Introduction to Green functions, the GW …nano-bio.ehu.es/files/kurth_green.pdfOne-particle Green functions Polarization propagator and two-particle Green functions GW approximation

One-particle Green functionsPolarization propagator and two-particle Green functions

GW approximationBethe-Salpeter equation (BSE)

Introduction to Green functions, the GWapproximation, and the Bethe-Salpeter equation

Stefan Kurth

1. Universidad del Paıs Vasco UPV/EHU, San Sebastian, Spain2. IKERBASQUE, Basque Foundation for Science, Bilbao, Spain

3. European Theoretical Spectroscopy Facility (ETSF), www.etsf.eu

CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE



Outline

One-particle Green functions

Polarization propagator and two-particle Green functions

The GW approximation

The Bethe-Salpeter equation (BSE)

Summary




Green functions in mathematicsOne-particle Green functionsPerturbation theory and Feynman diagramsSelf energy and Dyson equation

One-particle Green functions





Green functions in mathematics

consider inhomogeneous differential equation (1D for simplicity)

Dxy(x) = f(x)

where Dx is linear differential operator in x.

Example: damped harmonic oscillator Dx = d2

dx2+ γ d

dx + ω2

general solution of inhomogeneous equation:

y(x) = yhom(x) + yspec(x)

where yhom is solution of the homogeneous eqn. Dxyhom(x) = 0and yspec(x) is any special solution of the inhomogeneous equation.





Green functions in mathematics (cont.)

how to obtain a special solution of the inhomogeneous equation forany inhomogeneity f(x)?first find the solution of the following equation

DxG(x, x′) = δ(x− x′)

This defines the Green function G(x, x′) corresponding to theoperator Dx.

Once G(x, x′) is found, a special solution can be constructed by

yspec(x) =

∫dx′ G(x, x′)f(x′)

check: Dx

∫dx′ G(x, x′)f(x′) =

∫dx′ δ(x− x′)f(x′) = f(x)





Hamiltonian of interacting electrons

consider system of interacting electrons in static external potentialVext(r) described by Hamiltonian H

H = T + Vext + W =

∫d3x ψ†(x)

(−∇

2

2+ Vext(r)

)ψ(x)

+1

2

∫d3x

∫d3x′ ψ†(x)ψ†(x′)

1

|r− r′|ψ(x′)ψ(x)

x = (r, σ): space-spin coordinate

ψ†(x), ψ(x): electron creation and annihilation operators





One-particle Green functions at zero temperature

Time-ordered 1-particle Green function at zero temperature

iG(x, t;x′, t′) =〈ΨN

0 |T [ψ(x, t)H ψ†(x′, t′)H ]|ΨN

0 〉〈ΨN

0 |ΨN0 〉

|ΨN0 〉: N -particle ground state of H: H|ΨN

0 〉 = EN0 |ΨN0 〉

ψ(x, t)H = exp(iHt)ψ(x) exp(−iHt) :electron annihilation operator in Heisenberg picture

T : time-ordering operatorT [ψ(x, t)H ψ(x′, t′)†H ] =

θ(t− t′)ψ(x, t)H ψ(x′, t′)†H − θ(t′ − t)ψ(x′, t′)†H ψ(x, t)H





Green functions as propagator

t1 > t2 (t > t′)create electron at time t2 atposition r2 and propagate;

then annihilate electron at time t1at position r1

t2 > t1 (t′ > t)annihilate electron (create hole) at

time t1 at position r1;then create electron (annihilatehole) at time t2 at position r2





Observables from Green functions

Information which can be extracted from Green functions

ground-state expectation values of any single-particle operatorO =

∫d3x ψ†(x)o(x)ψ(x)

e.g., density operator n(r) =∑

σ ψ†(rσ)ψ(rσ)

ground-state energy of the system

Galitski-Migdal formula

EN0 = − i2

∫d3x lim

t′→t+limr′→r

(i∂

∂t− ∇

2

2

)G(rσ, t; r′σ, t′)

spectrum of system: direct photoemission, inversephotoemission





Spectral (Lehmann) representation of Green function

use completeness relation 1 =∑

N,k |ΨNk 〉〈ΨN

k | and Fouriertransform w.r.t. t− t′−→

Lehmann representation

G(x,x′;ω)

