Introduction to Green functions, the GW approximation, and the

Transcription

One-particle Green functions
Polarization propagator and two-particle Green functions
GW approximation
Bethe-Salpeter equation (BSE)
Introduction to Green functions, the GW
approximation, and the Bethe-Salpeter equation
Stefan Kurth
1. Universidad del Paı́s Vasco UPV/EHU, San Sebastián, Spain
2. IKERBASQUE, Basque Foundation for Science, Bilbao, Spain
3. European Theoretical Spectroscopy Facility (ETSF), www.etsf.eu
CEES 2015 Donostia-San Sebastián: S. Kurth
Introduction to Green functions, GW, and BSE
GW approximation
Outline
The GW approximation
The Bethe-Salpeter equation (BSE)
Summary
GW approximation
Green functions in mathematics
Perturbation theory and Feynman diagrams
Self energy and Dyson equation
GW approximation
consider inhomogeneous differential equation (1D for simplicity)
D̂x y(x) = f (x)
where D̂x is linear differential operator in x.
Example: damped harmonic oscillator D̂x =
d2
dx2
d
+ γ dx
+ ω2
general solution of inhomogeneous equation:
y(x) = yhom (x) + yspec (x)
where yhom is solution of the homogeneous eqn. D̂x yhom (x) = 0
and yspec (x) is any special solution of the inhomogeneous equation.
GW approximation
Green functions in mathematics (cont.)
how to obtain a special solution of the inhomogeneous equation for
any inhomogeneity f (x)?
first find the solution of the following equation
D̂x G(x, x0 ) = δ(x − x0 )
This defines the Green function G(x, x0 ) corresponding to the
operator D̂x .
Once G(x, x0 ) is found, a special solution can be constructed by
Z
yspec (x) =
check: D̂x
R
dx0 G(x, x0 )f (x0 )
dx0 G(x, x0 )f (x0 ) =
R
dx0 δ(x − x0 )f (x0 ) = f (x)
GW approximation
Hamiltonian of interacting electrons
consider system of interacting electrons in static external potential
Vext (r) described by Hamiltonian Ĥ
Z
Ĥ = T̂ + V̂ext + Ŵ =
1
+
2
Z
3
d x
Z
∇2
+ Vext (r) ψ̂(x)
d x ψ̂ (x) −
2
3
†
d3 x0 ψ̂ † (x)ψ̂ † (x0 )
1
ψ̂(x0 )ψ̂(x)
|r − r0 |
x = (r, σ): space-spin coordinate
ψ̂ † (x), ψ̂(x): electron creation and annihilation operators
GW approximation
One-particle Green functions at zero temperature
Time-ordered 1-particle Green function at zero temperature
iG(x, t; x0 , t0 ) =
† 0 0
N
hΨN
0 |T̂ [ψ̂(x, t)H ψ̂ (x , t )H ]|Ψ0 i
N
hΨN
0 |Ψ0 i
N
N
N
|ΨN
0 i: N -particle ground state of Ĥ: Ĥ|Ψ0 i = E0 |Ψ0 i
ψ̂(x, t)H = exp(iĤt)ψ̂(x) exp(−iĤt) :
electron annihilation operator in Heisenberg picture
T̂ : time-ordering operator
T̂ [ψ̂(x, t)H ψ̂(x0 , t0 )†H ] =
θ(t − t0 )ψ̂(x, t)H ψ̂(x0 , t0 )†H − θ(t0 − t)ψ̂(x0 , t0 )†H ψ̂(x, t)H
GW approximation
Green functions as propagator
t1 > t2 (t > t0 )
create electron at time t2 at
position r2 and propagate;
then annihilate electron at time t1
at position r1
t2 > t1 (t0 > t)
annihilate electron (create hole) at
time t1 at position r1 ;
then create electron (annihilate
hole) at time t2 at position r2
GW approximation
Observables from Green functions
Information which can be extracted from Green functions
ground-state
expectation values of any single-particle operator
R
Ô = d3 x ψ̂ † (x)o(x)ψ̂(x)
P
e.g., density operator n̂(r) = σ ψ̂ † (rσ)ψ̂(rσ)
ground-state energy of the system
Galitski-Migdal formula
Z
∂
i
∇2
3
