Theoretical spectroscopy beyond quasiparticles

A direct approach to the calculation of many-body Green s functions: Theoretical spectroscopy beyond quasiparticles Lucia Reining Palaiseau Theoretical Spectroscopy Group

Palaiseau Theoretical Spectroscopy Group & friends Jianqiang Zhou, Igor Reshetnyak, Claudia Roedl, Matteo Guzzo, Francesco Sottile, Matteo Gatti, Lucia Reining U. Washington: John Rehr, Joshua Kas Synchrotron SOLEIL: Fausto Sirotti, Matthieu Silly Synchrotron ESRF/U. Helsinki: Simo Huotari http://etsf.polytechnique.fr http://www.etsf.eu

A direct approach to the calculation of many-body Green s functions: Theoretical spectroscopy beyond quasiparticles Photoemission: quasiparticles and more Electron-boson coupling Electron-boson coupling in GW Self-energies and more Electron-boson coupling using a cumulant G Which boson?

Photoemission From http://www.ieap.uni-kiel.de/surface/ag-kipp/arpes/arpes.htm

Band structure of germanium LDA Exp Hartree-Fock Rohlfing et al., PRB 48, 17791 (1993) A. Svane, PRB 35, 5496 (1987) Picture from R.M. Martin. L. Reining, D.M. Ceperley Interacting Electrons: Theory and Computational Approaches, Cambridge University Press 2016

G(1,2) = -i <T[ (2)]> 1=(r1, 1,t1) Dyson equation: G =G0 + G0 G 12-37

+... ~ i WG GW L. Hedin (1965) W = ( v -1

Band structure of germanium Exp and GW LDA Rohlfing et al., PRB 48, 17791 (1993) A. Svane, PRB 35, 5496 (1987) Hartree-Fock

Couples to excitations!

Photoemission From Damascelli et al., RMP 75, 473 (2003) and http://www.ieap.uni-kiel.de/surface/ag-kipp/arpes/arpes.htm More than quasi-particles... +...

Electron boson coupling Sky J. Zhou, PhD thesis Palaiseau 2016

Bulk Silicon Valence bands Cohen and Chelikowsky: Electronic Structure and Optical Properties of Semiconductors Solid-State Sciences 75, Springer-Verlag 1988) Exp.: F. Sirotti et al., TEMPO beamline SOLEIL

A. Fujimori et al., PRL 69, 1796 (1992)

Convincing description in terms of electron-boson coupling... IF we have the boson we solve the coupled problem reasonably well

Electron boson coupling in GW +... ~ i WG GW L. Hedin (1965) W = ( v -1

T. C. Koethe et al., Phys. Rev. Lett. 97, 116402 (2006).

T. C. Koethe et al., Phys. Rev. Lett. 97, 116402 (2006). In GW: M. Gatti, F. Bruneval, V. Olevano and L. Reining, Phys. Rev. Lett. 99, 266402 (2007)

Screening (dynamical): W Metal Insulator

Gatti, Panaccione, Reining (PRL 2015)

Silicon EXPO Satellites Valence bands Cohen and Chelikowsky: Electronic Structure and Optical Properties of Semiconductors Solid-State Sciences 75, Springer-Verlag 1988) M. Guzzo et al., PRL 107, 166401 (2011)

Self energies and more Want: Want: G = 1/( = [G0 ]-1 - G-1 = ( G = a/( a /( = [G0 ]-1 - G-1 = ( a (1 a) a a a Poles in G must be physical Poles in must be different: for more poles, building a good is not obvious

Many-body perturbation theory 1=(r1, 1,t1)

HF GW

HF GW..

HF GW test of GW and beyond multiple solutions of Dyson equations, and how to do ok uniqueneness of the LW functional Lani et al., New J. Phys. 14, 013056 (2012) Berger et al., New J. Phys. 16, 113025 (2014) Stan et al., New J. Phys. 17, 093045 (2015)

HF GW To do better: ansatz where does not depend on the external potential.