=∑k

gk(x)g∗k(x′)

ω − (EN+1k − EN0 ) + iη

+∑k

fk(x)f∗k (x′)

ω + (EN−1k − EN0 )− iη

with quasiparticle amplitudes

fk(x) = 〈ΨN−1k |ψ(x)|ΨN

0 〉

gk(x) = 〈ΨN0 |ψ(x)|ΨN+1

k 〉





Spectral information contained in Green function

Green function contains spectral information on single-particleexcitations changing the number of particles by one! The poles ofthe GF give the corresponding excitation energies.

direct photoemission inverse photoemission





Spectral function

Spectral function

A(x,x′;ω) = − 1

πImGR(x,x′;ω) =

∑k

gk(x)g∗k(x′)δ(ω+EN0 −EN+1

k )+fk(x)f∗k (x′)δ(ω+EN−1k −EN0 )

A(x,x′;ω): local density of states





Perturbation theory for Green function

Green function G(x, t;x′, t′) = −i〈ΨN0 |T [ψ(x, t)H ψ(x′, t′)†H ]|ΨN

0 〉is a complicated object, it involves many-body ground state |ΨN

0 〉−→ perturbation theory to calculate Green function: splitHamitonian in two parts

H = H0 + W = T + Vext + W

treat interaction W as perturbation −→ machinery of many-bodyperturbation theory: Wick’s theorem, Gell-Mann-Low theorem,and, most importantly, Feynman diagrams





Feynman diagrams

Feynman diagrams: graphical representation of perturbation series

elements of diagrams:

Green function G0 of noninteracting system (H0)

Green function G of interacting system

Coulomb interaction vClb(x, t;x′, t′) = δ(t−t′)|r−r′|





Diagrammatic series for Green function

Perturbation series for G: sum of all connected diagrams

= + + +

+ ++ + . . . .+

+

Lots of diagrams!





Self energy: reducible and irreducible

Self energy insertion and reducible self energy

Self energy insertion: any part of a diagram which isconnected to the rest of the diagram by two G0-lines, oneincoming and one outgoing

Reducible self energy Σ: sum of all self-energy insertions

= + +

+ ++ + . . . .

++





Self energy: reducible and irreducible

Proper self energy insertion and irreducible (proper) self energy

Proper self energy insertion: any self energy insertion whichcannot be separated in two pieces by cutting a single G0-line

Irreducible self energy Σ: sum of all proper self-energyinsertions

= +

+ . . . .+





Dyson equation

= +

= +

+ . . . .

+

+ +

=

Dyson equation

G(x,x′;ω) = G0(x,x′;ω)

+

∫d3y

∫d3y′G0(x,y;ω)Σ(y,y′;ω)G(y′,x′;ω)





Skeletons and dressed skeletons

Skeleton diagram: self-energy diagram which does contain noother self-energy insertions except itself

Dressed skeleton: replace all G0-lines in a skeleton by G-lines −→irreduzible self energy: sum of all dressed skeleton diagrams

−→ Σ becomes functional of G: Σ = Σ[G]CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE




Equation of motion for Green function

Lehmann representation for G0

G0(x,x′;ω)

=∑k

θ(εk − εF )ϕk(x)ϕ∗k(x′)

ω − εk + iη+∑k

θ(εF − εk)ϕk(x)ϕ∗k(x′)

ω − εk − iη

act with operator ω − h0(x) = ω − (−∇2x2 + vext(x)) on G0

Equation of motion for non-interacting Green function G0

(ω − h0(x))G0(x,x′;ω) =

∑k

ϕk(x)ϕ∗k(x′) = δ(x− x′)

−→ G0 is a mathematical Green function !





