a direct approach to the calculation of many-body green’s...

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 & friendsJianqiang Zhou, Igor Reshetnyak, Claudia Roedl, Matteo Guzzo, Francesco Sottile, Matteo Gatti, Lucia Reining

U. Washington: John Rehr, Joshua Kas

Synchrotron SOLEIL: Fausto Sirotti, Matthieu SillySynchrotron ESRF/U. Helsinki: Simo Huotari

http://etsf.polytechnique.frhttp://www.etsf.eu

http://etsf.polytechnique.fr/

→ Photoemission: quasiparticles and more

→ Electron-boson coupling using a cumulant G

→ Electron-boson coupling in GW

→ Electron-boson coupling

→ Which boson?

→ Self-energies and more


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

→ Photoemission

Band structure of germanium

Rohlfing et al., PRB 48, 17791 (1993)A. Svane, PRB 35, 5496 (1987)

LDA Hartree-FockExp

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,t

1)

Dyson equation: G =G0 + G

0 G

12-37

→ ~ i WG “GW”

L. Hedin (1965)

W = -1( v

+ ….....

Band structure of germanium

Rohlfing et al., PRB 48, 17791 (1993)A. Svane, PRB 35, 5496 (1987)

LDA Hartree-FockExp and GW

Couples to excitations!

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...............

→ Photoemission

→ Electronboson coupling

Sky J. Zhou, PhD thesis Palaiseau 2016

Valencebands

Exp.: F. Sirotti et al., TEMPO beamline SOLEIL

Cohen and Chelikowsky: “Electronic Structure and Optical Properties of Semiconductors” Solid-State Sciences 75, Springer-Verlag 1988)

Bulk Silicon

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

→ ~ i WG “GW”

L. Hedin (1965)

W = -1( v

+ ….....

→ Electronboson coupling in GW

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)

MetalInsulator

Screening (dynamical): W

Gatti, Panaccione, Reining (PRL 2015)

Valencebands

Satellites

EXPO


M. Guzzo et al., PRL 107, 166401 (2011)

Silicon

Poles in G must be “physical”

Poles in must be different: for more poles, building a good is not obvious

Want: G = a/( – a/( –

= [G0 ]-1 - G-1

= ( – – – – – – a

– (1 – a)aa

– – a

→ Selfenergies and more

Want: G = 1/( –

= [G0 ]-1 - G-1

= ( – – –

Many-body perturbation theory

1=(r1,

1,t

1)

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

Valencebands

Satellites

EXPO


M. Guzzo et al., PRL 107, 166401 (2011) in collab. with J. Kas and J. Rehr, M. Silly and F. Sirotti

→ Big improvement in real calculations: silicon

Kheifets et al., PRB 68, 2003Lischner et al. PRL 110 (2013)Caruso et al., PRL 114 (2015)


IF



Do we???

Jianqiang (Sky) Zhou, PhD thesis (2016)

1 2

b

a

→ Electronboson coupling using a cumulant G

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)

Z=0.42

CRC improves satellites

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


IF



C. Roedl, F. Sottile, L. Reining, PRB 91, 045102 (2015)

CuODo we need a better W?

Gap???? Theo. 1.0 – 4.0 eV

Screening??? Other?

C. Rödl, F. Sottile, L. Reining, PRB 91, 045102 (2015)

CuODo we need a better W?

Gap: EXP 1.67 eV PBE+U+GW

02.2 eV

C. Roedl et al. (2016)

C. Roedl et al. (2016)

“Best W” for QP's

Extremely sensitive to low energy boson!

Gap: EXP 1.67 eV PBE+U+GW

02.2 eV

e-h problem: Bethe-Salpeter equation

Dressed hole

Dressed electron

e-h interaction

(or beyond static: see J. Zhou et al, JCP (2015))

Example for a drastic failure of RPA at small q: excitons

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

59[1] S. P. Kowalczyk et al. Phys. Rev. B, 1974

Li1s

F2s

F2p

60

Analysis


61

Analysis



IF



Sky J. Zhou et al, JCP (2015)

Sodium

Do we need a better W?

Sky J. Zhou et al, JCP (2015)

Sodium

Do we need a better W?

Is it W?

The boson changes with increasing coupling

Is life that simple?

The GW image

See also Romaniello, Guyot, Reining, J Chem Phys 131, 154111 (2009)AndRomaniello, Bechstedt, Reining, PRB 85, 155131 (2012)

The T-matrix image

HF

GW

To do better: ansatz

where does not depend on the external potential.

GW

This solves the self-screening problem

→ Photoemission: quasiparticles and more

→ Electron-boson coupling using a cumulant G

→ Electron-boson coupling in GW

→ Electron-boson coupling

→ Which boson?

→ Self-energies and more


Palaiseau Theoretical Spectroscopy Group & friendsJianqiang Zhou, Igor Reshetnyak, Claudia Roedl, Matteo Guzzo, Francesco Sottile, Matteo Gatti, Lucia Reining

U. Washington: John Rehr, Joshua Kas

Synchrotron SOLEIL: Fausto Sirotti, Matthieu SillySynchrotron ESRF/U. Helsinki: Simo Huotari

http://etsf.polytechnique.frhttp://www.etsf.eu

http://etsf.polytechnique.fr/

Suggested Reading

Onida, G., Reining, L., and Rubio, A., “Electronic excitations: density-functional versus many-bodyGreens-function approaches,” Rev. Mod. Phys. 74, 601, 2002. Review of ab initio calculationsof electronic excitations with accent on optical properties and a comparison between Bethe–Salpeter and TDDFT

Strinati, G., “Application of the Green’s function method to the study of the optical-properties ofsemiconductors,” Rivista del Nuovo Cimento 11, 1, 1988. Pedagogical review of the theoreticalframework underlying today’s Bethe–Salpeter calculations. Derivation of the main equationsand link to spectroscopy.

R.M. Martin, L. Reining, D.M. Ceperley, “Interacting Electrons: Theory and Computational Approaches”, Cambridge University Press 2016New book containing many-body perturbation theory, DMFT and QMC

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,” SolidState Phys. 54:1–218, 2000;Two nice and quite complete reviews on GW

a direct approach to the calculation of many-body green’s...

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