Exponential solution: cumulant expansion Sodium: Aryasetiawan et al., PRL 77, 1996 Silicon: Kheifets et al., PRB 68, 2003 Here: the first in a series of approximations link to GW prescription for ingredients

Big improvement in real calculations: silicon M. Guzzo et al., PRL 107, 166401 (2011) in collab. with J. Kas and J. Rehr, M. Silly and F. Sirotti EXPO Satellites Valence bands Kheifets et al., PRB 68, 2003 Lischner et al. PRL 110 (2013) Caruso et al., PRL 114 (2015) Cohen and Chelikowsky: Electronic Structure and Optical Properties of Semiconductors Solid-State Sciences 75, Springer-Verlag 1988)

Convincing description in terms of electron-boson coupling... IF we have the boson we solve the coupled problem reasonably well Do we???

Electron boson coupling using a cumulant G Jianqiang (Sky) Zhou, PhD thesis (2016) a 1 2 b

Coupling occupied and empty states: more correlation Homogeneous Electron Gas Kas, Rehr, Reining Phys. Rev. B 90, 085112 (2014)

Bonding Anti-bonding See also B. Gumhalter et al., PRB 94, 035103 (2016)

CRC improves satellites Z=0.42

Work in progress: Improved solution to multi-level e-boson coupling

Convincing description in terms of electron-boson coupling... IF we have the boson we solve the coupled problem reasonably well

Do we need a better W? CuO C. Roedl, F. Sottile, L. Reining, PRB 91, 045102 (2015)

Do we need a better W? 4.0 1. o e h T???? p a? r G e h t O??? g n i n e e Scr CuO V e.0 C. Rödl, F. Sottile, L. Reining, PRB 91, 045102 (2015)

Gap: EXP PBE+U+GW0 1.67 ev 2.2 ev C. Roedl et al. (2016)

Best W for QP's Extremely sensitive to low energy boson! Gap: EXP PBE+U+GW0 1.67 ev 2.2 ev C. Roedl et al. (2016)

Example for a drastic failure of RPA at small q: excitons Dressed hole e-h interaction Dressed electron e-h problem: Bethe-Salpeter equation (or beyond static: see J. Zhou et al, JCP (2015))

Rohlfing and Louie, PRL 81, 2312 (1998)

Excitonic effects in photoemission satellites [1] Marisa Scrocco Phys. Rev. B, 1985. Igor Reshetnyak, PhD thesis Palaiseau 2015

Overall comparison to experiments F2p F2s Li1s 59 [1] S. P. Kowalczyk et al. Phys. Rev. B, 1974

Analysis [1] Marisa Scrocco Phys. Rev. B, 1985. 60

Analysis [1] Marisa Scrocco Phys. Rev. B, 1985. 61

Convincing description in terms of electron-boson coupling... IF we have the boson we solve the coupled problem reasonably well

Sodium Do we need a better W? Sky J. Zhou et al, JCP (2015)

Is it W? Sodium Do we need a better W? Sky J. Zhou et al, JCP (2015)

The boson changes with increasing coupling

Is life that simple?

The GW image

The T-matrix image See also Romaniello, Guyot, Reining, J Chem Phys 131, 154111 (2009) And Romaniello, Bechstedt, Reining, PRB 85, 155131 (2012)

HF GW GW To do better: ansatz where does not depend on the external potential. This solves the self-screening problem

Suggested Reading L. Hedin, On correlation effects in electron spectroscopies and the GW approximation, J. Phys. C 11:R489 528, 1999. Short review, very good for photoemission! F. Aryasetiawan and O. Gunnarsson, The GW method, Rep. Prog. Phys. 61:237 312, 1998; and: W. G. Aulbur, L. Jonsson, and J. W. Wilkins, Quasiparticle calculations in solids, Solid State Phys. 54:1 218, 2000; Two nice and quite complete reviews on GW Strinati, G., Application of the Green s function method to the study of the optical-properties of semiconductors, Rivista del Nuovo Cimento 11, 1, 1988. Pedagogical review of the theoretical framework underlying today s Bethe Salpeter calculations. Derivation of the main equations and link to spectroscopy. Onida, G., Reining, L., and Rubio, A., Electronic excitations: density-functional versus many-body Greens-function approaches, Rev. Mod. Phys. 74, 601, 2002. Review of ab initio calculations of electronic excitations with accent on optical properties and a comparison between Bethe Salpeter and TDDFT R.M. Martin, L. Reining, D.M. Ceperley, Interacting Electrons: Theory and Computational Approaches, Cambridge University Press 2016 New book containing many-body perturbation theory, DMFT and QMC