N
d x lim lim
i −
E0 = −
G(rσ, t; r0 σ, t0 )
2
∂t
2
t0 →t+ r0 →r
spectrum of system: direct photoemission, inverse
photoemission
GW approximation
Spectral (Lehmann) representation of Green function
use completeness relation 1 =
transform w.r.t. t − t0 −→
P
N,k
N
|ΨN
k ihΨk | and Fourier
Lehmann representation
G(x, x0 ; ω)
=
X
gk (x)gk∗ (x0 )
k
ω − (EkN +1 − E0N ) + iη
+
X
fk (x)fk∗ (x0 )
k
ω + (EkN −1 − E0N ) − iη
with quasiparticle amplitudes
−1
fk (x) = hΨN
|ψ̂(x)|ΨN
0 i
k
N +1
i
gk (x) = hΨN
0 |ψ̂(x)|Ψk
GW approximation
Spectral information contained in Green function
Green function contains spectral information on single-particle
excitations changing the number of particles by one! The poles of
the GF give the corresponding excitation energies.
direct photoemission
inverse photoemission
GW approximation
Spectral function
Spectral function
1
A(x, x0 ; ω) = − Im GR (x, x0 ; ω) =
π
X
gk (x)gk∗ (x0 )δ(ω +E0N −EkN +1 )+fk (x)fk∗ (x0 )δ(ω +EkN −1 −E0N )
k
A(x, x0 ; ω): local density of states
GW approximation
Perturbation theory for Green function
0 0 †
N
Green function G(x, t; x0 , t0 ) = −ihΨN
0 |T̂ [ψ̂(x, t)H ψ̂(x , t )H ]|Ψ0 i
is a complicated object, it involves many-body ground state |ΨN
0 i
−→ perturbation theory to calculate Green function: split
Hamitonian in two parts
Ĥ = Ĥ0 + Ŵ = T̂ + V̂ext + Ŵ
treat interaction Ŵ as perturbation −→ machinery of many-body
perturbation theory: Wick’s theorem, Gell-Mann-Low theorem,
and, most importantly, Feynman diagrams
GW approximation
Feynman diagrams
Feynman diagrams: graphical representation of perturbation series
elements of diagrams:
Green function G0 of noninteracting system (Ĥ0 )
Green function G of interacting system
Coulomb interaction vClb (x, t; x0 , t0 ) =
δ(t−t0 )
|r−r0 |
GW approximation
Diagrammatic series for Green function
Perturbation series for G: sum of all connected diagrams
=
+
+
+
+
+
+
+
+
+
....
Lots of diagrams!
GW approximation
Self energy: reducible and irreducible
Self energy insertion and reducible self energy
Self energy insertion: any part of a diagram which is
connected to the rest of the diagram by two G0 -lines, one
incoming and one outgoing
Reducible self energy Σ̃: sum of all self-energy insertions
=
+
+
+
+
+
+
+
+
....
GW approximation
Self energy: reducible and irreducible
Proper self energy insertion and irreducible (proper) self energy
Proper self energy insertion: any self energy insertion which
cannot be separated in two pieces by cutting a single G0 -line
Irreducible self energy Σ: sum of all proper self-energy
insertions
=
+
+
+
....
GW approximation
Dyson equation
=
+
=
+
=
+
+
+
Dyson equation
G(x, x0 ; ω) = G0 (x, x0 ; ω)
Z
Z
3
+ d y d3 y 0 G0 (x, y; ω)Σ(y, y0 ; ω)G(y0 , x0 ; ω)
+....
GW approximation
Skeletons and dressed skeletons
Skeleton diagram: self-energy diagram which does contain no
other 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]
GW approximation
Equation of motion for Green function
Lehmann representation for G0
=
G0 (x, x0 ; ω)
X θ(εk − εF )ϕk (x)ϕ∗ (x0 )
k
k
ω − εk + iη
+
X θ(εF − εk )ϕk (x)ϕ∗ (x0 )
k
ω − εk − iη
k
2
act with operator ω − ĥ0 (x) = ω − (− ∇2x + vext (x)) on G0
Equation of motion for non-interacting Green function G0
