Many-Body Perturbation Theory The GW approximation · Overview lIntroduction lTheory l Green function l Feynman diagrams l GWapproximation lImplementation l Basis sets (FLAPW method)

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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]






Many-body Schrödinger equation:


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=">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</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


DFT

GW

exp

exp


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


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


PIJ(k, ⇤) =⇤

�,q

occ⇤

n

unocc⇤

n�

...

�1

⇤ + ��qn � ��

k+qn� + i⇥� ...

⇥


εF


PIJ(k, ⇤) =⇤

�,q

occ⇤

n

unocc⇤

n�

...

�1

⇤ + ��qn � ��

k+qn� + i⇥� ...

⇥

metal


metal

εF


PIJ(k, ⇤) =⇤

�,q

occ⇤

n

unocc⇤

n�

...

�1

⇤ + ��qn � ��

k+qn� + i⇥� ...

⇥


metal

εF


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.


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



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



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


• 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}


• 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


• SPEX: Generate special equidistant k-point setk, k', k+k’, and 0 must be elements


• SPEX: GW calculationè Quasiparticle energies spex.inp: JOB GW IBZ:(1-4)


• SPEX: Generate special equidistant k-point setk, q, k+q, and 0 must be elements

Start of iterations


• 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.

Documents

Many-Body Perturbation Theory The GW approximation · Overview lIntroduction lTheory l Green function l Feynman diagrams l GWapproximation lImplementation l Basis sets (FLAPW method)