




ℒ = sum sum 119872119872119904119904|119958119899119899119904119904|2(119905119905)2

119903119903119899119899=1

119873119873119899119899=1 minus 1

2sum sum sum 120601120601120572120572120572120572prime












12 Dynamical matrix


equation as follows

2








form

119906119906119899119899119899119899120572120572(119905119905) = 1119872119872119904119904

119890119890119899119899120572120572119890119890119894119894(119954119954∙119929119929119899119899minus120596120596119905119905) (17)




where

119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = sum 1

119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime



119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime 119890119890

minus119894119894119954119954∙119929119929119897119897119897119897 (19)















120596120596120582120582(minus119954119954) = 120596120596120582120582(119954119954) (112)


sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582

prime119899119899120572120572 = 120575120575120582120582120582120582prime (113)





3





119890119890119899119899120572120572120582120582 (minus119954119954)lowast

= 119890119890119899119899120572120572120582120582 (119954119954) (116)


119906119906119899119899119899119899120572120572(119905119905) = sum 119862119862120582120582(119954119954)119873119873119872119872119904119904

119890119890119899119899120572120572120582120582 (119954119954)119890119890minus119894119894120596120596119954119954120582120582119905119905119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 = sum 119876119876120582120582(119954119954119905119905)119873119873119872119872119904119904

119890119890119899119899120572120572120582120582 (119954119954)119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 (21)




119876119876120582120582(119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)lowast

= 119876119876120582120582(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (minus119954119954) (22)



[119876119876120582120582(119954119954 119905119905)]lowast = 119876119876120582120582(minus119954119954 119905119905) (23)



119879119879 = 12sum 1198721198721198991198991199061198991198991198991198991205721205722 (119905119905)119899119899119899119899120572120572 = 1

2sum 119872119872119904119904





12sum 119876120582120582(119954119954 119905119905)119876120582120582prime

lowast (119954119954 119905119905) sum 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime(119954119954)

lowast119899119899120572120572 =119954119954120582120582120582120582prime

12sum 119876120582120582(119954119954 119905119905)119876120582120582lowast(119954119954 119905119905)119954119954120582120582 =

12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582

ie

119879119879 = 12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582 (24)


1119873119873sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899 = 120575120575119954119954+119954119954prime120782120782 (25)

4





12120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899+119954119954prime∙119929119929119899119899prime

119873119873119872119872119899119899119872119872119899119899prime



=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime


119873119873119872119872119899119899119872119872119899119899prime



=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954prime∙119929119929119897119897


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899

119899119899=120575120575119902119902+119902119902prime=0


=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954119929119929119897119897



=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)

119899119899120572120572



=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)

119899119899120572120572


119954119954120582120582120582120582prime


121205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)

119954119954120582120582

ie

119880119880 = 12sum 1205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)119954119954120582120582 (26)




2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)119954119954120582120582 (27)


119867119867 = 119879119879 + 119881119881 = 12sum [119875119875120582120582(minus119954119954 119905119905)119875119875120582120582(119954119954 119905119905) + 120596120596120582120582

2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)]119954119954120582120582 (28)


119867119867 = 12sum 119875119875minus119954119954120582120582119875119875119954119954120582120582 + 120596120596119954119954120582120582

2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582 (29)






119876119876119954119954120582120582 = ℏ

2120596120596119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582

dagger 119875119875119954119954120582120582 = 119894119894ℏ120596120596119954119954120582120582

2119887119887119954119954120582120582




5

119887119887119954119954120582120582 119887119887119954119954prime120582120582prime



dagger = 0 (212)


119867119867 =12119875119875minus119954119954120582120582119875119875119954119954120582120582 +120596120596119954119954120582120582

2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582

=12minus

ℏ120596120596119954119954120582120582

2119887119887minus119954119954120582120582


2 ℏ2120596120596119954119954120582120582


dagger 119954119954120582120582

=ℏ4120596120596119954119954120582120582minus119887119887minus119954119954120582120582



119954119954120582120582



dagger

=ℏ4120596120596119954119954120582120582119887119887minus119954119954120582120582


dagger + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582

119954119954120582120582

=ℏ41205961205961199541199541205821205822119887119887minus119954119954120582120582


119954119954120582120582

= ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 +

12

119954119954120582120582


119867119867 = sum ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1

2119954119954120582120582 (213)


119906119906119899119899119899119899120572120572 = sum ℏ

2119873119873119872119872119904119904120596120596119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119887119887119954119954120582120582 + 119887119887minus119954119954120582120582

dagger 119954119954120582120582 (214)








119867119867 = 1198671198671 + 1198671198672 + 11986711986712 (31)



119867119867 = 119879119879 + 119881119881 = sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2119899119899119899119899120572120572 + 1

2sum sum sum 120601120601120572120572120572120572prime


6


119904119904 = 1199031199032

+ 1 1199031199032


119879119879 = sum sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2

1199031199032119899119899=1 + sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)

2119903119903119899119899=1199031199032+1

119899119899120572120572 (33)


119881119881 =12 120601120601120572120572120572120572prime


1199031199032

119899119899119899119899prime=1


119903119903

119899119899119899119899prime=1199031199032+1


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime119899119899119899119899prime


1199031199032

119899119899prime=1

119903119903

119899119899=1199031199032+1

= 11988111988111 + 11988111988122 + 11988111988112 + 11988111988121

Or equivalently

⎩⎪⎪⎨

⎪⎪⎧ 11988111988111 = 1

2sum sum sum 120601120601120572120572120572120572prime



11988111988122 = 12sum sum sum 120601120601120572120572120572120572prime





119903119903119899119899prime=1199031199032+1

1199031199032119899119899=1120572120572120572120572prime119899119899119899119899prime



1199031199032119899119899prime=1

119903119903119899119899=1199031199032+1120572120572120572120572prime119899119899119899119899prime

(34)




the following form

119906119906119899119899119899119899120572120572 =

⎩⎨

⎧ sum ℏ

2119873119873119872119872119904119904ω119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger 119954119954120582120582 119904119904 isin 1 1199031199032

sum ℏ

2119873119873119872119872119904119904120596120596119954119954prime120582120582prime119890119899119899120572120572120582120582


2+ 1 119903119903

(35)


dagger )




prime119899119899120572120572 = 120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582

lowast119890119899119899120572120572120582120582

prime119899119899120572120572 =

120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582

prime119899119899120572120572 = sum 119890119890119899119899120572120572120582120582

lowast119890119899119899120572120572120582120582


119876119876119954119954120582120582 equiv ℏ

2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger 119876119876119954119954120582120582 equiv ℏ

2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger (36)


7

119875119875119954119954120582120582 = 119894119894ℏω119954119954120582120582

2119886119886119954119954120582120582



Eq (35) reads

119906119906119899119899119899119899120572120572 = sum 119890119890119904119904119899119899120582120582 (119954119954)

119873119873119872119872119904119904119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1 119903119903

2

sum 119890119904119904119899119899120582120582 (119954119954)119873119873119872119872119904119904

119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1199031199032

+ 1 119903119903 (38)


119879119879 = 1198791198791 + 1198791198792 = 12sum 119875119875119954119954120582120582119875119875minus119954119954120582120582119954119954120582120582 + 1



11988111988111 =12 120601120601120572120572120572120572prime


119899119899119899119899prime


119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime

119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954) 119863119863119899119899120572120572119899119899


120572120572prime

1199031199032

119899119899prime=1

1199031199032

119899119899=1120572120572

=12 ω119954119954120582120582prime


119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime (119954119954)

lowast

1199031199032

119899119899=1

=120572120572

12ω119954119954120582120582

2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582

ie

11988111988111 = 12sum ω119954119954120582120582

2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582 (310)






11988111988122 = 12sum 1205961205961199541199541205821205822 119876119876119954119954120582120582119876119876minus119954119954120582120582119954119954120582120582 (311)


8

11988111988112 =12 120601120601120572120572120572120572prime


119899119899119899119899prime

119903119903

119899119899prime=1199031199032+1

119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime

120582120582prime (119954119954prime)

119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954


lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119890119890119899119899120572120572120582120582 (119954119954)119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954



lowast119881119881119899119899120572120572119899119899


1199031199032119899119899=1120572120572120572120572prime (312)

where


119872119872119904119904119872119872119904119904prime119897119897 120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime (313)




9

11988111988121 =12 120601120601120572120572120572120572prime


119899119899119899119899prime

119903119903

119899119899=1199031199032+1

119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime

120582120582prime (119954119954prime)

119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899prime=1120572120572120572120572prime

=12 120601120601120572120572prime120572120572


119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1


119872119872119899119899prime119872119872119899119899

1119873119873 119890119890


1199031199032

119899119899=1120572120572prime120572120572

=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582


1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582


119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime





119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime


dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954prime 119890119890119899119899120572120572120582120582 (119954119954prime)

lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954prime∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582prime119876119876119954119954120582120582

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954

119890119890119899119899120572120572120582120582 (119954119954)lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger dagger

3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954


lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119890119890119899119899120572120572120582120582 (119954119954)119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

lowast

=

⎩⎨

⎧12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954⎭⎬

⎫dagger

= 11988111988112dagger

Define

1198671198671 = 11987911987911 + 119881119881111198671198672 = 11987911987922 + 1198811198812211986711986712 = 11988111988112 + 11988111988121

(315)


⎩⎪⎨

⎪⎧ 1198671198671 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582

dagger 119886119886119954119954120582120582 + 1231199031199032

120582120582=1119954119954


23119903119903

120582120582prime=1+31199031199032119954119954


dagger3119903119903120582120582prime=31199031199032 +1

31199031199032120582120582=1 + h c 119954119954

(316)




10


119867119867 = 1198671198670 + 119867119867119862119862 (317)

where

1198671198670 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582

dagger 119886119886119954119954120582120582 + 123119903119903

120582120582=1119954119954


dagger3119903119903120582120582prime=31199031199032 +1

31199031199032120582120582=1 + ℎ 119888119888 119954119954

(318)




