Towards an Electronic Structure Method for Correlated Electron Systems based on Dynamical

Mean Field Theory G. Kotliar

Physics Department and Center for Materials Theory

Rutgers

Second International Workshop 2004Ordering Phenomena in Transition Metal Oxides

September 26 - 29, 2004, Wildbad Kreuth

Outline

• Can we build a controlled but practical many body scheme, for first principles electronic structure calculations of correlated solids ?

• Application : electronic and elastic properties of Pu, a strongly correlated element.

• Collaborators S. Savrasov (NJIT) and N. Zein (Kurchatov-Rutgers)

Density functional and Kohn Sham reference system

2 / 2 ( ) KS kj kj kjV r y e y- Ñ + =

( ')( )[ ( )] ( ) ' [ ]

| ' | ( )xc

KS ext

ErV r r V r dr

r r r

drr r

dr= + +

-ò

2( ) ( ) | ( ) |kj

kj kjr f rr e y=å

[ ]

•Kohn Sham spectra, proved to be an excelent starting point for doing perturbatio theory in

screened Coulomb interactions GW.

= W

= [ - ]-11CV

= G

- [ - ]KS crystV V10KSG 1G

GW approximation (Hedin )

LDA+GW: semiconducting gaps

Strongly Correlated Materials• Can we construct a conceptual framework and

computational tools, for studying strongly correlated materials, which will be as successful as the Fermi Liquid –LDA-GW program ?

• Is a local perspective reasonable ? accurate ?• Dynamical Mean Field Theory . Unify band

theory and atomic physics. Use an impurity model, (local degrees of freedom + free electron enviroment ) to describe the local spectra of a correlated system.

Two paths for ab-initio calculation of electronic

structure of strongly correlated materials

Correlation Functions Total Energies etc.

Model Hamiltonian

Crystal structure +Atomic positions

DMFT ideas can be used in both cases.

[V. Kancharla C. Bolech and GK PRB 67, 075110 (2003)][[M.CaponeM.Civelli V Kancharla C.Castellani and GK P. R B 69,195105 (2004) ]

U/t=4.

Testing CDMFT (G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys.

Rev. Lett. 87, 186401 (2001) ) with two sites in the Hubbard model in one dimension.

Comparison with other cluster methods.

Two paths for ab-initio calculation of electronic

structure of strongly correlated materials

Correlation Functions Total Energies etc.

Model Hamiltonian

Crystal structure +Atomic positions

DMFT ideas can be used in both cases.

Functional formulation. Chitra and Kotliar Phys. Rev. B 62, 12715 (2000)

and Phys. Rev.B (2001) . 

1 †1( ) ( , ') ( ') ( ) ( ) ( )

2Cx V x x x i x x xff f y y-+ +òò ò

†( ') ( )G x xy y=- < > ( ') ( ) ( ') ( )x x x x Wff ff< >- < >< >=

Ex. Ir>=|R, > Gloc=G(R, R ’) R,R’’

1 10

1 1[ , , , ] [ ] [ ] [ ] [ ] [ , ]

2 2C hartreeG W M P TrLn G M Tr G TrLn V P Tr P W E G W

Introduce Notion of Local Greens functions, Wloc, Gloc G=Gloc+Gnonloc .

Sum of 2PI graphs[ , ] [ , , 0, 0]EDMFT loc loc nonloc nonlocG W G W G W

One can also view as an approximation to an exact Spetral Density Functional of Gloc and Wloc.

1 10

1 10

loc

loc

G

W

G M

V P1

1

( )

1

( )

lock

locq C

GH k

Wv q

M

P

0 0G V,,intM PLocal Impurity Model

Input: ,M P

Output: Self-Consistent Solution

Spectral Density Functional Theory withinLocal Dynamical Mean Field Approximation

,loc locG W 1 10

1 10

loc

loc

G

W

G M

V P1

1

( )

1

( )

lock

locq C

GH k

Wv q

M

P

0 0G V,,intM PLocal Impurity Model

Input: ,M P

Output: Self-Consistent Solution

Spectral Density Functional Theory withinLocal Dynamical Mean Field Approximation

•Full implementation in the context of a a one orbital model. P Sun and G. Kotliar Phys. Rev. B 66, 85120 (2002).