Equation of motion for Green function (cont.)

act with ω − h0(x) on Dyson equation for G

Equation of motion for interacting Green function G

(ω− h0(x))G(x,x′;ω) = δ(x−x′) +

∫d3y′ Σ(x,y′;ω)G(y′,x′;ω)

or with time arguments(i∂

∂t− h0(x)

)G(x, t;x′, t′) = δ(x− x′)δ(t− t′)

+

∫d3y′

∫dt′′Σ(x, t;y′, t′′)G(y′, t′′,x′; t′)




Linear density responseTwo-particle Green function and polarization propagatorParticle-hole propagator: diagrammatic representationHedin’s equations

Polarization propagator andtwo-particle Green functions





Linear density response

Suppose we expose our interacting many-electron system to anexternal, time-dependent perturbation V (t) =

∫d3x δV (x, t)n(x)

we are interested in the change of the density

δn(x, t) = 〈ΨN (t)|n(x)|ΨN (t)〉 − 〈ΨN0 |n(x)|ΨN

0 〉

to linear order in δV (x, t)

time-dependent Schrodinger equation

i∂

∂t|ΨN (t)〉 =

(H + V (t)

)|ΨN (t)〉

in Heisenberg picture |ΨN (t)〉H = exp(iHt)|ΨN (t)〉 −→

i∂

∂t|ΨN (t)〉H = V (t)H |ΨN (t)〉H





Linear density response (cont.)

−→ to linear order in δV (x, t) we have

|ΨN (t)〉 = exp(−iHt)(

1− i∫ t

0dt′ V (t′)H

)|ΨN

0 〉

and for δn(x, t) =∫d3x′

∫∞0 dt′ χ(x, t;x′, t′)δV (x′, t′) with

linear density response function

iχ(x, t;x′, t′) = iΠR(x, t;x′, t′)

= θ(t− t′)〈ΨN0 |[ˆn(x, t)H , ˆn(x′, t′)H ]|ΨN

0 〉〈ΨN

0 |ΨN0 〉

with ˆn(x, t)H = n(x, t)H − 〈ΨN0 |n(x)|ΨN

0 〉





Linear density response (cont.)

Lehmann representation of linear density response function

χ(x,x′;ω) = ΠR(x,x′;ω) =∑k

〈ΨN0 |ˆn(x)|ΨN

k 〉〈ΨNk |ˆn(x′)|ΨN

0 〉ω − (ENk − EN0 ) + iη

−∑k

〈ΨN0 |ˆn(x′)|ΨN

k 〉〈ΨNk |ˆn(x)|ΨN

0 〉ω + (ENk − EN0 ) + iη

note: the poles of χ are at the optical excitation energies of thesystem, i.e., excitations for which the number of particles does notchange!





Macroscopic response in solids

Optical absorption from macroscopic dielectric function εM (ω)

δV ext(r, t) = V ext(q)e−i(ωt−qr) , q � G

In a periodic system, induced potential δV ind contains allcomponents with k = q + G

δV ind(r, t) = e−iωt∑G

V indG (q)ei(q+G)r

Fourier component of the total potential in a solid:

V totG (q) = δG,0V

ext(q)+V indG (q) = [δG,0 + vG(q)χG,0(q, ω)]V ext(q)

with response function in momentum representation

χG,G′(q, ω) =

∫dr1dr2e

i(q+G)r1χ(r1, r2, ω)e−i(q+G′)r2





Macroscopic response in solids

Macroscopic field and macroscopic dielectric function

Macroscopic (averaged) potential: V totM (q) = V tot

G=0(q)

Macroscopic dielectric function:

V ext(q) = εM (q, ω)V totM (q)

εM (q, ω) =1

1 + vG=0(q)χ0,0(q, ω)

optical absorption rate

Abs(ω) = limq→0

Im εM (q, ω)





Two-particle Green function and polarization propagator

Two-particle Green function

i2G(2)(x1, t1;x2, t2;x3, t3;x4, t4) =1

〈ΨN0 |ΨN

0 〉

〈ΨN0 |T [ψ(x1, t1)H ψ(x2, t2)H ψ

†(x3, t3)H ψ†(x4, t4)H ]|ΨN

0 〉

Polarization propagator

iΠ(x, t;x′, t′) =〈ΨN

0 |T [ˆn(x, t)H ˆn(x′, t′)H ]|ΨN0 〉

〈ΨN0 |ΨN

0 〉

relation between the two:

i2G(2)(x1, t1;x2, t2;x1, t+1 ;x2, t

+2 ) = iΠ(x1, t1;x2, t2)+n(x1)n(x2)





