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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
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Outline
One-particle Green functions
Polarization propagator and two-particle Green functions
The GW approximation
The Bethe-Salpeter equation (BSE)
Summary
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Green functions in mathematicsOne-particle Green functionsPerturbation theory and Feynman diagramsSelf energy and Dyson equation
One-particle Green functions
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Green functions in mathematicsOne-particle Green functionsPerturbation theory and Feynman diagramsSelf energy and Dyson equation
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.
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Green functions in mathematicsOne-particle Green functionsPerturbation theory and Feynman diagramsSelf energy and Dyson 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)
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Green functions in mathematicsOne-particle Green functionsPerturbation theory and Feynman diagramsSelf energy and Dyson equation
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
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Green functions in mathematicsOne-particle Green functionsPerturbation theory and Feynman diagramsSelf energy and Dyson equation
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
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Green functions in mathematicsOne-particle Green functionsPerturbation theory and Feynman diagramsSelf energy and Dyson equation
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
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Green functions in mathematicsOne-particle Green functionsPerturbation theory and Feynman diagramsSelf energy and Dyson equation
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
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Green functions in mathematicsOne-particle Green functionsPerturbation theory and Feynman diagramsSelf energy and Dyson equation
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 〉
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Green functions in mathematicsOne-particle Green functionsPerturbation theory and Feynman diagramsSelf energy and Dyson equation
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
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Green functions in mathematicsOne-particle Green functionsPerturbation theory and Feynman diagramsSelf energy and Dyson equation
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
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Green functions in mathematicsOne-particle Green functionsPerturbation theory and Feynman diagramsSelf energy and Dyson equation
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
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Green functions in mathematicsOne-particle Green functionsPerturbation theory and Feynman diagramsSelf energy and Dyson equation
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′|
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Green functions in mathematicsOne-particle Green functionsPerturbation theory and Feynman diagramsSelf energy and Dyson equation
Diagrammatic series for Green function
Perturbation series for G: sum of all connected diagrams
= + + +
+ ++ + . . . .+
+
Lots of diagrams!
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Green functions in mathematicsOne-particle Green functionsPerturbation theory and Feynman diagramsSelf energy and Dyson equation
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
= + +
+ ++ + . . . .
++
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Green functions in mathematicsOne-particle Green functionsPerturbation theory and Feynman diagramsSelf energy and Dyson equation
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
= +
+ . . . .+
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Green functions in mathematicsOne-particle Green functionsPerturbation theory and Feynman diagramsSelf energy and Dyson equation
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)
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Green functions in mathematicsOne-particle Green functionsPerturbation theory and Feynman diagramsSelf energy and Dyson equation
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
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Green functions in mathematicsOne-particle Green functionsPerturbation theory and Feynman diagramsSelf energy and Dyson equation
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 !
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Green functions in mathematicsOne-particle Green functionsPerturbation theory and Feynman diagramsSelf energy and Dyson equation
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′)
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Linear density responseTwo-particle Green function and polarization propagatorParticle-hole propagator: diagrammatic representationHedin’s equations
Polarization propagator andtwo-particle Green functions
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Linear density responseTwo-particle Green function and polarization propagatorParticle-hole propagator: diagrammatic representationHedin’s equations
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
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Linear density responseTwo-particle Green function and polarization propagatorParticle-hole propagator: diagrammatic representationHedin’s equations
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 〉
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Linear density responseTwo-particle Green function and polarization propagatorParticle-hole propagator: diagrammatic representationHedin’s equations
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!
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Linear density responseTwo-particle Green function and polarization propagatorParticle-hole propagator: diagrammatic representationHedin’s equations
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
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Linear density responseTwo-particle Green function and polarization propagatorParticle-hole propagator: diagrammatic representationHedin’s equations
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)
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Linear density responseTwo-particle Green function and polarization propagatorParticle-hole propagator: diagrammatic representationHedin’s equations
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)
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Linear density responseTwo-particle Green function and polarization propagatorParticle-hole propagator: diagrammatic representationHedin’s equations
Lehmann representation of polarization propagator
Π(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
compare with Lehmann representation of linear density response
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
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Linear density responseTwo-particle Green function and polarization propagatorParticle-hole propagator: diagrammatic representationHedin’s equations
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:
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Linear density responseTwo-particle Green function and polarization propagatorParticle-hole propagator: diagrammatic representationHedin’s equations
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
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Linear density responseTwo-particle Green function and polarization propagatorParticle-hole propagator: diagrammatic representationHedin’s equations
Effective interaction and dielectric function
Effective interaction
W =: ε−1vClb = vClb + vClbPW
Dielectric function
ε = 1− vClbP
Inverse dielectric function
ε−1 = 1 + vClbΠ
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Linear density responseTwo-particle Green function and polarization propagatorParticle-hole propagator: diagrammatic representationHedin’s equations
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
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Linear density responseTwo-particle Green function and polarization propagatorParticle-hole propagator: diagrammatic representationHedin’s equations
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
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Band gaps and quasiparticle bands in GWOptical absorption in GW
The GW approximation
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Band gaps and quasiparticle bands in GWOptical absorption in GW
GW approximation
In the GW approximation the vertex is approximated as: Γ ≈ 1
Hedin’s equations (exact) GW approximation
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Band gaps and quasiparticle bands in GWOptical absorption in GW
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)
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Band gaps and quasiparticle bands in GWOptical absorption in GW
GW propaganda slide: improvement of gaps over LDA
Schilfgaarde, PRL 96, 226402 (2006)
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Band gaps and quasiparticle bands in GWOptical absorption in GW
GW quasiparticle bands in copper
dashed: LDA, solid: GW, dots: expt.
