many-body perturbation theory the gw approximation · overview lintroduction ltheory l green...
TRANSCRIPT
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Many-Body Perturbation Theory The GW approximation
C. Friedrich
Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich, 52425 Jülich, Germany
GW
Hybrids (PBE0, HSE) OEP (EXX)
RPA total energy
Bethe-Salpeter eq.(magnons)
GT self-energy
COHSEX
GSOCWSOC
QSGW
Hubbard U(LDA+U, DMFT)
Wannier functions (interpolation) Hartree-Fock
Dielectric function
Overviewl Introduction
l Theoryl Green functionl Feynman diagramsl GW approximation
l Implementationl Basis sets (FLAPW method)l Exchange: Hartree-Fockl Correlation: Imaginary-frequency formulationl Analytic continuation vs. contour integration
l Applicationsl Siliconl Zinc Oxidel Metallic Nal HgTe
l Computational procedurel One-Shot GWl Full band structurel Self-consistent QSGW
l Summary
Density functional theory
Exchange and correlation potentialApproximations: LDA, GGA
Kohn-Sham (KS) equations:
è real ρ(r) è E0[ρ]
Exchange and correlation potentialApproximations: LDA, GGA
Kohn-Sham (KS) equations:
Density functional theory
Exchange and correlation potentialApproximations: LDA, GGA
Kohn-Sham (KS) equations:
Many-body Schrödinger equation:
Density functional theory
Central quantity is the single-particle Green function(probability amplitude for the propagation of a particle)
which contains poles at the excitation energies of themany-electron system (photoelectron spectroscopy),
seen by Fourier transformation
Excitation energy measured in photoemission spectroscopy
directinverse
TheoryGreen function
G(r, r0;!) =X
n
N+1n (r) N+1⇤
n (r0)
! � (EN+1n � EN
0 ) + i⌘+
X
n
N�1n (r) N�1⇤
n (r0)
! � (EN0 � EN�1
n )� i⌘<latexit sha1_base64="Ce/UbqkyAZ6LJJQQC2U2uVuJ6Uc=">AAAC2nicfVJdixMxFM2MX2v8qvroy2CR7VinzKyCggiLUvRpWcFuF5rukEnvdMMmmSHJCGWYFx8U8dVf5ps/w39gptti3XW9EDjck3NP7r3JSsGNjeOfnn/p8pWr17au4xs3b92+07l778AUlWYwYoUo9GFGDQiuYGS5FXBYaqAyEzDOTt60/PgjaMML9cEuSphKOlc854xal0o7v972iKT2OMtr3Tz5A7dfkkLCnIavMDGVTBUmuaasxqQ0PFVH9V4/aTak4SbxeJPZDnHjZMtqUW+41kbDND7aC/ucgKW4wf2LfKKLfKL/+7jq0XBdIYzWPjjtdONBvIzgPEhWoItWsZ92fpBZwSoJyjJBjZkkcWmnNdWWMwENJpWBkrITOoeJg4pKMNN6uZomeOQysyAvtDvKBsvspqKm0piFzNzNthNzlmuT/+Imlc1fTGuuysqCYqdGeSUCWwTtnoMZ18CsWDhAmeburQE7pm601v2GdgjJ2ZbPg4OdQfJ0sPP+WXf39WocW+gBeoh6KEHP0S56h/bRCDFv7DXeZ++LT/xP/lf/2+lV31tp7qO/wv/+G/EC3zU=</latexit>
TheoryFeynman diagrams
Self-energy: sum over allscattering diagrams
TheoryDyson equation
Σ is the electronic self-energy (scattering potential).
TheoryDyson equation
Σ is the electronic self-energy (scattering potential).
