## 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

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)
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where

(H.2)
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$νl=2πkBTl$ 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 $ν→0$.

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

(H.4)
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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 $α≠β$. 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)
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where we have introduced the local Green’s function

(H.6)
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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 $al2t∗2/16ℏ2$ when expressed in terms of the reduced nearest-neighbor hopping $t∗=2dt$.

We can now take the limit of $ν→0$. Writing

(H.7)
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and

(H.8)
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produces our final result

(H.9)
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where $Λtr(ω,T)$ is defined by

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

(H.11)
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(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:

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

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

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This term is small with respect to $Nc(ω)/ImΣ(ω)$, which diverges in the limit $T,ω→0$. 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.