119886119886119901119901(120591120591) = 119890119890119894119894ℏ119867119867119905119905119886119886119901119901119890119890

minus119894119894ℏ119867119867119905119905 = 119890119890

119867119867120591120591ℏ 119886119886119901119901119890119890

minus119867119867120591120591ℏ (41)



ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119867119867119886119886119953119953(120591120591) (42)


ℏpart119886119886119953119953(120591120591)120597120597120591120591

= 1198671198670119886119886119953119953(120591120591) = 1198671198670 1198901198901198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ = 1198671198670119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ minus 119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ 1198671198670

= 1198901198901198671198670120591120591ℏ 1198671198670119886119886119953119953 minus 1198861198861199531199531198671198670119890119890

minus1198671198670120591120591ℏ = 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ

ie

ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ (43)



2119953119953 119886119886119953119953prime = sum ℏ120596120596119953119953119886119886119953119953


dagger119886119886119953119953119953119953 =




ie


Therefore we have

ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119886119886119953119953(120591120591) (45)

11



dagger = ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 +

12

119953119953

119886119886119953119953primedagger = ℏ120596120596119953119953119886119886119953119953


119953119953

= ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953119886119886119953119953prime


dagger119886119886119953119953119953119953



dagger119886119886119953119953119953119953



dagger 119886119886119953119953dagger119886119886119953119953

119953119953


In summary we have

119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591

119886119886119953119953dagger(120591120591) = 119886119886119953119953

dagger(0)119890119890120596120596119953119953120591120591 = 119886119886119953119953dagger119890119890120596120596119953119953120591120591 (46)






dagger (120591120591prime) (47)



120588120588119867119867 = 119890119890minus120573120573119867119867119885119885119867119867 (48)








= minusTr 120588120588119867119867119890119890119867119867120591120591ℏ 119886119886119953119953119890119890

minus119867119867120591120591ℏ 119890119890

119867119867120591120591primeℏ 119886119886119953119953prime


minus119867119867120591120591primeℏ 119890119890

119867119867120591120591ℏ 119886119886119953119953119890119890


ℏ 119886119886119953119953primedagger




dagger (0)





12






119867119867120591120591ℏ 119886119886119953119953119890119890



ℏ 119886119886119953119953119890119890minus119867119867120591120591ℏ 119890119890

119867119867120591120591primeℏ


119867119867120591120591minus120591120591primeℏ 119886119886119953119953119890119890




dagger (0)








dagger (0)rang

= minusTr 120588120588119867119867119890119890119867119867(120591120591+120573120573ℏ)

ℏ 119886119886119953119953119890119890minus119867119867(120591120591+120573120573ℏ)

ℏ 119886119886119953119953prime = minusTr 120588120588119867119867119890119890120573120573119867119867119890119890

119867119867120591120591ℏ 119886119886119953119953119890119890



1119885119885119867119867


= minus1119885119885119867119867


Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119886119886119953119953(120591120591)

= minus1119885119885119867119867






ℏ 119886119886119953119953119890119890minus119867119867(120591120591minus120573120573ℏ)


119867119867120591120591ℏ 119886119886119953119953119890119890

minus119867119867120591120591ℏ 119890119890120573120573119867119867



= minus1119885119885119867119867


Tr119890119890minus120573120573119867119867119886119886119953119953(120591120591)119886119886119953119953prime(0)

= minus1119885119885119867119867


13



where 120596120596119899119899 = 2120587120587119899119899120573120573ℏ



120573120573ℏ (412)









primelang119886119886119953119953(0)119886119886119953119953prime





primelang119886119886119953119953prime





primelang119886119886119953119953prime




ie

119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =






dagger (0) = 120575120575119953119953119953119953prime


119890119890119860119860119861119861119890119890minus119860119860 = sum 1119899119899119860(119899119899)119861119861infin

119899119899=0 (414)



119891119891(119905119905) = 119891119891(0) + 119905119905119891119891prime(0) + 1199051199052

2119891119891primeprime(0) + ⋯ = sum 119905119905119899119899

119899119899119889119889119899119899119891119889119889119905119905119899119899

119905119905=0

infin119899119899=0 (415)


⎩⎪⎨



⋮119891119891(119899119899)(119905119905) = 119890119890119905119905119860119860119860(119899119899)119861119861119890119890minus119905119905119860119860

(416)

14


119890119890119905119905A119861119861119890119890minus119905119905119860119860 = sum 119905119905119899119899

119899119899119860(119899119899)119861119861infin

119899119899=0 (417)


follows


11198851198851198671198670


11198851198851198671198670


=11198851198851198671198670

Tr (minus120573120573)119899119899

1198991198991198671198670

(119899119899)119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)


dagger (0) we have

1198671198670(119899119899)119886119886119953119953prime


such that


11198851198851198671198670

Tr (minus120573120573)119899119899

1198991198991198671198670

(119899119899)119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)

=11198851198851198671198670

Tr minus120573120573ℏ120596120596119953119953prime

119899119899

119899119899119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)

= 119890119890minus120573120573ℏ120596120596119953119953prime11198851198851198671198670


dagger (0)rang





Hence we have


1119890119890120573120573ℏ120596120596119953119953minus1

equiv 120575120575119953119953prime119953119953119899119899119861119861120596120596119953119953 (419)


we obtain

119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =








119890119890119894119894120596120596119899119899minus120596120596119953119953120573120573ℏ minus

15

1 = minus1+ 1

119890119890120573120573ℏ120596120596119953119953minus1

119894119894120596120596119899119899minus120596120596119953119953119890119890119894119894


119894119894120596120596119899119899minus120596120596119953119953

119890119890minus120573120573ℏ120596120596119953119953minus1

119890119890120573120573ℏ120596120596119953119953minus1= minus 1

119894119894120596120596119899119899minus120596120596119953119953

1minus119890119890120573120573ℏ120596120596119953119953

119890119890120573120573ℏ120596120596119953119953minus1= 1

119894119894120596120596119899119899minus120596120596119953119953

ie

1198661198661199531199530(119894119894120596120596119899119899) == 1119894119894120596120596119899119899minus120596120596119953119953

(421)




119867119867119862119862119868119868 (120591120591) = 1198901198901198671198670120591120591ℏ 119867119867119862119862119890119890

minus1198671198670120591120591ℏ (51)


ℏ part119867119867119862119862119868119868 (120591120591)120597120597120591120591

= 1198671198670119867119867119862119862119868119868 (120591120591) (52)




119867119867120591120591ℏ 119886119886119953119953(0)119890119890minus

119867119867120591120591ℏ 119886119886119953119953prime

dagger (0)

= minusTr 120588120588119867119867119879119879120591120591 119890119890119867119867120591120591ℏ 119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119890119890minus

119867119867120591120591ℏ 119886119886119953119953prime

dagger (0)


where

119886119886119953119953119868119868 (120591120591) equiv 1198901198901198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus

1198671198670120591120591ℏ (53)


119880119880(1205911205911 1205911205912) equiv 11989011989011986711986701205911205911ℏ 119890119890minus

119867119867 (1205911205911minus1205911205912)ℏ 119890119890minus

11986711986701205911205912ℏ (54)


119880119880(1205911205911 1205911205912)119880119880(1205911205912 1205911205913) = 119880119880(1205911205911 1205911205913) (55)


ℏpart119880119880(120591120591 120591120591prime)

120597120597120591120591= 1198671198670119890119890

1198671198670120591120591ℏ 119890119890minus


1198671198670120591120591ℏ 119867119867119890119890minus


1198671198670120591120591primeℏ

= 1198901198901198671198670120591120591ℏ 1198671198670 minus 119867119867119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119890119890minus


1198671198670120591120591primeℏ = 119890119890

1198671198670120591120591ℏ minus119867119867119862119862119890119890

minus1198671198670120591120591ℏ 119880119880(120591120591 120591120591prime)

= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime)

16

ie

ℏ part119880119880120591120591120591120591prime120597120597120591120591

= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime) (56)


119880119880(120591120591 120591120591prime) = summinus1ℏ

119899119899

119899119899 int 1198891198891205911205911120591120591120591120591prime ⋯int 119889119889120591120591119899119899

120591120591120591120591prime 119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119899119899)infin

119899119899=0 (57)


119866119866119953119953119953119953prime(120591120591 0) = minus1119885119885119867119867



dagger (0)

Tr119890119890minus120573120573119867119867


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)=


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)





119866119866119953119953119953119953prime(120591120591 0) = minussum 1

119898119898minus1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)119886119886119953119953119868119868 (120591120591)119886119886


119898119898=0

sum 1119898119898minus

1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0infin

119898119898=0 (58)




120573120573ℏ0 lang119879119879120591120591119867119867119862119862


12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0+⋯

1minus1ℏint 1198891198891205911205911120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)rang0+12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0+⋯

(59)


119863119863119898119898 equiv 1119898119898minus 1

ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0 (510)




17







1198681198682 = minus1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0

= minus1ℏ 1198891198891205911205911120573120573ℏ



0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

= 119866119866119953119953119953119953prime0 (120591120591 0) minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888





(1) (120591120591) equiv

minus1ℏ int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime


1198681198683 = minus1ℏint 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime


(1) (120591120591) (513)


1198681198683 = 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0 = 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)119886119886119953119953(120591120591)rang0 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime


2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 +

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888



follows

18

12ℏ2

1198891198891205911205912120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 +1

2ℏ2 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

=12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

+12 minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0

minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

=12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

+12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

= minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime



1198681198683 = 119866119866119953119953119953119953prime(2) (120591120591) + 119866119866119953119953119953119953prime

(1) (120591120591)1198631198631 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198632 (514)


1198681198684 = minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0 =

minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913



minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913



minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913


dagger (0)rang0119888119888 +

minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 =

119866119866119953119953119953119953prime0 (120591120591 0) minus1

6ℏ3 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0+

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0

minus1ℏ int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888+

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime


120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0+ 119866119866119953119953119953119953prime

(3) (120591120591) = 119866119866119953119953119953119953prime(3) (120591120591) +

119866119866119953119953119953119953prime(2) (120591120591)1198631198631 + 119866119866119953119953119953119953prime