•After finishing the loop treat the graphs involving Gnonloc Wnonloc in perturbation theory. P.Sun and GK PRL (2004). Related work, Biermann Aersetiwan and Georges PRL 90,086402 (2003) .

 

EDMFT loop G. Kotliar and S. Savrasov in New Theoretical Approaches to Strongly Correlated G Systems, A. M. Tsvelik Ed. 2001 Kluwer Academic Publishers. 259-301 . cond-mat/0208241 S. Y. Savrasov, G. Kotliar, Phys. Rev. B 69, 245101 (2004)

Further Approximations. o The light, SP (or SPD) electrons are extended, well described by LDA .The heavy,

d(or f) electrons are localized treat by DMFT.LDA Kohn Sham Hamiltonian already contains an average interaction of the heavy electrons, subtract this out by shifting the heavy level (double counting term) .

o Truncate the W operator act on the H sector only. i.e.

• Replace W() by a static U. This quantity can be estimated by a constrained LDA calculation or by a GW calculation with light electrons only. e.g.

M.Springer and F.Aryasetiawan,Phys.Rev.B57,4364(1998) T.Kotani,J.Phys:Condens.Matter12,2413(2000). FAryasetiawan M Imada A Georges G Kotliar S Biermann and A Lichtenstein cond-matt (2004)

( , ', ) ( ') ( ) ( )( ( ) ) ( ')dcxc R H R Rr r r r V r r E rabe a ab bw d f w fS = - - S S -

( , ', ) ( ) ( ) ( ) ( ') ( ')R H R R R RW r r r r W r rabgde a b abgd g dw ff wff=S

or the U matrix can be adjusted empirically.• At this point, the approximation can be derived

from a functional (Savrasov and Kotliar 2001)

• FURTHER APPROXIMATION, ignore charge self consistency, namely set

LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997) See also . A Lichtenstein and M. Katsnelson PRB 57, 6884 (1988).

Reviews:Held, K., I. A. Nekrasov, G. Keller, V. Eyert, N. Blumer, A. K. �McMahan, R. T. Scalettar, T. Pruschke, V. I. Anisimov, and D. Vollhardt, 2003, Psi-k Newsletter #56, 65.

• Lichtenstein, A. I., M. I. Katsnelson, and G. Kotliar, in Electron Correlations and Materials Properties 2, edited by A. Gonis, N. Kioussis, and M. Ciftan (Kluwer Academic, Plenum Publishers, New York), p. 428.

• Georges, A., 2004, Electronic Archive, .lanl.gov, condmat/ 0403123 .

loc[ ]G

[ ] [ ]LDAVxc Vxc

Further approximations, use approximate impuirity solvers rational Interpolative Perturbative Theory. Savrasov Udovenko Villani Haule

and Kotliar . Cond-matt 0401539

Pu in the periodic table

actinides

Small amounts of Ga stabilize the phase (A. Lawson LANL)

Delta phase of Plutonium: Problems with LDA

o Many implementations.(Freeman, Koelling 1972)APW , ASA and FP-LMTO Soderlind et.al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999).all give an an equilibrium volume of the equilibrium volume of the phasephaseIs 35% lower Is 35% lower than experiment than experiment this is the largest discrepancy ever known in DFT based calculations. Negative shear modulus (Bouchet et. al.).

• LSDA predicts magnetic long range (Solovyev et.al.) Experimentally Pu is not magnetic.

• If one treats the f electrons as part of the core LDA overestimates the volume by 30%

Total Energy as a function of volume for Total Energy as a function of volume for Pu Pu W (ev) vs (a.u. 27.2 ev)

(Savrasov, Kotliar, Abrahams, Nature ( 2001)Non magnetic correlated state of fcc Pu.

iw

Zein Savrasov and Kotliar (2004)

Phonon Spectra

• Electrons are the glue that hold the atoms together. Vibration spectra (phonons) probe the electronic structure.

• Phonon spectra reveals instablities, via soft modes.

• Phonon spectrum of Pu had not been measured.

Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003

DMFT Phonons in fcc DMFT Phonons in fcc -Pu-Pu

  C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa)

Theory 34.56 33.03 26.81 3.88

Experiment 36.28 33.59 26.73 4.78

( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)

(experiments from Wong et.al, Science, 22 August 2003)

Epsilon Plutonium.