G = G0 +G0⌃G<latexit sha1_base64="3lo3KfMLwsHzpYNgkT1dOsmg43U=">AAAB+nicbVBNS8NAEJ34WetXqkcvi0UQhJJUQS9C0UM9VrQf0Iaw2W7bpbtJ2N0oJfanePGgiFd/iTf/jds2B219MPB4b4aZeUHMmdKO820tLa+srq3nNvKbW9s7u3Zhr6GiRBJaJxGPZCvAinIW0rpmmtNWLCkWAafNYHg98ZsPVCoWhfd6FFNP4H7IeoxgbSTfLlQvq75zYqpzx/oCo6pvF52SMwVaJG5GipCh5ttfnW5EEkFDTThWqu06sfZSLDUjnI7znUTRGJMh7tO2oSEWVHnp9PQxOjJKF/UiaSrUaKr+nkixUGokAtMpsB6oeW8i/ue1E9278FIWxommIZkt6iUc6QhNckBdJinRfGQIJpKZWxEZYImJNmnlTQju/MuLpFEuuael8u1ZsXKVxZGDAziEY3DhHCpwAzWoA4FHeIZXeLOerBfr3fqYtS5Z2cw+/IH1+QPK0pJo</latexit>
complex energy contains• excitation energies (real part)• excitation lifetimes (imaginary part)
The Dyson equation can be rewritten as the quasiparticle equation
h0(r) n(r) +
Z⌃(r, r0;En) n(r
0)d3r0 = En n(r)<latexit sha1_base64="WyR5kSijnwu0vIwhMMnYkZekCeE=">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</latexit>
v W
strongelectron
interaction v
weakquasiparticleinteraction W electron
Coulomb hole
Expansion up to linear order
The GW approximation contains electron exchange and a large part of electron correlation.
quasi-particle
in Coulomb interaction v → ΣHF = iG0v (Hartree-Fock)
in screened interaction W → ΣGW = iG0W (GW approximation)
TheorySelf-energy
-+
+
+
+
Quasiparticler'r v(r,r')
v(r,r'')
W(r,r')
nind(r'',r')
= + + + ...
TheorySelf-energy
P
= Pv + PvPv + PvPvPv + ...<latexit sha1_base64="aiDLBpSAMIZjVjYiAnqewuWAg5I=">AAAB/HicdVDLSgMxFM3UV62v0S7dBIsgCMNMa2lnIRTduBzBPqAdSiZN29DMgyRTGIb6K25cKOLWD3Hn35hpR1DRA8k9nHMvuTlexKiQpvmhFdbWNza3itulnd29/QP98Kgjwphj0sYhC3nPQ4IwGpC2pJKRXsQJ8j1Gut7sOvO7c8IFDYM7mUTE9dEkoGOKkVTSUC9fOvNzZ55fqhiGMdQrpmE36/VaE5qGadpVu6GIbdtWw4KWUjJUQA5nqL8PRiGOfRJIzJAQfcuMpJsiLilmZFEaxIJECM/QhPQVDZBPhJsul1/AU6WM4Djk6gQSLtXvEynyhUh8T3X6SE7Fby8T//L6sRw33ZQGUSxJgFcPjWMGZQizJOCIcoIlSxRBmFO1K8RTxBGWKq+SCuHrp/B/0qkaVs2o3l5UWld5HEVwDE7AGbBAA7TADXBAG2CQgAfwBJ61e+1Re9FeV60FLZ8pgx/Q3j4BCBGTvg==</latexit>
Lars Hedin, 1965
GW approximation corresponds to the 1st iteration starting from Σ=0:P P
TheoryHedin equations
(random-phase approximation)
Implementation
Dyson equation è quasiparticle equations: true excitation energies
energies of afictitious system
GW:
DFT:
Similarity motivates the use of perturbation theory
directsolution
linearizedsolution
renormalizationfactor
ImplementationBasis sets
Dyson equation è quasiparticle equations: true excitation energies
energies of afictitious system
GW:
DFT:
GaussiansPlane waves (Pseudopotential)PAWLMTOLAPW...
Basis set for wavefunctions
Auxiliary Gaussian set (density fitting)Plane wavesPlane wavesProduct basisMixed product basis...
Basis set for wavefunction products
ImplementationFLAPW method
Muffin-tin (MT) spheres:numerical MT functions
cutoff
In our GW implementation we use the mixed product basis, generated fromthe products of (1) interstitial plane waves (cutoff G'max)
and (2) MT functions (cutoff l'max)
