Cumulant Green s function approach for excited state and thermodynamic properties of cool to warm dense matter

Transcription

1 HoW exciting! Workshop Humboldt University Berlin 7 August, 2018 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

2 Inelastic losses and many-body effects in x-ray spectra Cumulant Green s function approach for excited state and thermodynamic properties of cool to warm dense matter 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

3 You can tell the quality of a many-body theory by how it treats the satellites L. Hedin

4 Motivation: Many-body effects in X-ray spectra Core-hole V c Excitonic effects, screening Self-energy Σ(E) Mean-free path, self-energy shifts Excitations ω p Debye-Waller σ 2 Inelastic losses, satellites Thermal vibrations QP Beyond QP

5 Mini-review Theory of X-ray and optical spectra ca 2009 Theoretical Spectroscopy L. Reining, (Ed, 2009) JJR et al., Comptes Rendus Physique 10, 548 (2009)

6 One-electron Green s function theory of XAS Golden rule for XAS via Wave Functions Ψ Paradigm shift quasiparticle approximation Golden rule via Green s Functions G = 1/( E h Σ ) Many body effects included in self energy Σ Σ(E) replaces V xc of DFT ε k = ε k 0 + Σ k

7 Approximate many-pole GW self energy Many-pole GW Self-energy Σ(E)* Extension of Hedin-Lundqvist GW plasmon-pole approx LiF loss fn Sum of plasmon-pole models matched to loss function Σ(E)= igw = Σ - i Γ Efficient GW method *J.J. Kas et. al, Phys Rev B 76, (2007) cf. J. Gesenhues, D. Nabok, M. Rohlfing, and C. Draxl, Phys. Rev. B 96, (2017)

8 Example: Quasi-particle GW self-energy Σ(E) DFT MgAl 2 O 4 µ(e) (arb u.) Self-energy shift Mean-free path damping E (ev) *J. J. Kas, J. Vinson, N. Trcera, D. Cabaret, E. L. Shirley, and J. J. Rehr, Journal of Physics: Conference Series 190, (2009)

9 II. Beyond QP: Satellites/multi-electron excitations Importance of satellites in XAS & XPS XAS CoO Reduction in peak height Satellite peaks

10 How? Green s function approach - which GF? GW or QPGW G(ω) = G 0 + G 0 Σ G Σ GW =igw Cumulant* G(t) = G 0 (t) e C(t) C ~ Im Σ GW ~ Im W Spectral function A k =δ(ω- ε k ) Better for QP & satellites A k = -(1/π) Im G k (ω)

11 Reviews: Cumulant expansion

12 Cumulant Green s Function Formalism* Cumulant Green s function in time domain Natural separation of QP, exchange, & correlation parts Landau cumulant (1944) Spectral function * L. Hedin, J. Phys.: Condens. Matter 11 R489 (1999) J.J. Kas, J. J. Rehr, and L. Reining, Phys. Rev. B 90, (2014)

13 Why does it work: Quasi-boson approximation IDEA: Neutral excitations (plasmons etc) are bosons 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 *D. C. Langreth, Phys. Rev. B 1, 471 (1970)

14 Results: multiple satellites in XPS of Si Si XPS Multiple Satellites Quasiparticle peaks Lucia Reining Problems: GW: one broad satellite at the wrong place 1-e Cumulant: multiple satellites BUT intensity too small

15 Improved theory: particle-hole cumulant Beyond one-particle theory: calculations with particle-hole excitations and all inelastic losses Extrinsic + Intrinsic - 2 x Interference

16 Extrinsic, Example: intrinsic particle-hole and interference cumulant terms XAS XAS of Al Satellite strengths S 0 2 = 1- total Particle-hole cumulant explains cancellation of extrinsic and intrinsic losses at threshold AND crossover: adiabatic to sudden approximation

17 Implementation: GW/BSE code OCEAN* Particle-hole Green s function approach LiF: F K edge Exp OCEAN FEFF9 *J. Vinson et al. Phys. Rev. B83, (2011); K. Gilmore et al. CPC 197,109 (2015)

18 Example: high accuracy XPS & XAS Phys Rev B 95,115112(2017) BSE + particle-hole cumulant F. Fossard, K. Gilmore, G. Hug, J J. Kas, J J Rehr, E L Shirley and F D Vila XPS EELS ~ XAS QP peak Satellites - OCEAN

19 Example: Real-time Cumulant for TMOs Langreth cumulant in time-domain* (RT-TDDFT) TiO 2 CT satellite *D. C. Langreth, Phys. Rev. B 1, 471 (1970)

20 Real-space interpretation of CT satellites RT TDDFT Cumulant Theory vs XPS Charge transfer fluctuations TiO 2 ω ct TiO 2 Interpretation: satellites arise from charge density fluctuations between ligand and metal at frequency ω CT due to suddenly turned-on core-hole

21 Example: f-electron system: CeO 2 * Ce 5s XPS of CeO 2 Spectral function Ce L 3 XAS of CeO 2 Spectral weights *J. Kas et al. Phys Rev B 94, (2016)

22 III. Finite-T cumulant Green s function Phys Rev Lett 109, (2017) Motivation: Interest in excited states & thermodynamics at finite-t and extreme conditions (WDM) T ~ T F Finite T occupation n k Need methods beyond Finite-T DFT 1 1 FT DFT: N.D. Mermin, Phys. Rev. 137, 1 (1965); FT DFT functionals V.V. Karasiev et al. Phys. Rev. Lett. 112, (2014)

23 Theory: Martin & Schwinger 1959

24 Finite-T dielectric & loss functions Finite-T RPA dielectric function f k =1/(e β(ε k -μ) +1) Loss-function: broadened & blue-shifted at high T ~

25 FT Quasi-particle energy and damping Self-energy Δ decreases Damping Δ INCREASES Classical limit: Σ 0 reached for ε k at high T BUT band gaps and band-structure are blurred, short-ranged metallic behavior

26 Finite-T Spectral-function GW-approx Single asymmetric peak at high T Breakdown of QP approx - smeared out band structure - short ranged propagators

27 FT Exchange-correlation energy and potentials Galitskii-Migdal-Koltun sum rule* ε(t) = E/N = ε H + ε xc Good agreement for ε xc and V xc with PIMC & FT DFT functionals *P. Martin and J. Schwinger, Phys. Rev. 115, 1342 (1959)

28 Finite-temperature Compton Profile* Suggested thermometer for WDM Comparison of cumulant (solid) & free electron (dotted) Many-body effects give effective temp T* correction *W. Schulke, G. Stutz, F.Wohlert, and A. Kaprolat, Phys. Rev. B 54, (1996)

29 Crossover: Exchange vs correlation energy ε x ε c Exchange decreases with T ; correlation dominates at high T

30 FT Exchange-correlation energy and free energy PIMC PIMC FT cumulant FT cumulant f xc ε xc

31 Finite-temperature TDDFT FT-TDDFT f xc /r s 2 *K. Burke, et al., Phys. Rev. B 93, ; Phys. Rev. Lett. 116, (2016).

32 Finite-T COHSEX Approximation = poles of W + poles of G COHSEX accurate to ~ 10% r s = COHSEX GW DFT

33 Conclusions Theory beyond DFT & QP essential for x-ray spectra QP δ(ω-ε k ) Spectral function: A k (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

34 Acknowledgments Supported by DOE BSE DE-FG02-97ER45623 and 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