(1) (120591120591)1198631198632 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198633



follows

119866119866119953119953119953119953prime0 (120591120591) minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0

+1

2ℏ2 1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime


= 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime

(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (1 + 1198631198631 + 1198631198632 + ⋯ )



19


(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (515)

where

119866119866119953119953119953119953prime

(1) (120591120591) = minus1ℏ int 1198891198891205911205911


dagger (0)rang0119888119888

119866119866119953119953119953119953prime(2) (120591120591) = 1

2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 (516)




119866119866119953119953119953119953prime(1) (120591120591) = minus

1ℏ

12 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591 119881119881119895119895119895119895prime(119948119948)119876119876119948119948119895119895(1205911205911)119876119876119948119948119895119895prime

dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


= minus12ℏ

1198891198891205911205911120573120573ℏ

0lang119879119879120591120591

ℏ119881119881119895119895119895119895prime(119948119948)2120596120596119948119948119895119895120596120596119948119948119895119895prime


dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948

+ ℏ119881119881119895119895119895119895prime

lowast (119948119948)



3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948





11989211989211 = minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886minus119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)rang0119888119888

= minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886minus119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)119886119886119948119948119895119895(1205911205911)rang0

= minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ



119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


20


11989211989211 =

⎩⎪⎨


4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582


2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)

4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582


2lt 120582120582 le 3119903119903

0 else

(517)


11989211989212 = minus14


4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886119948119948119895119895prime(1205911205911)119886119886119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)rang0119888119888

= minus14


4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886119954119954prime120582120582prime


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0

ie

11989211989212 =

⎩⎪⎨

⎪⎧



0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903



0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903

2lt 120582120582 le 3119903119903

0 else

(518)


can be written as

119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591) =

⎩⎪⎨

⎪⎧minus119881119881

120582120582120582120582prime(minus119954119954)

2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ

0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le3119903119903

2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582

(119954119954)

2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ

0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le3119903119903

2lt 120582120582 le 3119903119903

0 else

(519)


we have

1198891198891205911205911120573120573ℏ

01198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0) = 1198891198891205911205911120573120573ℏ

0

1120573120573ℏ

119890119890minus119894119894120596120596119899119899(120591120591minus1205911205911)1198661198661199541199541205821205820 (119894119894120596120596119899119899)

119899119899

1120573120573ℏ


119899119899prime

=1

(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591 1198891198891205911205911120573120573ℏ

0119890119890119894119894120596120596119899119899minus120596120596119899119899prime1205911205911 119866119866119954119954120582120582


119899119899119899119899prime=

=1


0 (119894119894120596120596119899119899prime)119899119899119899119899prime

=1120573120573ℏ


0 (119894119894120596120596119899119899)119899119899


21

119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591)

119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =

⎩⎪⎨



0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582 le 31199031199032

lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)


0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582prime le 31199031199032

lt 120582120582 le 3119903119903

0 else

(520)




Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =

⎩⎪⎨


2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)


2lt 120582120582 le 3119903119903

0 else

(521)



119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =

18ℏ2 1198891198891205911205911

120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0lang119879119879120591120591

⎣⎢⎢⎡ 11988111988111989511989511198951198951prime(119948119948120783120783)1198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951prime

dagger (1205911205911)3119903119903

1198951198951prime=31199031199032 +1

31199031199032

1198951198951=1119948119948120783120783


dagger (1205911205911)3119903119903

1198941198941prime=31199031199032 +1

31199031199032

1198941198941=1119948119948120783120783 ⎦⎥⎥⎤

⎣⎢⎢⎡ 11988111988111989511989521198951198952prime(119948119948120784120784)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime

dagger (1205911205912)3119903119903

1198951198952prime=31199031199032 +1

31199031199032

1198951198952=1119948119948120784120784


dagger (1205911205912)3119903119903

1198941198942prime=31199031199032 +1

31199031199032

1198941198942=1119948119948120784120784 ⎦⎥⎥⎤119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime

dagger (0)rang0119888119888 = 1198921198921 + 1198921198922 + 1198921198923 + 1198921198924

where

1198921198921 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


31199031199032 lt1198951198952

primele311990311990311994811994812078412078411989511989521198951198952prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime



(522)

1198921198922 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


1le1198941198942le31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime



(523)

1198921198923 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


1le1198951198952le31199031199032 lt1198951198952

primele311990311990311994811994812078412078411989511989521198951198952prime

1le1198941198941le31199031199032 lt1198941198941

primele311990311990311994811994812078312078311989411989411198941198941prime



(524)

22

1198921198924 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


lowast (119948119948120784120784)1le1198941198942le

31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198941198941le31199031199032 lt1198941198941

primele311990311990311994811994812078312078311989411989411198941198941prime



(525)



dagger we can


1198921198922 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)119881119881

11989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime

times 119868119868 (526)

Where










dagger (0)rang0c










⎩⎪⎨

⎪⎧ 1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951prime


dagger (0)rang0119888119888







(527)


23












0 (1205911205912 1205911205912)12057512057511989411989421198941198942prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582

+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime



0 (1205911205911 1205911205911)12057512057511989511989511198951198951prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582


0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582

= 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime



0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582




























24






























0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 11986611986611994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime




0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime




11989211989221 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198681198681 =

132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951


1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 + 1

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951prime

0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙


0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 =

⎩⎪⎨

⎪⎧

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912) ∙ 119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205912 1205911205911) ∙ 119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582

0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

0 else

and

25

11989211989224 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198681198684 =

132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903




32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903




0 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 =

⎩⎪⎨

⎪⎧120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime0 (1205911205912 0)119866119866119954119954120582120582

0 (120591120591 1205911205911) 120582120582 120582120582prime isin 1 31199031199032

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

0 else


(314)]



1198921198922 =

⎩⎪⎨

⎪⎧

120575120575119954119954119954119954prime

16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

120575120575119954119954119954119954prime

16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582

0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

0 else

(528)



calculated as

119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) = 41198921198922 =

⎩⎪⎨

⎪⎧120575120575119954119954119954119954prime

4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1


0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

120575120575119954119954119954119954prime

4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 31199031199032

+ 13119903119903

0 else

(529)



119866119866119954119954120582120582119954119954prime120582120582prime


(2) (119894119894120596120596119899119899)119899119899


(2) (120591120591)120573120573ℏ0

(530)


26

119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =

120575120575119954119954119954119954prime4

1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954120582120582prime

0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912)

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0

1120573120573ℏ


1198991198992

∙1120573120573ℏ

119890119890minus11989411989412059612059611989911989911205911205911119866119866119954119954120582120582prime0 1198941198941205961205961198991198991

1198991198991

∙1120573120573ℏ

119890119890minus119894119894120596120596119899119899(120591120591minus1205911205912)1198661198661199541199541205821205820 (119894119894120596120596119899119899)

119899119899

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1120573120573ℏ

119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954120582120582prime

0 11989411989412059612059611989911989911198661198661199541199541198951198951prime0 1198941198941205961205961198991198992

11989911989911989911989911198991198992⎩⎪⎨

⎪⎧ 1120573120573ℏ

1198891198891205911205911120573120573ℏ

0119890119890minus1198941198941205961205961198991198991minus12059612059611989911989921205911205911

times1120573120573ℏ

1198891198891205911205912120573120573ℏ

0119890119890minus1198941198941205961205961198991198992minus1205961205961198991198991205911205912

⎭⎪⎬

⎪⎫

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1120573120573ℏ

119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)

11989911989911989911989911198991198992

119866119866119954119954120582120582prime0 11989411989412059612059611989911989911198661198661199541199541198951198951prime

0 1198941198941205961205961198991198992120575120575119899119899111989911989921205751205751198991198991198991198992

=120575120575119954119954119954119954prime120573120573ℏ

119890119890minus119894119894120596120596119899119899120591120591

119899119899

1198661198661199541199541205821205820 (119894119894120596120596119899119899)

14

119881119881120582120582prime1198951198951prime(119954119954



3119903119903

1198951198951prime=31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899)119866119866119954119954120582120582prime

0 (119894119894120596120596119899119899)


119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582

0 (119894119894120596120596119899119899) 120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 119866119866119954119954120582120582prime

0 (119894119894120596120596119899119899) (531)



120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) (532)

and 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582

0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954120582120582prime(2) (119894119894120596120596119899119899) ∙ 119866119866119954119954120582120582prime


+ 13119903119903 we

have

Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) (533)


Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =

⎩⎪⎪⎨

⎪⎪⎧120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903

2

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

0 else

(534)

27



(1) (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) + ⋯

= 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582


0 (119894119894120596120596119899119899) + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime

(2) (119894119894120596120596119899119899)

∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) + ⋯ = 119866119866119954119954120582120582


0 (119894119894120596120596119899119899)

or





(2) (119894119894120596120596119899119899) + ⋯



Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) =

⎩⎪⎪⎪⎪⎨


120575120575119954119954119954119954prime

2

119881119881120582120582120582120582primelowast (119954119954)

120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903

minus120575120575119954119954119954119954prime

2

119881119881120582120582prime120582120582(119954119954)

120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582prime le 3119903119903

2lt 120582120582 le 3119903119903

120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903

2

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

(536)









Σ119954119954120582120582119903119903 (120596120596) via [13]

Γ119954119954120582120582(120596120596) = minus2ImΣ119954119954120582120582119954119954120582120582119903119903 (120596120596) (539)


we have

28

Σ119954119954120582120582119954119954120582120582119903119903 (119894119894120596120596119899119899) =

⎩⎪⎨

⎪⎧14sum

1198811198811205821205821198951198951prime(119954119954)

2

1205961205961199541199541198951198951prime120596120596119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

1

120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894

120582120582 isin 1 31199031199032

14sum 1198811198811198951198951120582120582(119954119954)2

1205961205961199541199541198951198951120596120596119954119954120582120582

311990311990321198951198951=1

1120596120596minus1205961205961199541199541198951198951+119894119894119894119894

120582120582 isin 31199031199032

+ 13119903119903

(540)