Phonon entropy drives the epsilon delta phase transition

• Epsilon is slightly more delocalized than delta, has SMALLER volume and lies at HIGHER energy than delta at T=0. But it has a much larger phonon entropy than delta.

• At the phase transition the volume shrinks but the phonon entropy increases.

• Estimates of the phase transition following Drumont and G. Ackland et. al. PRB.65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.

Transverse Phonon along (0,1,1) in epsilon Pu in self

consistent Born approximation.

Outline

• Can we build a controlled but practical many body scheme, for first principles electronic structure calculations of correlated solids ?

• Application : electronic and elastic properties of Pu, a strongly correlated element.

Conclusions/Questions/Discussion

• Is self consistency important ? [Or can we derive accurate model Hamiltonians, how do we separate scales ]

• what is currently limiting the accuracy of a given calculation. Is the neglect of a frequency dependent in U (W) ? Do we need to improve in the treatment of the self energy of the light electrons ? What is the minimal scale that is needed to take into account for a given material. How large of a cluster to use ? How about the impurity solver ? And the basis set ?

Outline

• Can we build a controlled but practical many body scheme, for first principles electronic structure calculations of correlated solids ?

• Application : electronic and elastic properties of Pu, a strongly correlated element.

Conclusion

• Spectral Density Functional. Connection between spectra and bonding. Microscopic theory of Pu, connecting its anomalies to the vicinity of a Mott point.

• Combining theory and experiment we can more than the sum of the parts. Next step in Pu, much better defined problem, discrepancy in (111 ) zone boundary, may be due to either the contribution of QP resonance, or the inclusion of nearest neighbor correlations. Both can be individually studied.

• Also needed, more experiment. Recent Neutron scattering .

Further approximations r ational Interpolative Perturbative Theory. Savrasov Udovenko Villani Haule and Kotliar . Cond-matt 0401539

cluster cluster exterior exteriorH H H H

H clusterH

Simpler "medium" Hamiltonian

cluster exterior exteriorH H

CDMFT G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001

Site Cell. Cellular DMFT. C-DMFT. G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001)

tˆ(K) hopping expressed in the superlattice notations.

•Other cluster extensions (DCA Jarrell Krishnamurthy, Katsnelson and Lichtenstein periodized scheme, Nested

Cluster Schemes , causality issues, O. Parcollet, G. Biroli and GK cond-matt 0307587 .

LDA+DMFT Self-Consistency loop. S. Savrasov and G. Kotliar

(2001) and cond-matt 0308053

G0 G

Im p u rityS o lver

S .C .C .

0( ) ( , , ) i

i

r T G r r i e w

w

r w+

= å

2| ( ) | ( )k xc k LMTOV H ka ac r c- Ñ + =

DMFT

U

E

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å

LDA+DMFT Formalism : V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359-7367 (1997). S. Y. Savrasov and G. Kotliar, Phys. Rev. B 69, 245101 (2004). V. Udovenko S. Savrasov K. Haule and G. Kotliar

Cond-mat 0209336

Comments on LDA+DMFT

• Gives the local spectra and the total energy simultaneously, treating QP and H bands on the same footing.

• Gives an approximate starting point, for perturbation theory in the non local part of the Coulomb interactions. [See for example, P. SunPhys. Rev. Lett. 92, 196402 (2004)]

• Good approximate starting point for optics.

• Topics for discussion. Test model Hamiltonian approaches. Could it be that

• Adjusting one parameter we get just one

• Quantity.

K. Haule , Pu- photoemission with DMFT using vertex corrected NCA.

Benchmarking SUNCA, V. Udovenko and K. Haule

Interpolative scheme with slave bosons.

Specific Heat Resistivity susceptibility.

Functional formulation. Chitra and Kotliar Phys. Rev. B 62, 12715 (2000)

and Phys. Rev.B (2001) . 

1 †1( ) ( , ') ( ') ( ) ( ) ( )

2Cx V x x x i x x xff f y y-+ +òò ò

†( ') ( ')G R Ry r y r=- < > ( ' ') ( ) ( ' ') ( )R R R R Wf r f r f r f r< >- < >< >=

Ir>=|R, >1 1

0

1 1[ , , , ] [ ] [ ] [ ] [ ] [ , ]

2 2C hartreeG W M P TrLn G M Tr G TrLn V P Tr P W E G W

Sum of all 2 PI graphs.