X
(ω − ĥ0 (x))G0 (x, x0 ; ω) =
ϕk (x)ϕ∗k (x0 ) = δ(x − x0 )
k
−→ G0 is a mathematical Green function !
GW approximation
Equation of motion for Green function (cont.)
act with ω − ĥ0 (x) on Dyson equation for G
Equation of motion for interacting Green function G
0
0
Z
(ω − ĥ0 (x))G(x, x ; ω) = δ(x − x ) + d3 y 0 Σ(x, y0 ; ω)G(y0 , x0 ; ω)
or with time arguments
∂
i − ĥ0 (x) G(x, t; x0 , t0 ) = δ(x − x0 )δ(t − t0 )
∂t
Z
Z
3 0
+ d y
dt00 Σ(x, t; y0 , t00 )G(y0 , t00 , x0 ; t0 )
GW approximation
Linear density response
Two-particle Green function and polarization propagator
Particle-hole propagator: diagrammatic representation
Hedin’s equations
Polarization propagator and
two-particle Green functions
GW approximation
Hedin’s equations
Suppose we expose our interacting many-electron
R system to an
external, time-dependent perturbation V̂ (t) = d3 x δV (x, t)n̂(x)
we are interested in the change of the density
N
δn(x, t) = hΨN (t)|n̂(x)|ΨN (t)i − hΨN
0 |n̂(x)|Ψ0 i
to linear order in δV (x, t)
time-dependent Schrödinger equation
∂
i |ΨN (t)i = Ĥ + V̂ (t) |ΨN (t)i
∂t
in Heisenberg picture |ΨN (t)iH = exp(iĤt)|ΨN (t)i −→
i
∂ N
|Ψ (t)iH = V̂ (t)H |ΨN (t)iH
∂t
GW approximation
Hedin’s equations
Linear density response (cont.)
−→ to linear order in δV (x, t) we have
Z t
N
0
0
|Ψ (t)i = exp(−iĤt) 1 − i
dt V̂ (t )H |ΨN
0 i
0
R∞
and for δn(x, t) = d3 x0 0 dt0 χ(x, t; x0 , t0 )δV (x0 , t0 ) with
R
linear density response function
iχ(x, t; x0 , t0 ) = iΠR (x, t; x0 , t0 )
= θ(t − t0 )
N
ˆ
ˆ 0 0
hΨN
0 |[ñ(x, t)H , ñ(x , t )H ]|Ψ0 i
N
N
hΨ0 |Ψ0 i
ˆ t)H = n̂(x, t)H − hΨN |n̂(x)|ΨN i
with ñ(x,
0
0
GW approximation
Hedin’s equations
Linear density response (cont.)
Lehmann representation of linear density response function
χ(x, x0 ; ω) = ΠR (x, x0 ; ω) =
N ihΨN |ñ(x
X hΨN |ñ(x)|Ψ
ˆ
ˆ 0 )|ΨN i
0
k
−
k
k
0
ω − (EkN − E0N ) + iη
Ni
X hΨN |ñ(x
ˆ 0 )|ΨN ihΨN |ñ(x)|Ψ
ˆ
0
k
k
k
0
ω + (EkN − E0N ) + iη
note: the poles of χ are at the optical excitation energies of the
system, i.e., excitations for which the number of particles does not
change!
GW approximation
Hedin’s equations
Macroscopic response in solids
Optical absorption from macroscopic dielectric function M (ω)
δV ext (r, t) = V ext (q)e−i(ωt−qr) ,
qG
In a periodic system, induced potential δV ind contains all
components with k = q + G
X
VGind (q)ei(q+G)r
δV ind (r, t) = e−iωt
G
Fourier component of the total potential in a solid:
VGtot (q) = δG,0 V ext (q)+VGind (q) = [δG,0 + vG (q)χG,0 (q, ω)] V ext (q)
with response function in momentum representation
Z
0
χG,G0 (q, ω) = dr1 dr2 ei(q+G)r1 χ(r1 , r2 , ω)e−i(q+G )r2
GW approximation
Hedin’s equations
Macroscopic response in solids
Macroscopic field and macroscopic dielectric function
Macroscopic (averaged) potential:
tot (q) = V tot (q)
VM
G=0
Macroscopic dielectric function:
tot
V ext (q) = M (q, ω)VM
(q)
M (q, ω) =
1
1 + vG=0 (q)χ0,0 (q, ω)
optical absorption rate
Abs(ω) = lim Im M (q, ω)
q→0
GW approximation
Hedin’s equations
Two-particle Green function
i2 G(2) (x1 , t1 ; x2 , t2 ; x3 , t3 ; x4 , t4 ) =
1
N
hΨN
0 |Ψ0 i
†
N
†
hΨN
0 |T̂ [ψ̂(x1 , t1 )H ψ̂(x2 , t2 )H ψ̂ (x3 , t3 )H ψ̂ (x4 , t4 )H ]|Ψ0 i
Polarization propagator
iΠ(x, t; x0 , t0 ) =
N
ˆ