Lehmann representation of polarization propagator

Π(x,x′;ω) =∑k

〈ΨN0 |ˆn(x)|ΨN


0 〉ω − (ENk − EN0 ) + iη

−∑k



0 〉ω + (ENk − EN0 )− iη

compare with Lehmann representation of linear density response

χ(x,x′;ω) = ΠR(x,x′;ω) =∑k

〈ΨN0 |ˆn(x)|ΨN


0 〉ω − (ENk − EN0 ) + iη

−∑k



0 〉ω + (ENk − EN0 ) + iη





Particle-hole propagator: diagrammatic representation

Definition of particle-hole propagator

The particle-hole propagator is the two-particle Green function witha time-ordering such that both the two latest and the two earliesttimes correspond to one creation and one annihilation operator

Diagrammatic representation:

x 1

, t 1

L(x1, t

1; x

2, t

2; x

3, t

3; x

4, t

4) +=

x 4

, t 4

x 3

, t 3

x 2

, t 2

x 4

, t 4

x 1

, t 1

x 3

, t 3

x 2

, t 2

Diagrammatic representation of polarization propagator:





Polarization propagator and irreducible polarizationinsertions

Irreducible polarization insertion

A diagram for the polarization propagator which cannot be reducedto lower-order diagrams for Π by cutting a single interaction line

Def:

−→ Dyson-like eqn. for fullpolarization propagator





Effective interaction and dielectric function

Effective interaction

W =: ε−1vClb = vClb + vClbPW

Dielectric function

ε = 1− vClbP

Inverse dielectric function

ε−1 = 1 + vClbΠ





Vertex insertions

Vertex insertion

(part of a) diagram with one external in- and one outgoing G0-lineand one external interaction line

Irreducible vertex insertion

A vertex insertion which has no self-energy insertions on the in-and outgoing G0-lines and no polarization insertion on the externalinteraction line





Irreducible vertex and Hedin’s equations

Irreducible vertex

Hedin’s equations (exact!)

L. Hedin, Phys. Rev. 139 (1965)

Hedin’s equations

Σ = vHart + iGWΓ

iP = GGΓ

G = G0 +G0ΣG

W = vClb + vClbPW

Γ = 1 +δΣ

δGGGΓ




Band gaps and quasiparticle bands in GWOptical absorption in GW

The GW approximation





GW approximation

In the GW approximation the vertex is approximated as: Γ ≈ 1

Hedin’s equations (exact) GW approximation





GW approximation

Perturbative GW corrections

h0(r)ϕi(r) + Vxc(r)ϕi(r) = εiϕi(r)

h0(r)φi(r) +

∫dr′ Σ(r, r′, ω = Ei) φi(r

′) = Ei φi(r)

First-order perturbative corrections with Σ = GW :

Ei − εi = 〈ϕi|Σ− Vxc|ϕi〉

Hybertsen and Louie, PRB 34, 5390 (1986);

Godby, Schluter and Sham, PRB 37, 10159 (1988)





GW propaganda slide: improvement of gaps over LDA

Schilfgaarde, PRL 96, 226402 (2006)





GW quasiparticle bands in copper

dashed: LDA, solid: GW, dots: expt.

Marini, Onida, Del Sole, PRL 88, 016403 (2001)





Absorption by independent Kohn-Sham particles

Independent transitions:

Im[ε(ω)] =8π2

ω2

∑ij |〈ϕj |e · v|ϕi〉|2δ(εj−εi−ω)





Absorption by independent Kohn-Sham particles


Im[ε(ω)] =8π2

ω2

∑ij |〈ϕj |e · v|ϕi〉|2δ(εj−εi−ω)

Particles are interacting!