Marini, Onida, Del Sole, PRL 88, 016403 (2001)
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Band gaps and quasiparticle bands in GWOptical absorption in GW
Absorption by independent Kohn-Sham particles
Independent transitions:
Im[ε(ω)] =8π2
ω2
∑ij |〈ϕj |e · v|ϕi〉|2δ(εj−εi−ω)
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Band gaps and quasiparticle bands in GWOptical absorption in GW
Absorption by independent Kohn-Sham particles
Independent transitions:
Im[ε(ω)] =8π2
ω2
∑ij |〈ϕj |e · v|ϕi〉|2δ(εj−εi−ω)
Particles are interacting!
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Band gaps and quasiparticle bands in GWOptical absorption in GW
Optical absorption in GW: Independent quasiparticles
Independent transitions:
Im[ε(ω)] =8π2
ω2
∑ij |〈ϕj |e·v|ϕi〉|2δ(Ej−Ei−ω)
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Band gaps and quasiparticle bands in GWOptical absorption in GW
Optical absorption in GW: Independent quasiparticles
Independent transitions:
Im[ε(ω)] =8π2
ω2
∑ij |〈ϕj |e·v|ϕi〉|2δ(Ej−Ei−ω)
Something still missing: the vertex, i.e., particle-hole interactions!
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Band gaps and quasiparticle bands in GWOptical absorption in GW
Absorption
Neutral excitations → poles of two-particle Green’s function Lin GW: excitonic effects, i.e., electron-hole interaction missing
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Band gaps and quasiparticle bands in GWOptical absorption in GW
Absorption
Neutral excitations → poles of two-particle Green’s function Lin GW: excitonic effects, i.e., electron-hole interaction missing
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Band gaps and quasiparticle bands in GWOptical absorption in GW
Absorption
Neutral excitations → poles of two-particle Green’s function Lin GW: excitonic effects, i.e., electron-hole interaction missing
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Derivation of BSEStandard approximations to BSE
The Bethe-Salpeter equation (BSE)
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Derivation of BSEStandard approximations to 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:
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Derivation of BSEStandard approximations to BSE
Derivation of the Bethe-Salpeter equation (2)
2nd step: Classification of scattering processes
Identify two-particle irreducible blocks
where γ(1234) is the electron-hole stattering amplitude
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Derivation of BSEStandard approximations to BSE
Derivation of the Bethe-Salpeter equation (3)
Final step: Summation of a geometric series
−→ Bethe-Salpeter equation
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Derivation of BSEStandard approximations to BSE
Derivation of the Bethe-Salpeter equation (3)
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
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Derivation of BSEStandard approximations to BSE
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)
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Derivation of BSEStandard approximations to BSE
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
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Derivation of BSEStandard approximations to BSE
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)
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Derivation of BSEStandard approximations to BSE
Results: Continuum excitons (Si)
Bulk silicon
G. Onida, L. Reining, and A. Rubio, RMP 74, 601 (2002)
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Derivation of BSEStandard approximations to BSE
Results: Bound excitons (solid Ar)
Solid argon
F. Sottile, M. Marsili, V. Olevano, and L. Reining, PRB 76, 161103(R) (2007)
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Derivation of BSEStandard approximations to BSE
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
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Derivation of BSEStandard approximations to BSE
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)
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Derivation of BSEStandard approximations to BSE
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)
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Derivation of BSEStandard approximations to BSE
Thanks
Ilya Tokatly and Matteo Gatti for some figures
YOU for your patience!
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE
One-particle Green functionsPolarization propagator and two-particle Green functions
GW approximationBethe-Salpeter equation (BSE)
Derivation of BSEStandard approximations to BSE
Thanks
Ilya Tokatly and Matteo Gatti for some figures
YOU for your patience!
CEES 2015 Donostia-San Sebastian: S. Kurth Introduction to Green functions, GW, and BSE