Manybody perturbation theory: From fundamental ideas to...
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Many-body perturbation theory: From fundamental ideas to use in practice Lucia Reining Palaiseau Theoretical Spectroscopy Group
Transcript of Manybody perturbation theory: From fundamental ideas to...
Manybody perturbation theory: From fundamental ideas to use in practice
Lucia ReiningPalaiseau Theoretical Spectroscopy Group
1) Aims2) Reminder TDDFT: induced potentials3) Add a particle: GW
* exchange* screening* dynamical screening
4) Neutral excitations in the (quasi)particle world: BSE * changes in transition energies
* mixing of transitions5) Problems6) Summary
Manybody perturbation theory:
From fundamental ideas to use in practice
Ab initio calculations of electronic excitations
550 600 650 700704
708
712
716
720
724Intensity
Photoelectron Kinetic Energy (eV)
Ph
oto
n E
ner
gy
(eV
) ?
Ab initio calculations of electronic excitations
H(x1,....x
N) = E (x
1,....x
N)
?
Ab initio calculations of electronic excitations
realisticfeasible
easy to interpret
Calculations
Ab initio calculations of electronic excitations
hν
Independent electrons and transitions
Im [0] ~ vc |<v|D|c>|2 (EcEv)
v
c
Energies and wavefunctions often from DFT – Kohn Sham
PCM: Reversible phase transition
rapid transition: ~10100ns large optical & electric contrast
although already applied still not
well understood
GeSbTe alloys:200nm
W. Welnic et al., Phys. Rev. Lett. 98, 236403 (2007)
calculations & experiment
spectroscopy on polycrystalline thin films of GeTe optical spectra calculations (RPA)
change in local order results in change of optical absorption.
Origin of strong change in absorption
No significant changes in jdos
Absorption given by Fermi's golden rule:
Matrix elements define
optical contrast!
Silicon:
3 – 5 eV
20 eV
Induced potentials
Excitation ?
Change of potentials
h
V
H+V
XC VH+ V
H+V
XC+V
XC
Excitation ?
Change of potentials
Induced Hartree: longrange and local field effects
RPA
h
V
H+V
XC VH+ V
H+V
XC+V
XC
TDDFTRPA: = 0 + 0 [ v ]
v (q+G) ~ 1/|q+G|2
G=0 → plasmonG ǂ 0 → crystal local field effects
VH/
Graphene, plasmon
R. Hambach, Diplomarbeit
Graphene, plasmon
C. Kramberger et al., PRL 100, 196803 (2008)
TDDFT = 0 + 0 [ v + fxc ]
Vxc/
TDLDA: VxcLDA(r,t)/ (r',t') = (rr')(tt')dVxc/d
VH/
RPA and TDLDA: sometimes quite wrong!
Move to another framework
RPA and TDLDA: sometimes quite wrong!
EcEv (minimum, eV)
DFTKS0.554.265.38.2
SiliconDiamondMgOAr
Exp.1.175.487.8314.2
KS underestimates gaps (transition energies)
v
c
Hole (N1) electrons
“Real” (quasi…) electrons:
Exchange
Relaxation dynamical correlations
(∇ 2/2 + Vext + VH + Vxc(r)) |n> = En |n>
(∇ 2/2 + Vext + VH + ) |n> = En |n>
Vxc(r) → (r,r’,En)
= iGW L. Hedin, PR 1965W= 1v !!!
G(1,2) = i <T[†(2)]> 1=(r1,
1,t
1)
Dyson equation: G=G0 + G
0 G
From Damascelli et al., RMP 75, 473 (2003)
A()~Im[G()]
Interaction = variation of “potential”
What does the system do (create eh pairs....)
VH(12)= i(12)G(33+)v
c(31)
x(12)= iG(12+)v
c(21)
Interaction = variation of “potential”
What does the system do (create eh pairs....)
VH(12)= i(12)G(33+)v
c(31)
x(12)= iG(12+)v
c(21)
System does nothing: HFOnly classical part of : L→i,
xc→ iG(v
c + v
c v
c) =iGW
What is so good about GW?
→ Fock exchange localizationL. Hedin, PR 139, A796 (1965)
Away from the LDA starting point!→ S. V. Faleev, M. van Schilfgaarde, and T. Kotani, PRL 93, 126406 (2004).→ Bruneval, Vast, and Reining, Phys. Rev. B 74, 045102 (2006).COHSEX→ T. Miyake et al., Phys. Rev. B 74, 245231 (2006).LDA+U→ F. Fuchs, et al., Phys. Rev. B 76, 115109 (2007). LDA+U, hybrids→ Hong Jiang et al., , Phys. Rev. Lett. 102, 126403 (2009). LDA+U→ P. Rinke et al., New J. Phys. 7, 126 (2005). KSEXX→ ...........................
Improve QP energies and wavefunctions
van Schilfgaarde, Kotani, Faleev,Phys. Rev. Lett. 96, 226402 (2006)
rutile monoclinic
Even qualitatively important VO2
Matteo Gatti et al.