Using the relation

Im 1

120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894



Γ119954119954120582120582(120596120596) =

⎩⎪⎨

⎪⎧1205871205872sum

1198811198811205821205821198951198951prime(119954119954)

2

1205961205961199541199541198951198951prime120596120596119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

120575120575120596120596 minus 1205961205961199541199541198951198951prime 120582120582 isin 1 31199031199032

1205871205872sum 1198811198811198951198951120582120582(119954119954)2

1205961205961199541199541198951198951120596120596119954119954120582120582

311990311990321198951198951=1

120575120575120596120596 minus 1205961205961199541199541198951198951 120582120582 isin 31199031199032

+ 13119903119903

(542)

Or equivalently

Γ119954119954120582120582(119864119864 = ℏ120596120596) =

⎩⎪⎨

⎪⎧120587120587ℏ3

2sum

1198811198811205821205821198951198951prime(119954119954)

2

1198641198641199541199541198951198951prime119864119864119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

120575120575119864119864 minus 1198641198641199541199541198951198951prime 120582120582 isin 1 31199031199032

120587120587ℏ3

2sum 1198811198811198951198951120582120582(119954119954)2

1198641198641199541199541198951198951119864119864119954119954120582120582

311990311990321198951198951=1

120575120575119864119864 minus 1198641198641199541199541198951198951 120582120582 isin 31199031199032

+ 13119903119903

(543)








int(⋯ )120588120588119864119864119954119954d119864119864119954119954 giving

Γ119954119954120582120582(119864119864) = 120587120587ℏ3


2119864119864119954119954119895119895prime=119864119864119954119954prime


2120588120588(119864119864)

119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864

119864119864119864119864119954119954120582120582 (544)







29


Γtot = 1119885119885119890119890minus120573120573119864119864119954119954120582120582Γ119954119954120582120582119864119864119954119954120582120582

119954119954120582120582

=120587120587ℏ3

2

119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582 119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864119954119954120582120582

1198641198641199541199541205821205822119954119954120582120582

=120587120587ℏ3

2

119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582 119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582

Finally we get

Γtot = Γ119954119954120582120582(119864119864) =120587120587ℏ3

2sum 119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582 (545)





to make



References


MainText


SI

1 Methodology

11 Model system












11 Basic notations

12 Dynamical matrix


















9


conductivity

AUTHOR INFORMATION



Notes


ACKNOWLEDGMENTS










MP1303

10

References


11


S1










1 Methodology





S2

1 Methodology

11 Model system












S3



















S4




120581120581CP = 119876119860119860Δ119879119879Δ119911119911

(S1)

















S1(b)]







S5















part119864119864t120597120597120597120597

= 119860119860119877119877119879119879bot(119905119905) minus 119879119879top(119905119905) (S3)






S6














119871119871 = |11989911989911199381199381 + 11989911989921199381199382| = 11988611988611989911989912 + 11989911989922 + 11989911989911198991198992 = |1198991198991minus1198991198992|1198861198862 sin(1205791205792) (S4)




2120587120587120575120575119894119894119895119895 such that

1199181199181 = 4120587120587radic3119871119871

radic32

minus12 1199181199182 = 4120587120587

radic3119871119871(01) (S5)

S7





120579120579 (deg) 119860119860 (nm2) 1198991198991 1198991198992 1198981198981 1198981198982

0 60383688 24 0 24 0

0696407 709613169 48 47 47 48

1121311 273718420 30 29 29 30

2000628 85648391 17 16 16 17

3006558 152322047 23 21 21 23

4048894 189013524 26 23 23 26

5085849 53255058 14 12 12 14

7926470 49376245 14 11 11 14

9998709 86277388 19 14 14 19

15178179 31554670 15 4 4 15

19932013 42876612 15 8 8 15

25039660 13942761 9 4 4 9

30158276 18974735 11 4 4 11

32204228 21805221 12 4 4 12









119868119868 = 1119881119881119887119887int 119891119891(119954119954)BZ d119954119954 asymp 1

119873119873sum 119908119908119894119894119891119891(119954119954119894119894)119894119894 (S6)

S8






⎩⎪⎨

⎪⎧ 1199541199541 = 1

9 1205971205973 1199541199542 = 2

9 1205971205973 1199541199543 = 1

18 512059712059718

1199541199544 = 518

512059712059718 1199541199545 = 1

9 21205971205979 1199541199546 = 2

9 21205971205979

1199541199547 = 118

312059712059718 1199541199548 = 1

9 1205971205979 1199541199549 = 1

18 12059712059718

(S7)


119908119908124689 = 112

119908119908357 = 16 (S8)


Γtot = 120587120587ℏ3

2sum 119908119908119896119896 sum

119890119890minus120573120573119864119864119954119954119948119948120582120582

119885119885

120588120588119864119864119954119954119948119948120582120582119881119881120582120582120582120582+31199031199032(119954119954119948119948)

2

1198641198641199541199541199481199481205821205822120582120582

9119896119896=1 (S9)





S9


















contribution [10]

S10



S11


S12


1









119955119955119899119899119899119899(119905119905) = 119929119929119899119899 + 119929119929s0 + 119958119958119899119899119899119899(119905119905) (11)



ℒ = sum sum 119872119872119904119904|119955119899119899119904119904|2(119905119905)2

119903119903119899119899=1

119873119873119899119899=1 minus 119881119881 (12)




ℒ = sum sum 119872119872119904119904|119958119899119899119904119904|2(119905119905)2

119903119903119899119899=1

119873119873119899119899=1 minus 1

2sum sum sum 120601120601120572120572120572120572prime












12 Dynamical matrix


equation as follows

2








form

119906119906119899119899119899119899120572120572(119905119905) = 1119872119872119904119904

119890119890119899119899120572120572119890119890119894119894(119954119954∙119929119929119899119899minus120596120596119905119905) (17)




where

119863119863119899119899120572120572119899119899prime120572120572prime(119954119954) = sum 1

119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime



119872119872119904119904119872119872119904119904prime120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime 119890119890

minus119894119894119954119954∙119929119929119897119897119897119897 (19)















120596120596120582120582(minus119954119954) = 120596120596120582120582(119954119954) (112)


sum 119890119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582

prime119899119899120572120572 = 120575120575120582120582120582120582prime (113)





3





119890119890119899119899120572120572120582120582 (minus119954119954)lowast

= 119890119890119899119899120572120572120582120582 (119954119954) (116)


119906119906119899119899119899119899120572120572(119905119905) = sum 119862119862120582120582(119954119954)119873119873119872119872119904119904

119890119890119899119899120572120572120582120582 (119954119954)119890119890minus119894119894120596120596119954119954120582120582119905119905119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 = sum 119876119876120582120582(119954119954119905119905)119873119873119872119872119904119904

119890119890119899119899120572120572120582120582 (119954119954)119890119890119894119894119954119954∙119929119929119899119899119954119954120582120582 (21)




119876119876120582120582(119954119954 119905119905)119890119890119899119899120572120572120582120582 (119954119954)lowast

= 119876119876120582120582(minus119954119954 119905119905)119890119890119899119899120572120572120582120582 (minus119954119954) (22)



[119876119876120582120582(119954119954 119905119905)]lowast = 119876119876120582120582(minus119954119954 119905119905) (23)



119879119879 = 12sum 1198721198721198991198991199061198991198991198991198991205721205722 (119905119905)119899119899119899119899120572120572 = 1

2sum 119872119872119904119904





12sum 119876120582120582(119954119954 119905119905)119876120582120582prime

lowast (119954119954 119905119905) sum 119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime(119954119954)

lowast119899119899120572120572 =119954119954120582120582120582120582prime

12sum 119876120582120582(119954119954 119905119905)119876120582120582lowast(119954119954 119905119905)119954119954120582120582 =

12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582

ie

119879119879 = 12sum 119876120582120582(119954119954 119905119905)119876120582120582(minus119954119954 119905119905)119954119954120582120582 (24)


1119873119873sum 119890119890119894119894119954119954+119954119954prime∙119929119929119899119899119899119899 = 120575120575119954119954+119954119954prime120782120782 (25)

4





12120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899+119954119954prime∙119929119929119899119899prime

119873119873119872119872119899119899119872119872119899119899prime



=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime


119873119873119872119872119899119899119872119872119899119899prime



=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954prime∙119929119929119897119897


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899

119899119899=120575120575119902119902+119902119902prime=0


=12120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954119929119929119897119897



=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)

119899119899120572120572



=12 [119876119876120582120582(119954119954 119905119905)119876119876120582120582prime(minus119954119954 119905119905)]119890119890119899119899120572120572120582120582 (119954119954)

119899119899120572120572


119954119954120582120582120582120582prime


121205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)

119954119954120582120582

ie

119880119880 = 12sum 1205961205961205821205822(119954119954)119876119876120582120582(119954119954 119905119905)119876119876120582120582(minus119954119954 119905119905)119954119954120582120582 (26)




2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)119954119954120582120582 (27)


119867119867 = 119879119879 + 119881119881 = 12sum [119875119875120582120582(minus119954119954 119905119905)119875119875120582120582(119954119954 119905119905) + 120596120596120582120582

2(119954119954)119876119876120582120582(minus119954119954 119905119905)119876119876120582120582(119954119954 119905119905)]119954119954120582120582 (28)


119867119867 = 12sum 119875119875minus119954119954120582120582119875119875119954119954120582120582 + 120596120596119954119954120582120582

2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582 (29)






119876119876119954119954120582120582 = ℏ

2120596120596119954119954120582120582119887119887119954119954120582120582 + 119887119887minus119954119954120582120582

dagger 119875119875119954119954120582120582 = 119894119894ℏ120596120596119954119954120582120582

2119887119887119954119954120582120582




5

119887119887119954119954120582120582 119887119887119954119954prime120582120582prime



dagger = 0 (212)


119867119867 =12119875119875minus119954119954120582120582119875119875119954119954120582120582 +120596120596119954119954120582120582

2 119876119876minus119954119954120582120582119876119876119954119954120582120582119954119954120582120582