ˆ 0 0
hΨN
0 |T̂ [ñ(x, t)H ñ(x , t )H ]|Ψ0 i
N
N
hΨ0 |Ψ0 i
relation between the two:
+
i2 G(2) (x1 , t1 ; x2 , t2 ; x1 , t+
1 ; x2 , t2 ) = iΠ(x1 , t1 ; x2 , t2 )+n(x1 )n(x2 )
GW approximation
Hedin’s equations
Lehmann representation of polarization propagator
Π(x, x0 ; ω) =
N ihΨN |ñ(x
X hΨN |ñ(x)|Ψ
ˆ
ˆ 0 )|ΨN i
0
ω−
k
−
k
(EkN
−
k
E0N )
0
+ iη
Ni
X hΨN |ñ(x
ˆ
0
ω+
k
k
N
(Ek −
k
E0N )
0
− iη
compare with Lehmann representation of linear density response
χ(x, x0 ; ω) = ΠR (x, x0 ; ω) =
N ihΨN |ñ(x
X hΨN |ñ(x)|Ψ
ˆ
ˆ 0 )|ΨN i
0
k
−
k
k
0
ω − (EkN − E0N ) + iη
Ni
X hΨN |ñ(x
ˆ
0
k
k
k
ω + (EkN − E0N ) + iη
0
GW approximation
Hedin’s equations
Definition of particle-hole propagator
The particle-hole propagator is the two-particle Green function with
a time-ordering such that both the two latest and the two earliest
times correspond to one creation and one annihilation operator
Diagrammatic representation:
x 4, t
L(x1 , t1 ; x2 , t2 ; x3 , t3 ; x4 , t4 )
4
x 1, t
1
=
x 4, t
4
x 1, t
1
x 2, t
2
x 3, t
3
+
x 2, t
2
x 3, t
3
Diagrammatic representation of polarization propagator:
GW approximation
Hedin’s equations
Polarization propagator and irreducible polarization
insertions
Irreducible polarization insertion
A diagram for the polarization propagator which cannot be reduced
to lower-order diagrams for Π by cutting a single interaction line
Def:
−→ Dyson-like eqn. for full
polarization propagator
GW approximation
Hedin’s equations
Effective interaction and dielectric function
Effective interaction
W =: ε−1 vClb = vClb + vClb P W
Dielectric function
ε = 1 − vClb P
Inverse dielectric function
ε−1 = 1 + vClb Π
GW approximation
Hedin’s equations
Vertex insertions
Vertex insertion
(part of a) diagram with one external in- and one outgoing G0 -line
and one external interaction line
Irreducible vertex insertion
A vertex insertion which has no self-energy insertions on the inand outgoing G0 -lines and no polarization insertion on the external
interaction line
GW approximation
Hedin’s equations
Irreducible vertex and Hedin’s equations
Irreducible vertex
Hedin’s equations (exact!)
Hedin’s equations
Σ = vHart + iGW Γ
iP = GGΓ
G = G0 + G0 ΣG
W = vClb + vClb P W
L. Hedin, Phys. Rev. 139 (1965)
Γ=1+
δΣ
GGΓ
δG
GW approximation
Band gaps and quasiparticle bands in GW
Optical absorption in GW
The GW approximation
GW approximation
GW approximation
In the GW approximation the vertex is approximated as: Γ ≈ 1
Hedin’s equations (exact)
GW approximation
GW approximation
GW approximation
Perturbative GW corrections
ĥ0 (r)ϕi (r) + Vxc (r)ϕi (r) = i ϕi (r)
Z
ĥ0 (r)φi (r) +
dr0 Σ(r, r0 , ω = Ei ) φi (r0 ) = Ei φi (r)
First-order perturbative corrections with Σ = GW :
Ei − i = hϕi |Σ − Vxc |ϕi i
Hybertsen and Louie, PRB 34, 5390 (1986);
Godby, Schlüter and Sham, PRB 37, 10159 (1988)
GW approximation
GW propaganda slide: improvement of gaps over LDA
Schilfgaarde, PRL 96, 226402 (2006)
GW approximation
GW quasiparticle bands in copper
dashed: LDA, solid: GW, dots: expt.
Marini, Onida, Del Sole, PRL 88, 016403 (2001)
GW approximation
Absorption by independent Kohn-Sham particles
Independent transitions:
Im[(ω)] =
P
8π 2
2
ij |hϕj |e · v̂|ϕi i| δ(εj − εi − ω)
ω2
GW approximation