Optical absorption in GW: Independent quasiparticles


Im[ε(ω)] =8π2

ω2

∑ij |〈ϕj |e·v|ϕi〉|2δ(Ej−Ei−ω)





Optical absorption in GW: Independent quasiparticles


Im[ε(ω)] =8π2

ω2

∑ij |〈ϕj |e·v|ϕi〉|2δ(Ej−Ei−ω)

Something still missing: the vertex, i.e., particle-hole interactions!





Absorption

Neutral excitations → poles of two-particle Green’s function Lin GW: excitonic effects, i.e., electron-hole interaction missing





Absorption






Absorption





Derivation of BSEStandard approximations to BSE

The Bethe-Salpeter equation (BSE)





Derivation of the Bethe-Salpeter equation (1)

Propagator of e-h pair in a many-body system:

1st step: Dressing one-particle propagators (Dyson equation)

Propagation of dressed interacting electron and hole:






2nd step: Classification of scattering processes

Identify two-particle irreducible blocks

where γ(1234) is the electron-hole stattering amplitude






Final step: Summation of a geometric series

−→ Bethe-Salpeter equation






Final step: Summation of a geometric series

−→ Bethe-Salpeter equation

Analytic form of the Bethe-Salpeter equation (j = {xj , tj})

L(1234) = L0(1234)+∫L0(1256)[v(57)δ(56)δ(78)− γ(5678)]L(7834)d5d6d7d8





The Bethe-Salpeter equation: Approximation

BSE determines electron-hole propagator L(1234), provided thatself-energy Σ(12) and e-h scattering amplitude γ(1234) are given.

Standard approximation:

Approximate Σ by GW diagram: Σ(12) = G(12)W (12)

== +

Approximate γ by W : γ(1234) = W (12)δ(13)δ(24)





The Bethe-Salpeter equation: Approximation

Approximate Bethe-Salpeter equation

Analytic form of the approximate Bethe-Salpeter equation

L(1234) = L0(1234) +

∫L0(1256)[v(57)δ(56)δ(78)−

W (56)δ(57)δ(68)]L(7834)d5d6d7d8

L0(1234) = G(12)G(43) and W (12) from GW calculation





BSE calculations

A three-step method

1 LDA calculation⇒ Kohn-Sham wavefunctions ϕi

2 GW calculation⇒ GW energies Ei and screened Coulomb interaction W

3 BSE calculation

solution of L = L0 + L0(v − γ)L

⇒ e-h propagator L(r1r2r3r4ω)

⇒ spectra εM (ω)





Results: Continuum excitons (Si)

Bulk silicon

G. Onida, L. Reining, and A. Rubio, RMP 74, 601 (2002)





Results: Bound excitons (solid Ar)

Solid argon

F. Sottile, M. Marsili, V. Olevano, and L. Reining, PRB 76, 161103(R) (2007)





Summary

Green functions: important concept in many-particle physics

Diagrammatic analysis of Green functions (deceptively)simple, actual calculation of specific diagrams much harder

GW approximation for self energy

Bethe-Salpeter equation: captures excitons in opticalabsorption





Literature

endless number of textbooks on Green functions

My favorites

E.K.U. Gross, E. Runge, O. Heinonen, Many-Particle Theory(Hilger, Bristol, 1991)

A.L. Fetter, J.D. Walecka, Quantum Theory of Many-ParticleSystems (McGraw-Hill, New York, 1971) and later edition byDover press

G. Stefanucci, R. van Leeuwen, Nonequilibrium Many-BodyTheory of Quantum Systems: A Modern Introduction(Cambridge, 2003)





Literature

on GW and BSE

G. Onida, L. Reining, A. Rubio, Rev. Mod. Phys. 74, 601(2002)

S. Botti, A. Schindlmayr, R. Del Sole, and L. Reining, Rep.Progr. Phys. 70, 357 (2007)





Thanks

Ilya Tokatly and Matteo Gatti for some figures

YOU for your patience!





Thanks

Ilya Tokatly and Matteo Gatti for some figures

YOU for your patience!


Introduction to Green functions, the GW …nano-bio.ehu.es/files/kurth_green.pdfOne-particle Green functions Polarization propagator and two-particle Green functions GW approximation

Documents