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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.271) Appendix H Correlation functions in the Fermi liquid regime: the DMFT solution

(p.271) Appendix H Correlation functions in the Fermi liquid regime: the DMFT solution

Source:
Modern Theory of Thermoelectricity
Publisher:
Oxford University Press

To find Λtr(ω, T) we use the Kubo formula for the static conductivity, which provides L11 as the zero-frequency limit of the charge current density–charge current density correlation function (Mahan, 1981):

(H.1)
L11αβ=limν0Im1νLˉ11αβ(ν)=limν0ReiνLˉ11αβ(ν),

where

(H.2)
L¯11αβ(iνl)=(VcellNi)0βdτeiνlτTτjα(τ)jβ(0),

νl=2πkBTl is the bosonic Matsubara frequency, the τ-dependence of the operator is with respect to the Hamiltonian in Eq. (A.13), α and β denote Cartesian coordinates, and we must analytically continue Lˉ11αβ(iνl) to the real axis Lˉ11αβ(ν) before taking the limit ν0.

Substituting the definition of the charge current density operator from Eq. (11.67) into Eq. (H.2) for Lˉ11αβ and taking the limit of infinite dimensions, in which the vertex corrections to the correlation function become vanishingly small (Khurana 1990, Zlatić and Horvatić, 1990), we obtain

(H.3)
Lˉ11αβ(iνl)=e2VcellNi0βdτeiνlτkvkαvkβGc(k,τ)Gc(k,τ).

where Gc(k,τ) is the imaginary time Green’s function of the c-electrons, which can be expressed as a Fourier series Gc(k,τ)=kBTnexp(iωnτ)Gc(k,iωn). Substituting into Eq. (H.3) and integrating over imaginary time provides the result

(H.4)
Lˉ11αβ(iνl)=e2VcellNikBTnkvkαvkβGc(k,iωn)Gc(k,iωn+iνl),

which has to be analytically continued to the real axis. Since the Green’s functions in Eq. (H.4) depend on k only through ϵ(k), which is an even function of k, while vkα (p.272) is odd, the summation over k vanishes for αβ. We consider only isotropic systems, where L11αα=L11. To perform the analytic continuation we follow closely Freericks and Zlatić (2001) and obtain, in the Fermi liquid regime, the result (evaluated explicitly for a three-dimensional system)

(H.5)
L11=e2VcellNivF231πdωlimν0f(ω)f(ω+ν)ν×Re[Gc(ω)Gc(ω+ν)ν+Σc(ω)Σc(ω+ν)Gc(ω)Gc(ω+ν)ν+Σc(ω)Σc(ω+ν)].

where we have introduced the local Green’s function

(H.6)
Gc(ω)=kGc(k,ω)

and have replaced the square of the α-component of velocity by its Fermi surface average vF2/d (d is the spatial dimension, which we can take to be equal to 3 for real systems). This step is justified because the energy integration in Eq. (H.5) is restricted to a narrow interval around the Fermi energy, where the integrand is singular, so that the main contribution to the k-summation comes from the k-points close to the Fermi surface; on the infinite-dimensional hypercubic lattice, the integral can be performed exactly and one finds that the average square velocity is equal to al2t2/162 when expressed in terms of the reduced nearest-neighbor hopping t=2dt.

We can now take the limit of ν0. Writing

(H.7)
limν0Gc(ω)Gc(ω+ν)ν+Σc(ω)Σc(ω+ν)=Gc(ω)ω11Σc(ω)/ω

and

(H.8)
limν0Gc(ω)Gc(ω+ν)ν+Σc(ω)Σc(ω+ν)=ImGc(ω)ImΣc(ω)

produces our final result

(H.9)
L11=e2dωf(ω)ωΛtr(ω,T),

where Λtr(ω,T) is defined by

(H.10)
Λtr(ω)=1VcellNivF231πImGc(ω)ImΣc(ω)+ReGc(ω)/ω1Σc(ω)/ω.

To estimate the relative importance of the two terms in Eq. (H.10), we introduce the Hilbert transform of Nc(ω):

(H.11)
Hc(ω)=1πVcellNiReGc(ω).

(p.273) In the Fermi liquid regime, where Nc(ω) is δ-function-like, the slope of Hc(ω) is very large; it is proportional to the c-electron enhancement factor:

Re1Σc(ω)ω=Zc1.

On the other hand, ImΣc(ω)/ω is small around ω=0, because ImΣc(ω) is close to its maximum value. Using

ImΣc(ω)ωZc1.

we neglect [ImΣc(ω)/ω]2 in the denominator of the second term for Λ(ω) in Eq. (H.10) and approximate

ReGc(ω)/ω1Σc(ω)/ωZcHc(ω)ωNc(ω)ωImΣc(ω)ω.

This term is small with respect to Nc(ω)/ImΣ(ω), which diverges in the limit T,ω0. Keeping only the singular term in Eq. (H.10) we obtain the result of Eq. (14.6) used in the main text.

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.