a direct approach to the calculation of many-body green's...
TRANSCRIPT
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2440-12
16th International Workshop on Computational Physics and Materials Science: Total Energy and Force Methods
Lucia Reining
10 - 12 January 2013
Ecole Polytechnique, Palaiseau France
A direct approach to the calculation of many-body Green's functions: beyond quasiparticles
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A direct approach to the calculation of many-body Green' s functions: beyond quasiparticles
Giovanna Lani, Pina Romaniello, Matteo Guzzo, Lucia Reining
Palaiseau Theoretical Spectroscopy Group & friends
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Many-body perturbation theory: GW
An exponential solution for G
What's ongoing?
Outlook
An alternative strategy
Insight from a simple model
A direct approach to the calculation of many-body Green' s functions: beyond quasiparticles
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v
c
Hole - (N-1) (excited) electrons
Photoemission:
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From Damascelli et al., RMP 75, 473 (2003)
A(ω)~Im[G(ω)]
Coupling to other excitations!
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Many-body perturbation theory
G(1,2) = -i <T[ψ(1)ψ (2)]>
1=(r1,σ
1,t
1)
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Many-body perturbation theory
~ GG HF
Dyson equation: G=G0 + G
0 Σ G
Σ ~ i vcG
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Many-body perturbation theory: GW
1. Linearization VH [ϕ] = V
H
0 + vc χϕ .....
.......leads to screening: W = ε-1vc
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Σ ~ i WG “GW”
Lani et al., New J. Phys. 14, 013056 (2012)
Many-body perturbation theory: GW
G=G0 + G
0 Σ G
~ GG
1. Linearization VH [ϕ] = V
H
0 + vc χϕ .....
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With optimized QP energies and wavefunctions
van Schilfgaarde, Kotani, Faleev,Phys. Rev. Lett. 96, 226402 (2006)
great bandstructure
Dyson equation no T=0 bandstructure problem.
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With optimized QP energies and wavefunctions
van Schilfgaarde, Kotani, Faleev,Phys. Rev. Lett. 96, 226402 (2006)
great bandstructure
What about the rest?
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From Damascelli et al., RMP 75, 473 (2003)
Σ=iGW
Hedin 1965 +......
GW: Electron-plasmon coupling
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Valence satellites in “simple” silicon
GW: QP ok, satellite is completely wrong
Experiment might also have limitations
- - - Dashed: GWKheifets et al., PRB 68, 2003
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XPS Spectrum of bulk Si
F. Sirotti and M. Silly, Synchrotron Soleil, France
Valencebands
Satellites
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XPS Spectrum of bulk Si
F. Sirotti and M. Silly, Synchrotron Soleil, France
Valencebands
Satellites
Quasiparticle bands
Ei
)
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XPS Spectrum of bulk Si
F. Sirotti and M. Silly, Synchrotron Soleil, France
Valencebands
Satellites
Plasmon satellites
) )
Peaked at 17 eV
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Bottom valence: a plasmaron!
2 solutions for Ei
)
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Total G0W
0 Spectral Function
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Plasmaron peaks stronger than plasmon peaks (Can mask plasmon contribution)
Artefact! Blomberg, Bergerse, Can. J. Phys. 50, 2286 (1972); Kus, Blomberg, Can. J. Phys. 51, 102 (1973)
No replicas within G0W
0 (See e.g. for sodium
Aryasetiawan et al., PRL 77, 1996)
G0W
0 Spectral Function:
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Plasmaron peaks stronger than plasmon peaks (Can mask plasmon contribution)
Artefact! Blomberg, Bergerse, Can. J. Phys. 50, 2286 (1972); Kus, Blomberg, Can. J. Phys. 51, 102 (1973)
No replicas within G0W
0 (See e.g. for sodium
Aryasetiawan et al., PRL 77, 1996)
G0W
0 Spectral Function:
GW pasmarons: suspicious!
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1. Linearization VH = V
H
0 + vc χϕ .....
~ GG GW
Lani et al., New J. Phys. 14, 013056 (2012)
An alternative strategy?
Dyson equation: G=G0 + G
0 Σ G
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1. Linearization VH = V
H
0 + vc χϕ .....
Lani et al., New J. Phys. 14, 013056 (2012)
An alternative strategy
Solve differential equation!