=12minus

ℏ120596120596119954119954120582120582

2119887119887minus119954119954120582120582


2 ℏ2120596120596119954119954120582120582


dagger 119954119954120582120582

=ℏ4120596120596119954119954120582120582minus119887119887minus119954119954120582120582



119954119954120582120582



dagger

=ℏ4120596120596119954119954120582120582119887119887minus119954119954120582120582


dagger + 119887119887119954119954120582120582dagger 119887119887119954119954120582120582

119954119954120582120582

=ℏ41205961205961199541199541205821205822119887119887minus119954119954120582120582


119954119954120582120582

= ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 +

12

119954119954120582120582


119867119867 = sum ℏ120596120596119954119954120582120582 119887119887119954119954120582120582dagger 119887119887119954119954120582120582 + 1

2119954119954120582120582 (213)


119906119906119899119899119899119899120572120572 = sum ℏ

2119873119873119872119872119904119904120596120596119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119887119887119954119954120582120582 + 119887119887minus119954119954120582120582

dagger 119954119954120582120582 (214)








119867119867 = 1198671198671 + 1198671198672 + 11986711986712 (31)



119867119867 = 119879119879 + 119881119881 = sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2119899119899119899119899120572120572 + 1

2sum sum sum 120601120601120572120572120572120572prime


6


119904119904 = 1199031199032

+ 1 1199031199032


119879119879 = sum sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)2

1199031199032119899119899=1 + sum 1198721198721199041199041199061198991198991199041199041198991198992 (119905119905)

2119903119903119899119899=1199031199032+1

119899119899120572120572 (33)


119881119881 =12 120601120601120572120572120572120572prime


1199031199032

119899119899119899119899prime=1


119903119903

119899119899119899119899prime=1199031199032+1


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime119899119899119899119899prime


1199031199032

119899119899prime=1

119903119903

119899119899=1199031199032+1

= 11988111988111 + 11988111988122 + 11988111988112 + 11988111988121

Or equivalently

⎩⎪⎪⎨

⎪⎪⎧ 11988111988111 = 1

2sum sum sum 120601120601120572120572120572120572prime



11988111988122 = 12sum sum sum 120601120601120572120572120572120572prime





119903119903119899119899prime=1199031199032+1

1199031199032119899119899=1120572120572120572120572prime119899119899119899119899prime



1199031199032119899119899prime=1

119903119903119899119899=1199031199032+1120572120572120572120572prime119899119899119899119899prime

(34)




the following form

119906119906119899119899119899119899120572120572 =

⎩⎨

⎧ sum ℏ

2119873119873119872119872119904119904ω119954119954120582120582119890119890119899119899120572120572120582120582 119890119890119894119894119954119954∙119929119929119899119899119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger 119954119954120582120582 119904119904 isin 1 1199031199032

sum ℏ

2119873119873119872119872119904119904120596120596119954119954prime120582120582prime119890119899119899120572120572120582120582


2+ 1 119903119903

(35)


dagger )




prime119899119899120572120572 = 120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582

lowast119890119899119899120572120572120582120582

prime119899119899120572120572 =

120575120575120582120582120582120582prime sum 119890119899119899120572120572120582120582 lowast119890119890119899119899120572120572120582120582

prime119899119899120572120572 = sum 119890119890119899119899120572120572120582120582

lowast119890119899119899120572120572120582120582


119876119876119954119954120582120582 equiv ℏ

2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger 119876119876119954119954120582120582 equiv ℏ

2ω119954119954120582120582119886119886119954119954120582120582 + 119886119886minus119954119954120582120582

dagger (36)


7

119875119875119954119954120582120582 = 119894119894ℏω119954119954120582120582

2119886119886119954119954120582120582



Eq (35) reads

119906119906119899119899119899119899120572120572 = sum 119890119890119904119904119899119899120582120582 (119954119954)

119873119873119872119872119904119904119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1 119903119903

2

sum 119890119904119904119899119899120582120582 (119954119954)119873119873119872119872119904119904

119890119890119894119894119954119954∙119929119929119899119899119876119876119954119954120582120582119954119954120582120582 119904119904 isin 1199031199032

+ 1 119903119903 (38)


119879119879 = 1198791198791 + 1198791198792 = 12sum 119875119875119954119954120582120582119875119875minus119954119954120582120582119954119954120582120582 + 1



11988111988111 =12 120601120601120572120572120572120572prime


119899119899119899119899prime


119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1199031199032

119899119899119899119899prime=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime

119954119954120582120582120582120582prime119890119890119899119899120572120572120582120582 (119954119954) 119863119863119899119899120572120572119899119899


120572120572prime

1199031199032

119899119899prime=1

1199031199032

119899119899=1120572120572

=12 ω119954119954120582120582prime


119890119890119899119899120572120572120582120582 (119954119954)119890119890119899119899120572120572120582120582prime (119954119954)

lowast

1199031199032

119899119899=1

=120572120572

12ω119954119954120582120582

2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582

ie

11988111988111 = 12sum ω119954119954120582120582

2 119876119876119954119954120582120582119876119876minus119954119954120582120582prime119954119954120582120582 (310)






11988111988122 = 12sum 1205961205961199541199541205821205822 119876119876119954119954120582120582119876119876minus119954119954120582120582119954119954120582120582 (311)


8

11988111988112 =12 120601120601120572120572120572120572prime


119899119899119899119899prime

119903119903

119899119899prime=1199031199032+1

119890119890119899119899120572120572120582120582 (119954119954)119890119899119899prime120572120572prime

120582120582prime (119954119954prime)

119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime


120582120582prime (119954119954prime)1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954prime120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876minus119954119954120582120582prime



119904119904 119904119904prime119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954


lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119890119890119899119899120572120572120582120582 (119954119954)119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954



lowast119881119881119899119899120572120572119899119899


1199031199032119899119899=1120572120572120572120572prime (312)

where


119872119872119904119904119872119872119904119904prime119897119897 120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime (313)




9

11988111988121 =12 120601120601120572120572120572120572prime


119899119899119899119899prime

119903119903

119899119899=1199031199032+1

119890119899119899120572120572120582120582 (119954119954)119890119890119899119899prime120572120572prime

120582120582prime (119954119954prime)

119872119872119899119899119872119872119899119899prime


119954119954120582120582119954119954prime120582120582prime

1199031199032

119899119899prime=1120572120572120572120572prime

=12 120601120601120572120572prime120572120572


119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1


119872119872119899119899prime119872119872119899119899

1119873119873 119890119890


1199031199032

119899119899=1120572120572prime120572120572

=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582


1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime120601120601120572120572120572120572prime


119890119890119894119894119954119954∙119929119929119899119899minus119929119929119899119899prime

119872119872119899119899119872119872119899119899prime119899119899

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954prime120582120582prime119876119876119954119954120582120582


119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

1119873119873 119890119890

119894119894(119954119954+119954119954prime)∙119929119929119899119899prime

119899119899prime

119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime





119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime


dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954prime 119890119890119899119899120572120572120582120582 (119954119954prime)

lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954prime∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582prime119876119876119954119954120582120582

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954

119890119890119899119899120572120572120582120582 (119954119954)lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890minus119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897


119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

=12 119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger dagger

3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954


lowast120601120601120572120572120572120572prime

119897119897119904119904 119904119904prime

119890119890119894119894119954119954∙119929119929119897119897

119872119872119899119899119872119872119899119899prime119897119897

119890119890119899119899120572120572120582120582 (119954119954)119903119903

119899119899prime=1199031199032+1

1199031199032

119899119899=1120572120572120572120572prime

lowast

=

⎩⎨

⎧12 119881119881120582120582120582120582prime(119954119954)119876119876119954119954120582120582119876119876119954119954120582120582prime

dagger3119903119903

120582120582prime=31199031199032 +1

31199031199032

120582120582=1119954119954⎭⎬

⎫dagger

= 11988111988112dagger

Define

1198671198671 = 11987911987911 + 119881119881111198671198672 = 11987911987922 + 1198811198812211986711986712 = 11988111988112 + 11988111988121

(315)


⎩⎪⎨

⎪⎧ 1198671198671 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582

dagger 119886119886119954119954120582120582 + 1231199031199032

120582120582=1119954119954


23119903119903

120582120582prime=1+31199031199032119954119954


dagger3119903119903120582120582prime=31199031199032 +1

31199031199032120582120582=1 + h c 119954119954

(316)




10


119867119867 = 1198671198670 + 119867119867119862119862 (317)

where

1198671198670 = sum sum ℏ120596120596119954119954120582120582 119886119886119954119954120582120582

dagger 119886119886119954119954120582120582 + 123119903119903

120582120582=1119954119954


dagger3119903119903120582120582prime=31199031199032 +1

31199031199032120582120582=1 + ℎ 119888119888 119954119954

(318)




119886119886119901119901(120591120591) = 119890119890119894119894ℏ119867119867119905119905119886119886119901119901119890119890

minus119894119894ℏ119867119867119905119905 = 119890119890

119867119867120591120591ℏ 119886119886119901119901119890119890

minus119867119867120591120591ℏ (41)



ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119867119867119886119886119953119953(120591120591) (42)


ℏpart119886119886119953119953(120591120591)120597120597120591120591

= 1198671198670119886119886119953119953(120591120591) = 1198671198670 1198901198901198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ = 1198671198670119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ minus 119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ 1198671198670

= 1198901198901198671198670120591120591ℏ 1198671198670119886119886119953119953 minus 1198861198861199531199531198671198670119890119890

minus1198671198670120591120591ℏ = 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ

ie

ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ (43)



2119953119953 119886119886119953119953prime = sum ℏ120596120596119953119953119886119886119953119953


dagger119886119886119953119953119953119953 =




ie


Therefore we have

ℏ part119886119886119953119953(120591120591)

120597120597120591120591= 119890119890

1198671198670120591120591ℏ 1198671198670119886119886119953119953119890119890

minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119890119890

1198671198670120591120591ℏ 119886119886119953119953119890119890

minus1198671198670120591120591ℏ = minusℏ120596120596119953119953119886119886119953119953(120591120591) (45)