• Experimentally delta Pu stable. It has a negative coefficient of thermal expansion.

• Delta Pu has the largest shear anisotropy of all elements , in spite of its cubic fcc structure,fcc Al, c44/c’=1.2, in Pu C44/C’ ~ 6.

Functional formulation. Chitra and Kotliar Phys. Rev. B 62, 12715 (2000)

and Phys. Rev.B (2001) . 

1 †1( ) ( , ') ( ') ( ) ( ) ( )

2Cx V x x x i x x xff f y y-+ +òò ò

†( ') ( ')G R Ry r y r=- < > ( ' ') ( ) ( ' ') ( )R R R R Wf r f r f r f r< >- < >< >=

Ir>=|R, >1 1

0

1 1[ , , , ] [ ] [ ] [ ] [ ] [ , ]

2 2C hartreeG W M P TrLn G M Tr G TrLn V P Tr P W E G W

Sum of all 2 PI graphs.

• Introduce localized basis set (e.g. LMTO’s)

• Make local (i.e. cluster approximation)

• Treat non local pieces in perturbation theory.

( ) ( ) ( ) ( )RR R RG r r i G i r r

1 2 3 4 1 2 3 4( ) ( ) ( ) ( ) ( ) ( )R R R R R R R RW r r i W i r r r r

[ , ] [ , , 0, 0]EDMFT loc loc nonloc nonlocG W G W G W

DMFT eqs for Gloc and Wloc

LL HL

LH HH

H H

H H

© ¬ª ª ª « ®

Functional formulation. Chitra and Kotliar Phys Rev B62, 12715 (2000) and Phys. Rev. B Phys. Rev. B 63, 115110 (2001).

1 †1( ) ( , ') ( ') ( ) ( ) ( )

2Cx V x x x i x x xff f y y-+ +òò ò

†( , ') ( ' ') ( )G r r R Ry r y r=- < > ( ') ( ) ( ') ( ) ( , ')R R R R W r rf r f r f r f r< >- < >< >=

Ir>=|R, >1 1 1 1

0

1 1[ , ] [ ] [ ] [ , ]

2 2 C hartreeG W TrLnG Tr G G G TrLnW Tr V W W E G W

Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev.

E = Ei - EfQ =ki - kf

Experiments . Joe Wong, Michael Krisch, Daniel L. Farber, Florent Occelli, Adam J. Schwartz, Tai-C. Chiang, Mark Wall Carl Boro Ruqing Xu, Science, Vol 301,

Issue 5636, 1078-1080 , 22 August 2003.

Alpha and delta Pu

LDA+DMFT functional

2 *log[ / 2 ( ) ( )]

( ) ( ) ( ) ( )

1 ( ) ( ')( ) ( ) ' [ ]

2 | ' |

[ ]

R R

n

n KS

KS n n

i

LDAext xc

DC

R

Tr i V r r

V r r dr Tr i G i

r rV r r dr drdr E

r r

G

a b ba

w

w c c

r w w

r rr r

- +Ñ - - S -

- S +

+ + +-

F - F

åò

ò òå

Sum of local 2PI graphs with local U matrix and local G

1[ ] ( 1)

2DC G Un nF = - ( )0( ) iab

abi

n T G i ew

w+

= å

KS ab [ ( ) G V ( ) ]LDA DMFT a br r

LDA+DMFT Self-Consistency loop

G0 G

Im p u rityS o lver

S .C .C .

0( ) ( , , ) i

i

r T G r r i e w

w

r w+

= å

2| ( ) | ( )k xc k LMTOV H ka ac r c- Ñ + =

DMFT

U

E

0( , , )HHi

HH

i

n T G r r i e w

w

w+

= å

Electronic Structure of Materials

• Issues of principle.• How do we think bout the electronic excitations

of a given material ? • Is it possible to obtain practical computational

schemes of the free energies and excitations of a material ? What quantities do we have a reasonable expectations to be able to compute ? with which methods ?

• Issues of implementation.