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Two problems:
* solve the set of differential equations
* pick the good solution!
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Insight from a simple model
“1 point”
Great for questions like: how to pick the good solution?
Ex.: Does a self-consistent GW calculation always converge
* to the same result, for any starting point?
* to the physical solution, if there is more than one?
G = GH + G
Hϕ G + i G
HW δ G/δ ϕ
Lani et al., New J. Phys. 14, 013056 (2012)
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G = GH + G
Hϕ G - G
HW δ G/δ ϕ
G = GH - G
H(W G)G
Quadratic equation: 2 solutions, G1 and G
2!
G = GH /(1+G
HW G)
Solution 1: iterate Gn+1 = G
H /(1+G
HW Gn)
continuous fraction. For all starting points: Converges to G
1 with G
1[W 0] = G
H
GW
“exact”
Which one will we find?
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Determined by G[W 0] = GH
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G = GH + G
Hϕ G - G
HW δ G/δ ϕ
G = GH - G
H(W G)G
Quadratic equation: 2 solutions, G1 and G
2!
Solution 1: iterate Dyson equation as usualcontinuous fraction, well behaved W 0 limit.
Solution 2 ??? G = 1/(WG) – 1(WGH)
iterate Gn+1 = 1/(WGn) – 1(WGH)
continuous fraction. For all starting points:
Converges to G2 with G
2[W 0] divergent!
GW
“exact”
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G = GH + G
Hϕ G - G
HW δ G/δ ϕ
G = GH - G
H(W G)G
Quadratic equation: 2 solutions, G1 and G
2!
Solution 1: iterate Dyson equation as usualcontinuous fraction, well behaved W 0 limit.
Solution 2 ??? G = 1/(WG) – 1(WGH)
iterate Gn+1 = 1/(WGn) – 1(WGH)
continuous fraction. For all starting points:
Converges to G2 with G
2[W 0] divergent!
GW
“exact”
The way you iterate!!!!
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1-point model is marvellous playground:
* Benchmarks
* New approximations beyond GW
* insight : how to solve DE, how to pick solution.
More physics?
Self-consistency for GW0 is a good thing
Expect that standard self-consistent GW0 is ok
Expect more delicate situation beyond GW0
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Solve linearized differential equation with times
G = GH + G
Hϕ G + i G
HW δ G/δ ϕ
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Solve linearized differential equation with times
G = GH + G
Hϕ G + i G
HW δ G/δ ϕ
cf U(ω ) !!!
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Exponential solution: cumulant expansion
Sodium: Aryasetiawan et al., PRL 77, 1996)
Kheifets et al., PRB 68, 2003Silicon
Here: one possible simple approximation
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Spectral Function from exponential:
Theory: exp
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Valencebands
Satellites
Spectrum, exponential versus GW and experiment:
EXPO
M. Guzzo et al., PRL 107, 166401 (2011)
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Graphite
Satellites?
See e.g. Vos et al., PRB 63, 033108 (2001)Sattler et al., PRB 63, 155204 (2001)McFeely et al., PRB 9, 5268 (1974)
Exp: Eberlein et al., Phys. Rev. B 77, 233406 (2008)
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M. Guzzo et al., PhD thesis. Exp: SOLEIL TEMPO beamline.
Graphite
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Exp: Eberlein et al., Phys. Rev. B 77, 233406 (2008)
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Exponential calculation, spectral function of graphite and graphene
Graphite Graphene
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Very good description of plasmon satellites
Can make predictions
There is more in life.........
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Solution of the (almost) full equations......
What's ongoing ?
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What's ongoing ?
We have: explicit G(1,2,[ϕ ,W,q])
We have sum rules for q* from the Differential eq.* from the W 0 condition
We have symmetry constraints* from rules on fctl derivatives
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Outlook
Full solution (ok ok, some approxs....)
coupling of other excitations
2-particle G
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Palaiseau Theoretical Spectroscopy Group & friends
Giovanna Lani, Matteo Guzzo, Lorenzo Sponza,Francesco Sottile, Matteo Gatti, Christine Giorgetti, Lucia Reining
Toulouse: Pina Romaniello, Arjan Berger
U. Washington: John Rehr, Joshua Kas
Synchrotron SOLEIL: Fausto Sirotti, Matthieu Silly
http://etsf.polytechnique.frhttp://www.etsf.eu