Absorption by independent Kohn-Sham particles
Im[(ω)] =
P
8π 2
2
ij |hϕj |e · v̂|ϕi i| δ(εj − εi − ω)
ω2
Particles are interacting!
GW approximation
Optical absorption in GW: Independent quasiparticles
Im[(ω)] =
P
8π 2
2
ij |hϕj |e·v̂|ϕi i| δ(Ej −Ei −ω)
ω2
GW approximation
Optical absorption in GW: Independent quasiparticles
Im[(ω)] =
P
8π 2
2
ij |hϕj |e·v̂|ϕi i| δ(Ej −Ei −ω)
ω2
Something still missing: the vertex, i.e., particle-hole interactions!
GW approximation
Absorption
Neutral excitations → poles of two-particle Green’s function L
in GW: excitonic effects, i.e., electron-hole interaction missing
GW approximation
Absorption
GW approximation
Absorption
GW approximation
Derivation of BSE
Standard approximations to BSE
The Bethe-Salpeter equation (BSE)
GW approximation
Derivation of 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:
GW approximation
Derivation of BSE
2nd step: Classification of scattering processes
Identify two-particle irreducible blocks
where γ(1234) is the electron-hole stattering amplitude
GW approximation
Derivation of BSE
Final step: Summation of a geometric series
−→ Bethe-Salpeter equation
GW approximation
Derivation of BSE
Final step: Summation of a geometric series
−→ Bethe-Salpeter equation
Analytic form of the Bethe-Salpeter equation (j = {xj , tj })
L(1234) = L0 (1234)+
Z
L0 (1256)[v(57)δ(56)δ(78) − γ(5678)]L(7834)d5d6d7d8
GW approximation
Derivation of BSE
The Bethe-Salpeter equation: Approximation
BSE determines electron-hole propagator L(1234), provided that
self-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)
GW approximation
Derivation of BSE
The Bethe-Salpeter equation: Approximation
Approximate Bethe-Salpeter equation
Analytic form of the approximate Bethe-Salpeter equation
Z
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
GW approximation
Derivation of 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(r1 r2 r3 r4 ω)
⇒ spectra M (ω)
GW approximation
Derivation of BSE
Results: Continuum excitons (Si)
Bulk silicon
G. Onida, L. Reining, and A. Rubio, RMP 74, 601 (2002)
GW approximation
Derivation of BSE
Results: Bound excitons (solid Ar)
Solid argon
F. Sottile, M. Marsili, V. Olevano, and L. Reining, PRB 76, 161103(R) (2007)
GW approximation
Derivation of 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 optical
absorption
GW approximation
Derivation of 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-Particle
Systems (McGraw-Hill, New York, 1971) and later edition by
Dover press
G. Stefanucci, R. van Leeuwen, Nonequilibrium Many-Body
Theory of Quantum Systems: A Modern Introduction
(Cambridge, 2003)
GW approximation
Derivation of 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)
GW approximation
Derivation of BSE
Thanks
Ilya Tokatly and Matteo Gatti for some figures
YOU for your patience!
GW approximation
Derivation of BSE
Thanks
Ilya Tokatly and Matteo Gatti for some figures
YOU for your patience!

Introduction to Green functions, the GW approximation, and the

Transcription

Similar documents

BSE - Bering Sea Eccotech

Quarterly Newsletter - Bering Sea Eccotech

The BSE Flow Vol. 2 No. 2

2004 - Michigan Engineering

PDF File

May

o - ANMI

THE BSE FLOW Vol. 2 No. 4