cumulant green’s function approach for excited state and...
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Cumulant Green’s function approach for excited state and thermodynamic properties of cool to warm dense matter
J. J. Rehr & J. J. Kas
University of Washington and SLAC
HoW exciting! Workshop Humboldt University Berlin 7 August, 2018
Inelastic losses and many-body effects in x-ray spectra
• TALK: Cumulant Green’s function approach When and why you might need to go beyond DFT & quasi-particle approximations in excited states and x-ray spectra
I. Introduction Quasi-particle theory & GW approx II. Inelastic losses & satellites Cumulant Green’s functions III. Finite- T effects Exchange-correlation in spectra & thermodynamics
Cumulant Green’s function approach for excited state and thermodynamic properties of cool to warm dense matter
You can tell the quality of a many-body theory by how it treats the satellites L. Hedin
● Core-hole Vc Excitonic effects, screening
● Self-energy Σ(E) Mean-free path, self-energy shifts
● Excitations ωp Inelastic losses, satellites
● Debye-Waller σ2 Thermal vibrations
Motivation: Many-body effects in X-ray spectra
QP
Beyond QP
Mini-review
Theory of X-ray and optical spectra
ca 2009
JJR et al., Comptes Rendus Physique 10, 548 (2009)
Theoretical Spectroscopy L. Reining, (Ed, 2009)
Golden rule via Green’s Functions G = 1/( E – h′ – Σ )
Golden rule for XAS via Wave Functions
Ψ Paradigm shift – quasiparticle approximation
Many body effects included in self energy Σ Σ(E) replaces Vxc of DFT εk = εk
0 + Σk
One-electron Green’s function theory of XAS
Many-pole GW Self-energy Σ(E)*
Extension of Hedin-Lundqvist GW plasmon-pole approx Sum of plasmon-pole models
matched to loss function
*J.J. Kas et. al, Phys Rev B 76, 195116 (2007)
cf. J. Gesenhues, D. Nabok, M. Rohlfing, and C. Draxl, Phys. Rev. B 96, 245124 (2017)
LiF loss fn
Efficient GW method
Σ(E)= iGW = Σ′ - i Γ
Approximate many-pole GW self energy
*J. J. Kas, J. Vinson, N. Trcera, D. Cabaret, E. L. Shirley, and J. J. Rehr, Journal of Physics: Conference Series 190, 012009 (2009)
µ(E)
(arb
u.)
E (eV)
Self-energy shift
MgAl2O4 DFT
Example: Quasi-particle GW self-energy Σ(E)
Mean-free path damping
II. Beyond QP: Satellites/multi-electron excitations
Importance of satellites in XAS & XPS
XAS
Satellite peaks
Reduction in peak height
CoO
G(ω) = G0+ G0 Σ G G(t) = G0 (t) eC(t) GW or QPGW Cumulant*
ΣGW =iGW
Spectral function Ak=δ(ω- εk)
C ~ | Im ΣGW| ~ | Im W |
How ? Green’s function approach - which GF ?
Better for QP & satellites Ak= -(1/π) Im Gk(ω)
Reviews: Cumulant expansion
Cumulant Green’s function in time domain
Cumulant Green’s Function Formalism*
* L. Hedin, J. Phys.: Condens. Matter 11 R489 (1999) J.J. Kas, J. J. Rehr, and L. Reining, Phys. Rev. B 90, 085112 (2014)
Natural separation of QP, exchange, & correlation parts
Landau cumulant (1944)
Spe
ctra
l fun
ctio
n
Theorem:* Cumulant representation of core-hole Green’s function is EXACT* for core electrons coupled to bosons Cumulant formalism represents a mapping between e-e interactions e-boson couplings
IDEA: Neutral excitations (plasmons etc) are bosons
*D. C. Langreth, Phys. Rev. B 1, 471 (1970)
Why does it work: Quasi-boson approximation
Multiple Satellites Quasiparticle peaks
Lucia Reining
Problems: GW: one broad satellite at the wrong place 1-e Cumulant: multiple satellites BUT intensity too small
Si XPS
Results: multiple satellites in XPS of Si
Beyond one-particle theory: calculations with particle-hole excitations and all inelastic losses
Extrinsic + Intrinsic - 2 x Interference
+ - -
Improved theory: particle-hole cumulant