T. Kotani and M. van Schilfgaarde, Solid State Commun. 121, 461 (2002).
up to l+l' è l'max = 2lmax
è G'max = 2Gmax
Interstitial region:interstitial plane waves
cutoffei(k+G)r
<latexit sha1_base64="72e2CZaUCNF3ayFY1rUSOram1sE=">AAACGXicbZDLSsNAFIYn9VbrLerSTbAIFaEkVdBl0YUuK9gLtLFMppN26MwkzEyEEvIabnwVNy4Ucakr38ZJmoK2/jDw8Z9zmHN+L6REKtv+NgpLyyura8X10sbm1vaOubvXkkEkEG6igAai40GJKeG4qYiiuBMKDJlHcdsbX6X19gMWkgT8Tk1C7DI45MQnCCpt9U0b38c9BtVIsJgklQw9Px4nJzO8To5nKJKk1DfLdtXOZC2Ck0MZ5Gr0zc/eIEARw1whCqXsOnao3BgKRRDFSakXSRxCNIZD3NXIIcPSjbPLEutIOwPLD4R+XFmZ+3sihkzKCfN0Z7qjnK+l5n+1bqT8CzcmPIwU5mj6kR9RSwVWGpM1IAIjRScaIBJE72qhERQQKR1mGoIzf/IitGpV57Rauz0r1y/zOIrgAByCCnDAOaiDG9AATYDAI3gGr+DNeDJejHfjY9paMPKZffBHxtcPCgyhjA==</latexit>
|k+G| Gmax<latexit sha1_base64="BUNLnLRb80txcxiT1f6G1670RMU=">AAACE3icbZDLSsNAFIYnXmu9RV26GSyCKJSkCrosuqjLCvYCTQiT6aQdOpOEmYlY0ryDG1/FjQtF3Lpx59uYpBG09YeBj/+cw5zzuyGjUhnGl7awuLS8slpaK69vbG5t6zu7bRlEApMWDlggui6ShFGftBRVjHRDQRB3Gem4o6us3rkjQtLAv1XjkNgcDXzqUYxUajn68cTiSA1dLx4lJz/YSCYWI7Dh5IbgMUf3SdnRK0bVyAXnwSygAgo1Hf3T6gc44sRXmCEpe6YRKjtGQlHMSFK2IklChEdoQHop+ogTacf5TQk8TJ0+9AKRPl/B3P09ESMu5Zi7aWe2pJytZeZ/tV6kvAs7pn4YKeLj6UdexKAKYBYQ7FNBsGLjFBAWNN0V4iESCKs0xiwEc/bkeWjXquZptXZzVqlfFnGUwD44AEfABOegDq5BE7QABg/gCbyAV+1Re9betPdp64JWzOyBP9I+vgH5I57U</latexit>
ImplementationMixed product basis
exact
converged
exact
converged
spex.inp: keyword “GCUT”.
spex.inp: keyword “LCUT”.
The self-energy can be decomposed into an exchange and a correlation term:
The exchange contribution is given analytically by the Hartree-Fock expression
Density matrix
v
ImplementationExchange Self-energy
The quantities (e.g. the polarization function P) are much smoother along the imaginary frequency axis than along the
real axis.
Poles of P
spex.inp: keyword “HILBERT”.spex.inp: keyword “MESH”.
ImplementationImaginary-frequency formulation
Analytic Continuation Contour integration
l Accurate evaluation of Σc
l More parameters necessaryl Takes more time
l Easy to implementl Fast computationl Analytic continuation critical
Poles of W
Poles of GPoles of G
Poles of W
spex.inp: keyword “CONTINUE”. spex.inp: keyword “CONTOUR”.
ImplementationCorrelation Self-energy
⌃c(!) =i
2⇡
Z 1
�1G0(! + !0)W c(!0)d!0
<latexit sha1_base64="XhI9AfMAehFo3pWnHJC6w29C3Go=">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</latexit>
LDADFT
ApplicationsBulk Silicon
DFTexp
exp
ApplicationsBulk Silicon
DFT
GW
exp
exp
ApplicationsBulk Silicon
GW
• Si was the first material to which GW wasapplied (Hybertsen, Louie 1985; Godby,Schlüter, Sham 1986).
• The one-shot GW calculation yields moreaccurate band gaps:
• Quasiparticle self-consistent GW (QSGW) tends to overestimate the gaps:
DFT GW exp.direct: 2.53 3.20 3.40 eVindirect: 0.47 1.04 1.17 eV
QSGWdirect: 3.60 eVindirect: 1.34 eV
exp
exp DFT
P
ApplicationsBulk Silicon
QSGW: S. V. Faleev, M. van Schilfgaarde, and T. Kotani, Phys. Rev. Lett. 93, 126406 (2004).
spex.inp: keyword “NBAND”.
