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