Satellite strengths XAS of Al
Particle-hole cumulant explains cancellation of extrinsic and intrinsic losses at threshold AND crossover: adiabatic to sudden approximation
Extrinsic, intrinsic and interference terms
S02 = 1- total
Example: particle-hole cumulant in XAS
LiF: F K edge
*J. Vinson et al. Phys. Rev. B83, 115106 (2011); K. Gilmore et al. CPC 197,109 (2015)
Exp
OCEAN
FEFF9
Implementation: GW/BSE code OCEAN*
Particle-hole Green’s function approach
XPS
F. Fossard, K. Gilmore, G. Hug, J J. Kas, J J Rehr, E L Shirley and F D Vila
Phys Rev B 95,115112(2017)
Example: high accuracy XPS & XAS
QP peak
Satellites - OCEAN
EELS ~ XAS
BSE + particle-hole cumulant
Langreth cumulant in time-domain* (RT-TDDFT)
TiO2
*D. C. Langreth, Phys. Rev. B 1, 471 (1970)
Example: Real-time Cumulant for TMOs
CT satellite
RT TDDFT Cumulant Theory vs XPS
Interpretation: satellites arise from charge density fluctuations between ligand and metal at frequency ωCT due to suddenly turned-on core-hole
Charge transfer fluctuations
ωct
Real-space interpretation of CT satellites
TiO2
TiO2
Ce L3 XAS of CeO2
Spectral function
Spectral weights
Ce 5s XPS of CeO2
Example: f-electron system: CeO2*
*J. Kas et al. Phys Rev B 94, 035156 (2016)
III. Finite-T cumulant Green’s function
Need methods beyond Finite-T DFT1
Phys Rev Lett 109, 176403 (2017)
Motivation: Interest in excited states & thermodynamics at finite-T and extreme conditions (WDM) T ~ TF
1FT DFT: N.D. Mermin, Phys. Rev. 137, 1 (1965); FT DFT functionals V.V. Karasiev et al. Phys. Rev. Lett. 112, 076403 (2014)
Finite T occupation nk
Theory: Martin & Schwinger 1959
Loss-function: broadened & blue-shifted at high T
Finite-T RPA dielectric function
Finite-T dielectric & loss functions
~
fk=1/(e β(εk -μ) +1)
FT Quasi-particle energy and damping
Classical limit: Σ 0 reached for εk at high T BUT band gaps and band-structure are blurred, short-ranged metallic behavior
Damping Δ″ INCREASES Self-energy Δ decreases
Finite-T Spectral-function
… GW-approx Single asymmetric peak at high T Breakdown of QP approx - smeared out band structure - short ranged propagators
FT Exchange-correlation energy and potentials
Galitskii-Migdal-Koltun sum rule* ε(T) = E/N
Good agreement for εxc and Vxc with PIMC & FT DFT functionals *P. Martin and J. Schwinger, Phys. Rev. 115, 1342 (1959)
= εH + εxc
Finite-temperature Compton Profile*
*W. Schulke, G. Stutz, F.Wohlert, and A. Kaprolat, Phys. Rev. B 54, 14381 (1996)
Many-body effects give “effective temp” T* correction
Comparison of cumulant (solid) & free electron (dotted)
Suggested thermometer for WDM
Crossover: Exchange vs correlation energy
εx
εc
Exchange decreases with T ; correlation dominates at high T
FT Exchange-correlation energy and free energy
PIMC PIMC
FT cumulant FT cumulant
fxc
εxc
Finite-temperature TDDFT
*K. Burke, et al., Phys. Rev. B 93, 195132; Phys. Rev. Lett. 116, 233001 (2016).
FT-TDDFT fxc /rs2
Finite-T COHSEX Approximation
= poles of W + poles of G
COHSEX GW ● DFT COHSEX accurate to ~ 10% r s
= 1
2
3 4
Theory beyond DFT & QP essential for x-ray spectra
QP δ(ω-εk) Spectral function: Ak(ω) Particle-hole cumulant explains QP and satellite effects: Finite T cumulant Green’s function yields excited states and thermodynamic properties from cool to WDM High T physics: short-ranged & correlation dominated
Conclusions
Supported by DOE BSE DE-FG02-97ER45623 and TIMES @ SLAC Special thanks to L. Reining G. Bertsch E. Shirley J. Vinson K. Gilmore J. Sky Zhou F. Vila S. Story T. Blanton M. Guzzo M. Verstraete Tun Tan F. Aryasetiwan T. Fujikawa C. Draxl
Acknowledgments
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