Scattering into unoccupied states
ApplicationsBand convergence
Large scatter of band-gap values from one-shot GW calculations (exp: 3.4 eV):2.44 eV (FLAPW) [M. Usuda et al., Phys. Rev. B 66, 125101 (2002)]2.12 eV (PAW) [M. Shishkin and G. Kresse, PRB 75, 235102 (2007)2.14 eV (PAW) [F. Fuchs et al., Phys. Rev. B 76, 115109 (2007)]2.6 eV (PW-PP) [P. Gori et al., Phys. Rev. B 81, 125207 (2010)]3.4 eV (PW-PP) [B.-C. Shih et al., Phys. Rev. Lett. 105, 146401 (2010)]2.83 eV (FLAPW) [C. Friedrich et al., Phys. Rev. B 83, 081101 (2011)]…
false convergence in the case of underconverged W
W
“standard”: 200 bands
occupied
ApplicationsZinc Oxide - band convergence
Shih et al., PRL 105, 146401Friedrich et al., PRB 83, 801101
semiconductor
εF
The polarization function is sum over virtual transitions inthe non-interacting reference system.
PIJ(k, ⇤) =⇤
�,q
occ⇤
n
unocc⇤
n�
...
�1
⇤ + ��qn � ��
k+qn� + i⇥� ...
⇥
ApplicationsGW for metals
semiconductor
εF
The polarization function is sum over virtual transitions inthe non-interacting reference system.
PIJ(k, ⇤) =⇤
�,q
occ⇤
n
unocc⇤
n�
...
�1
⇤ + ��qn � ��
k+qn� + i⇥� ...
⇥
ApplicationsGW for metals
εF
The polarization function is sum over virtual transitions inthe non-interacting reference system.
PIJ(k, ⇤) =⇤
�,q
occ⇤
n
unocc⇤
n�
...
�1
⇤ + ��qn � ��
k+qn� + i⇥� ...
⇥
metal
ApplicationsGW for metals
metal
εF
The polarization function is sum over virtual transitions inthe non-interacting reference system.
PIJ(k, ⇤) =⇤
�,q
occ⇤
n
unocc⇤
n�
...
�1
⇤ + ��qn � ��
k+qn� + i⇥� ...
⇥
ApplicationsGW for metals
metal
εF
The polarization function is sum over virtual transitions inthe non-interacting reference system.
PIJ(k, ⇤) =⇤
�,q
occ⇤
n
unocc⇤
n�
...
�1
⇤ + ��qn � ��
k+qn� + i⇥� ...
⇥
Virtual transitions of zero energyjust across the Fermi surfaceproduce the Drude term
for kè0.
ApplicationsGW for metals
l Vanishing of the density of states at theFermi energy in HF is exactlycompensated by the GW correlationself-energy.
l GW band width smaller than KS bandwidth due to increased effective(quasiparticle) mass.
ApplicationsSodium
GW approximation including SOC
Spin-orbit coupling (SOC) arises from the interaction of the electron spin with theB field that is created in the rest frame of the electron due to its motion throughthe material à coupling of spatial and spin degrees of freedom:
F. Aryasetiawan, S. Biermann, PRL 100, 116402 (2008).R. Sakuma, C. Friedrich, T. Miyake, S. Blügel, F. Aryasetiawan, PRB 84, 085144 (2011).