11



dagger = ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953 +

12

119953119953

119886119886119953119953primedagger = ℏ120596120596119953119953119886119886119953119953


119953119953

= ℏ120596120596119953119953 119886119886119953119953dagger119886119886119953119953119886119886119953119953prime


dagger119886119886119953119953119953119953



dagger119886119886119953119953119953119953



dagger 119886119886119953119953dagger119886119886119953119953

119953119953


In summary we have

119886119886119953119953(120591120591) = 119886119886119953119953(0)119890119890minus120596120596119953119953120591120591 = 119886119886119953119953119890119890minus120596120596119953119953120591120591

119886119886119953119953dagger(120591120591) = 119886119886119953119953

dagger(0)119890119890120596120596119953119953120591120591 = 119886119886119953119953dagger119890119890120596120596119953119953120591120591 (46)






dagger (120591120591prime) (47)



120588120588119867119867 = 119890119890minus120573120573119867119867119885119885119867119867 (48)








= minusTr 120588120588119867119867119890119890119867119867120591120591ℏ 119886119886119953119953119890119890

minus119867119867120591120591ℏ 119890119890

119867119867120591120591primeℏ 119886119886119953119953prime


minus119867119867120591120591primeℏ 119890119890

119867119867120591120591ℏ 119886119886119953119953119890119890


ℏ 119886119886119953119953primedagger




dagger (0)





12






119867119867120591120591ℏ 119886119886119953119953119890119890



ℏ 119886119886119953119953119890119890minus119867119867120591120591ℏ 119890119890

119867119867120591120591primeℏ


119867119867120591120591minus120591120591primeℏ 119886119886119953119953119890119890




dagger (0)








dagger (0)rang

= minusTr 120588120588119867119867119890119890119867119867(120591120591+120573120573ℏ)

ℏ 119886119886119953119953119890119890minus119867119867(120591120591+120573120573ℏ)

ℏ 119886119886119953119953prime = minusTr 120588120588119867119867119890119890120573120573119867119867119890119890

119867119867120591120591ℏ 119886119886119953119953119890119890



1119885119885119867119867


= minus1119885119885119867119867


Tr119890119890minus120573120573119867119867119886119886119953119953prime(0)119886119886119953119953(120591120591)

= minus1119885119885119867119867






ℏ 119886119886119953119953119890119890minus119867119867(120591120591minus120573120573ℏ)


119867119867120591120591ℏ 119886119886119953119953119890119890

minus119867119867120591120591ℏ 119890119890120573120573119867119867



= minus1119885119885119867119867


Tr119890119890minus120573120573119867119867119886119886119953119953(120591120591)119886119886119953119953prime(0)

= minus1119885119885119867119867


13



where 120596120596119899119899 = 2120587120587119899119899120573120573ℏ



120573120573ℏ (412)









primelang119886119886119953119953(0)119886119886119953119953prime





primelang119886119886119953119953prime





primelang119886119886119953119953prime




ie

119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =






dagger (0) = 120575120575119953119953119953119953prime


119890119890119860119860119861119861119890119890minus119860119860 = sum 1119899119899119860(119899119899)119861119861infin

119899119899=0 (414)



119891119891(119905119905) = 119891119891(0) + 119905119905119891119891prime(0) + 1199051199052

2119891119891primeprime(0) + ⋯ = sum 119905119905119899119899

119899119899119889119889119899119899119891119889119889119905119905119899119899

119905119905=0

infin119899119899=0 (415)


⎩⎪⎨



⋮119891119891(119899119899)(119905119905) = 119890119890119905119905119860119860119860(119899119899)119861119861119890119890minus119905119905119860119860

(416)

14


119890119890119905119905A119861119861119890119890minus119905119905119860119860 = sum 119905119905119899119899

119899119899119860(119899119899)119861119861infin

119899119899=0 (417)


follows


11198851198851198671198670


11198851198851198671198670


=11198851198851198671198670

Tr (minus120573120573)119899119899

1198991198991198671198670

(119899119899)119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)


dagger (0) we have

1198671198670(119899119899)119886119886119953119953prime


such that


11198851198851198671198670

Tr (minus120573120573)119899119899

1198991198991198671198670

(119899119899)119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)

=11198851198851198671198670

Tr minus120573120573ℏ120596120596119953119953prime

119899119899

119899119899119886119886119953119953primedagger (0)

infin

119899119899=0

119890119890minus1205731205731198671198670119886119886119953119953(0)

= 119890119890minus120573120573ℏ120596120596119953119953prime11198851198851198671198670


dagger (0)rang





Hence we have


1119890119890120573120573ℏ120596120596119953119953minus1

equiv 120575120575119953119953prime119953119953119899119899119861119861120596120596119953119953 (419)


we obtain

119866119866119953119953119953119953prime0 (120591120591 120591120591prime) =








119890119890119894119894120596120596119899119899minus120596120596119953119953120573120573ℏ minus

15

1 = minus1+ 1

119890119890120573120573ℏ120596120596119953119953minus1

119894119894120596120596119899119899minus120596120596119953119953119890119890119894119894


119894119894120596120596119899119899minus120596120596119953119953

119890119890minus120573120573ℏ120596120596119953119953minus1

119890119890120573120573ℏ120596120596119953119953minus1= minus 1

119894119894120596120596119899119899minus120596120596119953119953

1minus119890119890120573120573ℏ120596120596119953119953

119890119890120573120573ℏ120596120596119953119953minus1= 1

119894119894120596120596119899119899minus120596120596119953119953

ie

1198661198661199531199530(119894119894120596120596119899119899) == 1119894119894120596120596119899119899minus120596120596119953119953

(421)




119867119867119862119862119868119868 (120591120591) = 1198901198901198671198670120591120591ℏ 119867119867119862119862119890119890

minus1198671198670120591120591ℏ (51)


ℏ part119867119867119862119862119868119868 (120591120591)120597120597120591120591

= 1198671198670119867119867119862119862119868119868 (120591120591) (52)




119867119867120591120591ℏ 119886119886119953119953(0)119890119890minus

119867119867120591120591ℏ 119886119886119953119953prime

dagger (0)

= minusTr 120588120588119867119867119879119879120591120591 119890119890119867119867120591120591ℏ 119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119890119890minus

119867119867120591120591ℏ 119886119886119953119953prime

dagger (0)


where

119886119886119953119953119868119868 (120591120591) equiv 1198901198901198671198670120591120591ℏ 119886119886119953119953(0)119890119890minus

1198671198670120591120591ℏ (53)


119880119880(1205911205911 1205911205912) equiv 11989011989011986711986701205911205911ℏ 119890119890minus

119867119867 (1205911205911minus1205911205912)ℏ 119890119890minus

11986711986701205911205912ℏ (54)


119880119880(1205911205911 1205911205912)119880119880(1205911205912 1205911205913) = 119880119880(1205911205911 1205911205913) (55)


ℏpart119880119880(120591120591 120591120591prime)

120597120597120591120591= 1198671198670119890119890

1198671198670120591120591ℏ 119890119890minus


1198671198670120591120591ℏ 119867119867119890119890minus


1198671198670120591120591primeℏ

= 1198901198901198671198670120591120591ℏ 1198671198670 minus 119867119867119890119890minus

1198671198670120591120591ℏ 119890119890

1198671198670120591120591ℏ 119890119890minus


1198671198670120591120591primeℏ = 119890119890

1198671198670120591120591ℏ minus119867119867119862119862119890119890

minus1198671198670120591120591ℏ 119880119880(120591120591 120591120591prime)

= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime)

16

ie

ℏ part119880119880120591120591120591120591prime120597120597120591120591

= minus119867119867119862119862119868119868 (120591120591)119880119880(120591120591 120591120591prime) (56)


119880119880(120591120591 120591120591prime) = summinus1ℏ

119899119899

119899119899 int 1198891198891205911205911120591120591120591120591prime ⋯int 119889119889120591120591119899119899

120591120591120591120591prime 119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119899119899)infin

119899119899=0 (57)


119866119866119953119953119953119953prime(120591120591 0) = minus1119885119885119867119867



dagger (0)

Tr119890119890minus120573120573119867119867


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)=


dagger (0)

Tr119890119890minus1205731205731198671198670119880119880(120573120573ℏ 0)





119866119866119953119953119953119953prime(120591120591 0) = minussum 1

119898119898minus1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)119886119886119953119953119868119868 (120591120591)119886119886


119898119898=0

sum 1119898119898minus

1ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0infin

119898119898=0 (58)




120573120573ℏ0 lang119879119879120591120591119867119867119862119862


12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0+⋯

1minus1ℏint 1198891198891205911205911120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)rang0+12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862

119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0+⋯

(59)


119863119863119898119898 equiv 1119898119898minus 1

ℏ119898119898int 1198891198891205911205911120573120573ℏ0 ⋯int 119889119889120591120591119898119898

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)⋯119867119867119862119862119868119868 (120591120591119898119898)rang0 (510)




17







1198681198682 = minus1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0

= minus1ℏ 1198891198891205911205911120573120573ℏ



0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

= 119866119866119953119953119953119953prime0 (120591120591 0) minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888





(1) (120591120591) equiv

minus1ℏ int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862(1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime


1198681198683 = minus1ℏint 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime


(1) (120591120591) (513)


1198681198683 = 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0 = 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)119886119886119953119953(120591120591)rang0 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime


2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)rang0 +

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 + 12ℏ2 int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 +

12ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888



follows

18

12ℏ2

1198891198891205911205912120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 +1

2ℏ2 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

=12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

+12 minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0

minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

=12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

+12 minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888

= minus1ℏ 1198891198891205911205912

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)rang0

minus1ℏ 1198891198891205911205911

120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime



1198681198683 = 119866119866119953119953119953119953prime(2) (120591120591) + 119866119866119953119953119953119953prime

(1) (120591120591)1198631198631 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198632 (514)


1198681198684 = minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0 =

minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913



minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913



minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913


dagger (0)rang0119888119888 +

minus16ℏ3 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888 =

119866119866119953119953119953119953prime0 (120591120591 0) minus1

6ℏ3 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0+

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)rang0

minus1ℏ int 1198891198891205911205911

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0119888119888+

12ℏ2 int 1198891198891205911205912

120573120573ℏ0 int 1198891198891205911205913

120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205912)119867119867119862119862119868119868 (1205911205913)119886119886119953119953(120591120591)119886119886119953119953prime


