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Modern Theory of Thermoelectricity$

Veljko Zlatić and René Monnier

Print publication date: 2014

Print ISBN-13: 9780198705413

Published to Oxford Scholarship Online: June 2014

DOI: 10.1093/acprof:oso/9780198705413.001.0001

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(p.235) Appendix C Derivation of the spectral representation for the single-particle Green’s function

(p.235) Appendix C Derivation of the spectral representation for the single-particle Green’s function

Source:
Modern Theory of Thermoelectricity
Publisher:
Oxford University Press

We want to prove the following identities:

(C.1)
Gr,r(τ)={1πdωImGr,rR(ω)eωτ[1f(ω)],τ>0,1πdωImGr,rR(ω)eωτ[f(ω)],τ<0.

In the basis that diagonalizes the Hamiltonian, the Matsubara Green’s function reads

(C.2)
Gr, r(τ)=mneβ(EmΩ)eτ(EnEm)m|cr|nn|cr|m

The Fourier transform provides the Lehmann representation on the imaginary frequency axis:

Gr, r(ωl)=0βdτeiωlτGr, r(τ)=mneβ(EmΩ)1+eβ(EnEm)iωl(EnEm)|m|cr|n|2,

where ωl=(2l+1)π/β is the fermionic frequency appearing as a simple pole in the expansion of the Fermi distribution function

(C.3)
f(ω)=1eβω+1=12+1βn=+1(2n+1)πi/βω.

By introducing the spectral function

(C.4)
Ar, r(ω)=mneβ(EmΩ)(1+eβω)2πδωEnEm|m|cr|n|2,

the Matsubara Green’s function can be written as

(C.5)
Gr, r(ωl)=dω2πAr, r(ω)iωlω.

(p.236) The retarded Green’s function is defined for real time by the anticommutator

(C.6)
Gr, rR(tt)=iΘ(tt)cr(t)cr(t)+cr(t)cr(t).

Its Fourier transform is given by

(C.7)
Gr, rR(ν)=i0dtei(ν+iδ)tcr(t)cr(0)+cr(0)cr(t),

where we have shifted the time arguments by t and introduced the positive infinitesimal δ to ensure convergence at large times. Expressing, as before, the correlation function in the basis of eigenstates of the Hamiltonian, we finally get

(C.8)
Gr, rR(ν)=dω2πAr, r(ω)νω+iδ.

Since Ar, r(ω) is positive-definite for all ω, it follows that Gr, rR(ν) can be analytically continued in the upper part of the complex frequency plane.

Using ImGr, rR(ω)=Ar, r(ω)/2, we express the Matsubara Green’s function on the imaginary frequency axis as

(C.9)
Gr, r(ωl)=1πdωImGr, rR(ω)iωlω,

and that on the imaginary time axis (βτβ) as

(C.10)
Gr, r(τ)=1πβdωneiωnτiωnωImGr, rR(ω).

The Matsubara frequencies coincide with the poles of the Fermi function, and the sum over them is obtained from an integral along the contours C+ and C (see Fig. C.1). For τ<0, we calculate

(C.11)
1βneiωnτiωnω=12πiC+dzezτzωf(z)+Cdzezτzωf(z),=12πidxexτf(x)1x+i0+ω1xi0+ω,

which holds because the Fermi function f(z)=1/(1+eβz) provides convergence for z+, the factor ezτ provides boundedness for z when τ<0, and the integrand vanishes on the semicircle z=|R|(cosϕ+isinϕ) for |R| in all four quadrants. Thus, for τ<0, we obtain

(C.12)
1βneiωnτiωnω=eωτf(ω).

(p.237) For τ>0, we define τ=τ and z=z, and we get for the integrals along the real axis

(C.13)
1βneiωnτiωnω=12πidxexτf(x)1xi0++ω1x+i0++ω=eωτf(ω),

where now the pole has been circled in the positive sense. Reintroducing τ, this leads to

(C.14)
1βneiωnτiωnω=eωτf(ω)=eωτ[1f(ω)].

The results in Eqs. (C.12) and (C.14) can be substituted into Eq. (C.10) to yield the spectral representation of the single-particle Green’s function given by Eq. (C.1).

Appendix C Derivation of the spectral representation for the single-particle Green’s function

Fig. C.1 (A) The sum over Matsubara frequencies is given by the sum of the line integrals over the circles around the Matsubara frequencies. (B) Deforming the circles, we obtain two line integrals over the contours C+ and C running in opposite directions. The horizontals parts of C+ and C are running just above and below the real axis.

Notes:

Modern Theory of Thermoelectricity. First Edition. Veljko Zlatić and René Monnier

© Veljko Zlatić and René Monnier 2014. Published in 2014 by Oxford University Press.