��⇥(r, r�;⇥) =i
2�
�G�⇥(r, r�;⇥ + ⇥�)W (r, r�;⇥�)ei⇤⌅�
d⇥�
G�⇥(r, r�;⌅) =�
kn
⌃kn(r, �)⌃kn(r�, ⇥)⌅ � ⇧kn ± i⇤
P (r, r�;⇥) = � 12�
�
�⇥
⇥G�⇥(r, r�;⇥ + ⇥�)G⇥�(r�, r;⇥�)d⇥�i
LDA+SOCGW+SOCGSOCWSOC
QSGW+SOCQSGSOCWSOC
Experiment
E0
-1.16-0.59-0.60-0.43-0.46
-0.30
Δ0
0.740.740.830.750.93
0.91
Bulk HgTe
calculations with semicores Hg 5p and Te 4d
Γ6
Γ7
Γ8
spin-orbitsplitting Δ0Energy
gapE0
Bulk HgTe
Γ6
Γ7
Γ8
spin-orbitsplitting Δ0
Energygap E0
calculations with semicores Hg 5p and Te 4d
LDA+SOCGW+SOCGSOCWSOC
QSGW+SOCQSGSOCWSOC
Experiment
E0
-1.16-0.59-0.60-0.43-0.46
-0.30
Δ0
0.740.740.830.750.93
0.91
Bulk HgTe
Γ6
Γ7
Γ8
spin-orbitsplitting Δ0
Energygap E0
calculations with semicores Hg 5p and Te 4d
LDA+SOCGW+SOCGSOCWSOC
QSGW+SOCQSGSOCWSOC
Experiment
E0
-1.16-0.59-0.60-0.43-0.46
-0.30
Δ0
0.740.740.830.750.93
0.91
ApplicationsMany-body effects: lifetime broadening and plasma satellites
x10
DFTGW GW
Plasmaron bands(electrons+plasmons)
Renormalization due toformation of electron-hole pairs(density fluctuations)
• FLEUR: Self-consistent field calculationè Density, Exchange-correlation potential
• SPEX: Generate special equidistant k-point setk, k', k+k’, and 0 must be elements
• FLEUR: Diagonalize Hamiltonian on new k points (non iterative)è Kohn-Sham energies and wavefunctions
• SPEX: GW calculationè Quasiparticle energies
Computational procedureOne-Shot GW
k
k'k+k'
spex.inp: JOB GW X:(1-4,6)
spex.inp:
ITER
ATE
k-point set
• FLEUR: Self-consistent field calculationè Density, Exchange-correlation potential
• SPEX: Define q-point pathè {q}
Loop over q (high-symmetry line in Brillouin zone)
• SPEX: Generate two sets of equidistant k-pointsè {k}, {k+q}
• FLEUR: Diagonalize Hamiltonian on new k points (non iterative)è Kohn-Sham energies and wavefunctions
• SPEX: GW calculation (W has to be calculated only once)è Quasiparticle energies at q
spex.inp: keyword “RESTART”.
“spe
x.ba
nd”
Computational procedureGW band structure
spex.inp: KPTPATH (L,G,X)
k-point setshifted k-point set
Computational procedureWannier interpolation
spex.inp: KPTPATH (L,G,X)
spex.inp: section “WANNIER”:
SECTION WANNIERORBITALS 1 4 (sp3)MAXIMIZEINTERPOL
END
• FLEUR: Self-consistent field calculationè Density, Exchange-correlation potential
• SPEX: Generate special equidistant k-point setk, k', k+k’, and 0 must be elements
• FLEUR: Diagonalize Hamiltonian on new k points (non iterative)è Kohn-Sham energies and wavefunctions
• SPEX: GW calculationè Quasiparticle energies spex.inp: JOB GW IBZ:(1-4)
• FLEUR: Self-consistent field calculationè Density, Exchange-correlation potential
• SPEX: Generate special equidistant k-point setk, q, k+q, and 0 must be elements
Start of iterations
• FLEUR: Diagonalize Hamiltonian on new k points (non iterative)è Kohn-Sham energies and wavefunctions
• SPEX: GW calculation on all k points and many bands (costly)è Quasiparticle energies and self-energy matrix
• FLEUR: Self-consistent field calculation (with self-energy!)è Density, Exchange-correlation potential
“spe
x.se
lfc”
Computational procedureSelf-consistent GW (QSGW)
spex.inp: JOB GW FULL IBZ:(1-4)
• Configuration% ./configure --with-wan --prefix=$HOMEgenerates the Makefile for the computer system
• Compilation% makeproduces the three executables spex.inv, spex.noinv, and spex.extr;the launcher script "spex" always calls the correct executable
• Installation% make installinstalls executables and scripts into $HOME/bin
• The PATH environment variable should contain $HOME/binIf not: % export PATH=$PATH:$HOME/bin
Compilation and Installation
Summary
● Excitation energies and lifetimes of the (N+1) and (N-1)-electronsystem can be readily obtained from the one-particle Green function.These excitation energies form the band structure in solids.
● The Green function obeys an integral Dyson equation which may berewritten as a quasiparticle equation with the self-energy as ascattering potential that takes into account all exchange andcorrelation effects beyond the Hartree potential.
● The GW approximation constitutes the expansion of the self-energyup to linear order in the screened interaction W.
● It is usually implemented as a perturbative correction on a DFT bandstructure. But a self-consistent solution (QSGW) is possible, too.