120573120573ℏ0 lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)rang0+ 119866119866119953119953119953119953prime

(3) (120591120591) = 119866119866119953119953119953119953prime(3) (120591120591) +

119866119866119953119953119953119953prime(2) (120591120591)1198631198631 + 119866119866119953119953119953119953prime

(1) (120591120591)1198631198632 + 119866119866119953119953119953119953prime0 (120591120591 0)1198631198633



follows

119866119866119953119953119953119953prime0 (120591120591) minus

1ℏ 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119886119886119953119953(120591120591)119886119886119953119953prime

dagger (0)rang0

+1

2ℏ2 1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0lang119879119879120591120591119867119867119862119862119868119868 (1205911205911)119867119867119862119862119868119868 (1205911205912)119886119886119953119953(120591120591)119886119886119953119953prime


= 119866119866119953119953119953119953prime0 (120591120591) + 119866119866119953119953119953119953prime

(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (1 + 1198631198631 + 1198631198632 + ⋯ )



19


(1) (120591120591) + 119866119866119953119953119953119953prime(2) (120591120591) + ⋯ (515)

where

119866119866119953119953119953119953prime

(1) (120591120591) = minus1ℏ int 1198891198891205911205911


dagger (0)rang0119888119888

119866119866119953119953119953119953prime(2) (120591120591) = 1

2ℏ2 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912


dagger (0)rang0119888119888 (516)




119866119866119953119953119953119953prime(1) (120591120591) = minus

1ℏ

12 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591 119881119881119895119895119895119895prime(119948119948)119876119876119948119948119895119895(1205911205911)119876119876119948119948119895119895prime

dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


= minus12ℏ

1198891198891205911205911120573120573ℏ

0lang119879119879120591120591

ℏ119881119881119895119895119895119895prime(119948119948)2120596120596119948119948119895119895120596120596119948119948119895119895prime


dagger (1205911205911)3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948

+ ℏ119881119881119895119895119895119895prime

lowast (119948119948)



3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948





11989211989211 = minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886minus119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)rang0119888119888

= minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886minus119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)119886119886119948119948119895119895(1205911205911)rang0

= minus14

119881119881119895119895119895119895prime(119948119948)

120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ



119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


20


11989211989211 =

⎩⎪⎨


4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582


2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)

4120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ0 119866119866119954119954120582120582


2lt 120582120582 le 3119903119903

0 else

(517)


11989211989212 = minus14


4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886119948119948119895119895prime(1205911205911)119886119886119948119948119895119895


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (0)rang0119888119888

= minus14


4120596120596119948119948119895119895120596120596119948119948119895119895prime 1198891198891205911205911120573120573ℏ

0lang119879119879120591120591119886119886119954119954prime120582120582prime


3119903119903

119895119895prime=31199031199032 +1

31199031199032

119895119895=1119948119948


dagger (1205911205911)119886119886119954119954120582120582(120591120591)rang0

ie

11989211989212 =

⎩⎪⎨

⎪⎧



0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903



0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205911)120575120575119954119954119954119954prime 1 le 120582120582prime le 3119903119903

2lt 120582120582 le 3119903119903

0 else

(518)


can be written as

119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591) =

⎩⎪⎨

⎪⎧minus119881119881

120582120582120582120582prime(minus119954119954)

2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ

0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582 le3119903119903

2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582

(119954119954)

2120596120596119954119954120582120582120596120596119954119954120582120582primeint 1198891198891205911205911120573120573ℏ

0 1198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)120575120575119954119954119954119954prime 1 le 120582120582prime le3119903119903

2lt 120582120582 le 3119903119903

0 else

(519)


we have

1198891198891205911205911120573120573ℏ

01198661198661199541199541205821205820 (120591120591 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0) = 1198891198891205911205911120573120573ℏ

0

1120573120573ℏ

119890119890minus119894119894120596120596119899119899(120591120591minus1205911205911)1198661198661199541199541205821205820 (119894119894120596120596119899119899)

119899119899

1120573120573ℏ


119899119899prime

=1

(120573120573ℏ)2119890119890minus119894119894120596120596119899119899120591120591 1198891198891205911205911120573120573ℏ

0119890119890119894119894120596120596119899119899minus120596120596119899119899prime1205911205911 119866119866119954119954120582120582


119899119899119899119899prime=

=1


0 (119894119894120596120596119899119899prime)119899119899119899119899prime

=1120573120573ℏ


0 (119894119894120596120596119899119899)119899119899


21

119866119866119954119954120582120582119954119954prime120582120582prime(1) (120591120591)

119866119866119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =

⎩⎪⎨



0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582 le 31199031199032

lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)


0 (119894119894120596120596119899119899)120575120575119954119954119954119954prime 1 le 120582120582prime le 31199031199032

lt 120582120582 le 3119903119903

0 else

(520)




Σ119954119954120582120582119954119954prime120582120582prime(1) (119894119894120596120596119899119899) =

⎩⎪⎨


2120596120596119954119954120582120582120596120596119954119954120582120582prime120575120575119954119954119954119954prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903

minus119881119881120582120582prime120582120582(119954119954)


2lt 120582120582 le 3119903119903

0 else

(521)



119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =

18ℏ2 1198891198891205911205911

120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0lang119879119879120591120591

⎣⎢⎢⎡ 11988111988111989511989511198951198951prime(119948119948120783120783)1198761198761199481199481207831207831198951198951(1205911205911)1198761198761199481199481207831207831198951198951prime

dagger (1205911205911)3119903119903

1198951198951prime=31199031199032 +1

31199031199032

1198951198951=1119948119948120783120783


dagger (1205911205911)3119903119903

1198941198941prime=31199031199032 +1

31199031199032

1198941198941=1119948119948120783120783 ⎦⎥⎥⎤

⎣⎢⎢⎡ 11988111988111989511989521198951198952prime(119948119948120784120784)1198761198761199481199481207841207841198951198952(1205911205912)1198761198761199481199481207841207841198951198952prime

dagger (1205911205912)3119903119903

1198951198952prime=31199031199032 +1

31199031199032

1198951198952=1119948119948120784120784


dagger (1205911205912)3119903119903

1198941198942prime=31199031199032 +1

31199031199032

1198941198942=1119948119948120784120784 ⎦⎥⎥⎤119886119886119954119954120582120582(120591120591)119886119886119954119954prime120582120582prime

dagger (0)rang0119888119888 = 1198921198921 + 1198921198922 + 1198921198923 + 1198921198924

where

1198921198921 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


31199031199032 lt1198951198952

primele311990311990311994811994812078412078411989511989521198951198952prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime



(522)

1198921198922 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


1le1198941198942le31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime



(523)

1198921198923 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


1le1198951198952le31199031199032 lt1198951198952

primele311990311990311994811994812078412078411989511989521198951198952prime

1le1198941198941le31199031199032 lt1198941198941

primele311990311990311994811994812078312078311989411989411198941198941prime



(524)

22

1198921198924 = 18ℏ2 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0


lowast (119948119948120784120784)1le1198941198942le

31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198941198941le31199031199032 lt1198941198941

primele311990311990311994811994812078312078311989411989411198941198941prime



(525)



dagger we can


1198921198922 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)119881119881

11989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032 lt1198941198942

primele311990311990311994811994812078412078411989411989421198941198942prime

1le1198951198951le31199031199032 lt1198951198951

primele311990311990311994811994812078312078311989511989511198951198951prime

times 119868119868 (526)

Where










dagger (0)rang0c










⎩⎪⎨

⎪⎧ 1198681198681 = lang1198791198791205911205911198861198861199481199481207831207831198951198951(1205911205911)1198861198861199481199481207831207831198951198951prime


dagger (0)rang0119888119888







(527)


23












0 (1205911205912 1205911205912)12057512057511989411989421198941198942prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582

+ 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime



0 (1205911205911 1205911205911)12057512057511989511989511198951198951prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582


0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582

= 11986611986611994811994812078312078311989511989510 (1205911205911 1205911205912)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 ∙ 119866119866119954119954prime120582120582prime



0 (1205911205911 0)120575120575119948119948120783120783119954119954prime1205751205751198951198951120582120582prime ∙ 1198661198661199541199541205821205820 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582




























24






























0 (120591120591 1205911205911)120575120575minus1199481199481207831207831199541199541205751205751198951198951120582120582 ∙ 11986611986611994811994812078412078411989411989420 (1205911205912 1205911205912)12057512057511989411989421198941198942prime




0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 ∙ 119866119866minus11994811994812078312078311989511989510 (1205911205911 1205911205911)12057512057511989511989511198951198951prime




11989211989221 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198681198681 =

132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951


1198661198661199541199541205821205820 (120591120591 1205911205911)1205751205751199481199481207831207831199541199541205751205751198951198951prime120582120582 + 1

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198661198661199481199481207831207831198951198951prime

0 (1205911205912 1205911205911)1205751205751199481199481207831207831199481199481207841207841205751205751198941198942prime1198951198951prime ∙


0 (120591120591 1205911205912)1205751205751199481199481207841207841199541199541205751205751198941198942120582120582 =

⎩⎪⎨

⎪⎧

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912) ∙ 119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205912 1205911205911) ∙ 119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582

0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

0 else

and

25

11989211989224 = 132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903

11994811994812078312078311989511989511198951198951prime1198681198684 =

132 int 1198891198891205911205911

120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903




32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum sum

11988111988111989511989511198951198951prime(119948119948120783120783)11988111988111989411989421198941198942prime

lowast (119948119948120784120784)

12059612059611994811994812078312078311989511989511205961205961199481199481207831207831198951198951prime 12059612059611994811994812078412078411989411989421205961205961199481199481207841207841198941198942prime