T. C. Koethe et al., Phys. Rev. Lett. 97, 116402 (2006).
T. C. Koethe et al., Phys. Rev. Lett. 97, 116402 (2006).
T. C. Koethe et al., Phys. Rev. Lett. 97, 116402 (2006).
M. Gatti, F. Bruneval, V. Olevano and L. Reining, Phys. Rev. Lett. 99, 266402 (2007)
LDA insulator → metal ....
M. Gatti, F. Bruneval, V. Olevano and L. Reining, Phys. Rev. Lett. 99, 266402 (2007)
G0W
0 insulator → metal !!!!!
M. Gatti, F. Bruneval, V. Olevano and L. Reining, Phys. Rev. Lett. 99, 266402 (2007)
sc GW insulator = insulator !!!
What is so good about GW?
→ Fock exchange
→ Screening
v
c
localization
relaxation
...of practical importance: example photovoltaics
J. Vidal et al., collaboration EDF, PRL 104, 056401 (2010)
Bandgaps in function of structure (CuS)
Hybrids ~ approximate GW with almost fixed screening
What is so good about GW?
→ Fock exchange
→ Screening
→ Screening (dynamical)
W() leads to imaginary part
Broadening (lifetime), satellites
Screening (dynamical): W
MetalInsulator
R. Hambach, PhD thesis
H. Abe et al., Jpn. J. Appl. Phys. 36, 165 (1997)
M. Gatti et al., 2010
Screening (dynamical): W
EcEv (minimum, eV)
DFTKS0.554.265.38.2
SiliconDiamondMgOAr
Σ =iGW1.195.647.814.0
Exp.1.175.487.8314.2
GW contains screening of hole (or electron) correct energies et al.
Even good bandstructure → sometimes quite wrong!
But back to absorption.......
Absorption ?
v
c
Electronhole interaction
Excitonic effects
BetheSalpeter Equation
Solve a 2particle equation: Bethe Salpeter Eq.
Dysonlike equation: BSE
TDDFT = 0 + 0 [ v + fxc ]
Vxc/
TDLDA: VxcLDA(r,t)/ (r',t') = (rr')(tt')dVxc/d
VH/
Solve a 2particle equation: Bethe Salpeter Eq.
Dysonlike equation: BSE
Im [] ~ vc |<v|D|c>|2 (EcEv)
(Hel + Hhole + Helhole ) A = E A
Im [] ~ | vc<v|D|c> Avc|2 (E)
>Mixing of transitions
>Modification of excitation energies
GW + Bethe Salpeter Equation
Im [] ~ vc |<v|D|c>|2 (EcEv)
(Hel + Hhole + Helhole ) A = E A
Im [] ~ | vc<v|D|c> Avc|2 (E)
>Mixing of transitions
>Modification of excitation energies
From VH/ + /G
F. Bruneval et al., PRL 97, 267601 (2006))
Excitons in Cu2O
V. Garbuio, et al., PRL. 97, 137402 (2006)
Excitons in water
Excitons in Na4
Electronhole interaction for good absorption spectra
Is there anything BAD about GW?
→ Fock exchange
→ Screening
→ Screening (dynamical)
→ Widely valid and efficient
→ Room for speedup
G(1,2) = i <T[†(2)]> 1=(r1,
1,t
1)
Dyson equation: G=G0 + G
0 G
More consistent in orders of W
Interaction = variation of “potential”
What does the system do (create eh pairs....)
System does nothing: HFOnly classical part of : L→i,
xc→ iGW
Additional particle interacts in a classical way
2 site 1 electron Hubbard modelP. Romaniello et al., J. Chem. Phys 2009, 131, 154111
2 site 1 electron Hubbard model
We should consider 2 particles at a time
Interaction = variation of “potential”
What does the system do (create eh pairs....)
System does nothing: HFOnly classical part of : L→i,
xc→ iGW
Better : e.g. Tmatrix approach
Springer, Aryasetiawan, Karlsson, PRL 80, 2389 (1998)
6 eV satellite in Ni: 2hole bound state
We are used to solve 2particle equations: BSE
Dysonlike equation: BSE
5) Summary
Keywords:
* quasiparticles * induced potentials* screening* localization* satellites* Dyson equations* .....breakdowns....
http://www.etsf.eu
Palaiseau Theoretical Spectroscopy Group
4. Look at the total energy
Adiabatic connection fluctuation dissipation theorem (ACFDT)
........in DFT framework
Total energy in GW framework
.......evaluated at G=Gs, equivalent to RPA ACFDT
RPA+ : because RPA has deficiencies
Uniform gas correlation energy too negative.Can be corrected by LDA like term. S. Kurth and J. P. Perdew, Phys. Rev. B 59, 10 461(1999); M. Lein, E. K. U. Gross, and J. P. Perdew,
Phys. Rev. B 61, 13 431 (2000).
Sum over empty states: how to speed up
Replace i by one representative energy?
Bruneval, Gonze, PRB 78, 085125 (2008). SiC
From sum rules
Argon, no empty states summed. Berger et al., PRB 82, 041103R 2010
Very simple to be implemented, imediate speedup.
C. Kramberger et al., PRL 100, 196803 (2008)
Isolated carbon nanotube, plasmon
Nanotubes and graphene, plasmon dispersion
C. Kramberger et al., PRL 100, 196803 (2008)