1le1198941198942le31199031199032lt1198941198942primele3119903119903

11994811994812078412078411989411989421198941198942prime1le1198951198951le

31199031199032lt1198951198951primele3119903119903




0 (1205911205912 1205911205911)12057512057511994811994812078312078311994811994812078412078412057512057511989511989511198941198942 =

⎩⎪⎨

⎪⎧120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime0 (1205911205912 0)119866119866119954119954120582120582

0 (120591120591 1205911205911) 120582120582 120582120582prime isin 1 31199031199032

120575120575119954119954119954119954prime

32 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime

0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

0 else


(314)]



1198921198922 =

⎩⎪⎨

⎪⎧

120575120575119954119954119954119954prime

16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

120575120575119954119954119954119954prime

16 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


lowast (119954119954)


31199031199031198951198951prime=

31199031199032+1 1198661198661199541199541198951198951prime

0 (1205911205912 1205911205911)119866119866119954119954prime120582120582prime0 (1205911205911 0)119866119866119954119954120582120582

0 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

0 else

(528)



calculated as

119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) = 41198921198922 =

⎩⎪⎨

⎪⎧120575120575119954119954119954119954prime

4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1


0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912) 120582120582 120582120582prime isin 1 31199031199032

120575120575119954119954119954119954prime

4 int 1198891198891205911205911120573120573ℏ0 int 1198891198891205911205912

120573120573ℏ0 sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (1205911205911 1205911205912)119866119866119954119954prime120582120582prime

0 (1205911205912 0)1198661198661199541199541205821205820 (120591120591 1205911205911) 120582120582 120582120582prime isin 31199031199032

+ 13119903119903

0 else

(529)



119866119866119954119954120582120582119954119954prime120582120582prime


(2) (119894119894120596120596119899119899)119899119899


(2) (120591120591)120573120573ℏ0

(530)


26

119866119866119954119954120582120582119954119954prime120582120582prime(2) (120591120591) =

120575120575119954119954119954119954prime4

1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1198661198661199541199541198951198951prime0 (1205911205912 1205911205911)119866119866119954119954120582120582prime

0 (1205911205911 0)1198661198661199541199541205821205820 (120591120591 1205911205912)

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1198891198891205911205911120573120573ℏ

0 1198891198891205911205912120573120573ℏ

0

1120573120573ℏ


1198991198992

∙1120573120573ℏ

119890119890minus11989411989412059612059611989911989911205911205911119866119866119954119954120582120582prime0 1198941198941205961205961198991198991

1198991198991

∙1120573120573ℏ

119890119890minus119894119894120596120596119899119899(120591120591minus1205911205912)1198661198661199541199541205821205820 (119894119894120596120596119899119899)

119899119899

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1120573120573ℏ

119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)119866119866119954119954120582120582prime

0 11989411989412059612059611989911989911198661198661199541199541198951198951prime0 1198941198941205961205961198991198992

11989911989911989911989911198991198992⎩⎪⎨

⎪⎧ 1120573120573ℏ

1198891198891205911205911120573120573ℏ

0119890119890minus1198941198941205961205961198991198991minus12059612059611989911989921205911205911

times1120573120573ℏ

1198891198891205911205912120573120573ℏ

0119890119890minus1198941198941205961205961198991198992minus1205961205961198991198991205911205912

⎭⎪⎬

⎪⎫

=120575120575119954119954119954119954prime

4


lowast (119954119954)


3119903119903

1198951198951prime=31199031199032 +1

1120573120573ℏ

119890119890minus1198941198941205961205961198991198991205911205911198661198661199541199541205821205820 (119894119894120596120596119899119899)

11989911989911989911989911198991198992

119866119866119954119954120582120582prime0 11989411989412059612059611989911989911198661198661199541199541198951198951prime

0 1198941198941205961205961198991198992120575120575119899119899111989911989921205751205751198991198991198991198992

=120575120575119954119954119954119954prime120573120573ℏ

119890119890minus119894119894120596120596119899119899120591120591

119899119899

1198661198661199541199541205821205820 (119894119894120596120596119899119899)

14

119881119881120582120582prime1198951198951prime(119954119954



3119903119903

1198951198951prime=31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899)119866119866119954119954120582120582prime

0 (119894119894120596120596119899119899)


119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582

0 (119894119894120596120596119899119899) 120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 119866119866119954119954120582120582prime

0 (119894119894120596120596119899119899) (531)



120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) (532)

and 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) = 119866119866119954119954120582120582

0 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954120582120582prime(2) (119894119894120596120596119899119899) ∙ 119866119866119954119954120582120582prime


+ 13119903119903 we

have

Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) (533)


Σ119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) =

⎩⎪⎪⎨

⎪⎪⎧120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903

2

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

0 else

(534)

27



(1) (119894119894120596120596119899119899) + 119866119866119954119954120582120582119954119954prime120582120582prime(2) (119894119894120596120596119899119899) + ⋯

= 1198661198661199541199541205821205820 (119894119894120596120596119899119899)120575120575119954119954119954119954prime120575120575120582120582120582120582prime + 119866119866119954119954120582120582


0 (119894119894120596120596119899119899) + 1198661198661199541199541205821205820 (119894119894120596120596119899119899) ∙ Σ119954119954120582120582119954119954prime120582120582prime

(2) (119894119894120596120596119899119899)

∙ 119866119866119954119954prime120582120582prime0 (119894119894120596120596119899119899) + ⋯ = 119866119866119954119954120582120582


0 (119894119894120596120596119899119899)

or





(2) (119894119894120596120596119899119899) + ⋯



Σ119954119954120582120582119954119954prime120582120582prime(119894119894120596120596119899119899) =

⎩⎪⎪⎪⎪⎨


120575120575119954119954119954119954prime

2

119881119881120582120582120582120582primelowast (119954119954)

120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582 le 3119903119903

2lt 120582120582prime le 3119903119903

minus120575120575119954119954119954119954prime

2

119881119881120582120582prime120582120582(119954119954)

120596120596119954119954120582120582120596120596119954119954120582120582prime 1 le 120582120582prime le 3119903119903

2lt 120582120582 le 3119903119903

120575120575119954119954119954119954prime

4sum


1205821205821198951198951prime

lowast (119954119954)


31199031199031198951198951prime=

31199031199032 +1

1198661198661199541199541198951198951prime0 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 1 3119903119903

2

120575120575119954119954119954119954prime

4sum


1205961205961199541199541198951198951120596120596119954119954120582120582120596120596119954119954prime120582120582prime

311990311990321198951198951=1

11986611986611995411995411989511989510 (119894119894120596120596119899119899) 120582120582 120582120582prime isin 3119903119903

2+ 13119903119903

(536)









Σ119954119954120582120582119903119903 (120596120596) via [13]

Γ119954119954120582120582(120596120596) = minus2ImΣ119954119954120582120582119954119954120582120582119903119903 (120596120596) (539)


we have

28

Σ119954119954120582120582119954119954120582120582119903119903 (119894119894120596120596119899119899) =

⎩⎪⎨

⎪⎧14sum

1198811198811205821205821198951198951prime(119954119954)

2

1205961205961199541199541198951198951prime120596120596119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

1

120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894

120582120582 isin 1 31199031199032

14sum 1198811198811198951198951120582120582(119954119954)2

1205961205961199541199541198951198951120596120596119954119954120582120582

311990311990321198951198951=1

1120596120596minus1205961205961199541199541198951198951+119894119894119894119894

120582120582 isin 31199031199032

+ 13119903119903

(540)

Using the relation

Im 1

120596120596minus1205961205961199541199541198951198951prime +119894119894119894119894



Γ119954119954120582120582(120596120596) =

⎩⎪⎨

⎪⎧1205871205872sum

1198811198811205821205821198951198951prime(119954119954)

2

1205961205961199541199541198951198951prime120596120596119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

120575120575120596120596 minus 1205961205961199541199541198951198951prime 120582120582 isin 1 31199031199032

1205871205872sum 1198811198811198951198951120582120582(119954119954)2

1205961205961199541199541198951198951120596120596119954119954120582120582

311990311990321198951198951=1

120575120575120596120596 minus 1205961205961199541199541198951198951 120582120582 isin 31199031199032

+ 13119903119903

(542)

Or equivalently

Γ119954119954120582120582(119864119864 = ℏ120596120596) =

⎩⎪⎨

⎪⎧120587120587ℏ3

2sum

1198811198811205821205821198951198951prime(119954119954)

2

1198641198641199541199541198951198951prime119864119864119954119954120582120582

31199031199031198951198951prime=

31199031199032 +1

120575120575119864119864 minus 1198641198641199541199541198951198951prime 120582120582 isin 1 31199031199032

120587120587ℏ3

2sum 1198811198811198951198951120582120582(119954119954)2

1198641198641199541199541198951198951119864119864119954119954120582120582

311990311990321198951198951=1

120575120575119864119864 minus 1198641198641199541199541198951198951 120582120582 isin 31199031199032

+ 13119903119903

(543)








int(⋯ )120588120588119864119864119954119954d119864119864119954119954 giving

Γ119954119954120582120582(119864119864) = 120587120587ℏ3


2119864119864119954119954119895119895prime=119864119864119954119954prime


2120588120588(119864119864)

119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864

119864119864119864119864119954119954120582120582 (544)







29


Γtot = 1119885119885119890119890minus120573120573119864119864119954119954120582120582Γ119954119954120582120582119864119864119954119954120582120582

119954119954120582120582

=120587120587ℏ3

2

119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582 119881119881120582120582119895119895prime(119954119954)2119864119864119954119954119895119895prime=119864119864119954119954120582120582

1198641198641199541199541205821205822119954119954120582120582

=120587120587ℏ3

2

119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582 119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582

Finally we get

Γtot = Γ119954119954120582120582(119864119864) =120587120587ℏ3

2sum 119890119890minus120573120573119864119864119954119954120582120582

119885119885

120588120588119864119864119954119954120582120582119881119881120582120582120582120582+31199031199032(119954119954)

2

1198641198641199541199541205821205822119954119954120582120582 (545)





to make



References


MainText


SI

1 Methodology

11 Model system












11 Basic notations

12 Dynamical matrix


















controllable t hermal conductivity in twisted homogeneous

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