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arXiv:cond-mat/0509163 · Physical Review B

Nonanalytic corrections to the specific heat of a three-dimensional Fermi liquid

Andrey V. Chubukov, Dmitrii L. Maslov, Andrew J. Millis

Published Jan 27, 2006• 50 citations
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Abstract

We revisit the issue of the leading nonanalytic corrections to the temperature dependence of the specific heat coefficient, $\ensuremath{\gamma}(T)=C(T)∕T$, for a system of interacting fermions in three dimensions. We show that the leading temperature dependence of the specific heat coefficient $\ensuremath{\gamma}(T)\ensuremath{-}\ensuremath{\gamma}(0)\ensuremath{\propto}{T}^{3}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}T$ comes from two physically distinct processes. The first process involves a thermal excitation of a single particle-hole pair, whose components interact via a nonanalytic dynamic vertex. The second process involves an excitation of three particle-hole pairs which interact via the analytic static fixed-point vertex. We show that the single-pair contribution is expressed via the backscattering amplitude of quasiparticles at the Fermi surface. The three-pair contribution does not have a simple expression in terms of scattering in particular directions. We clarify the relation between these results and previous literature on both 3D and 2D systems, and discuss the relation between the nonanalyticities in $\ensuremath{\gamma}$ and those in spin susceptibilities.

DOI
10.1103/physrevb.73.045128

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We revisit the issue of the leading nonanalytic corrections to the temperature dependence of the specific heat coefficient, \ensuremathγ(T)=C(T)∕T\ensuremath{\gamma}(T)=C(T)∕T\ensuremathγ(T)=C(T)∕T, for a system of interacting fermions in three dimensions. We show that the leading temperature dependence of the specific heat coefficient \ensuremathγ(T)\ensuremath−\ensuremathγ(0)\ensuremath∝T3lnT\ensuremath{\gamma}(T)\ensuremath{-}\ensuremath{\gamma}(0)\ensuremath{\propto}{T}^{3}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}T\ensuremathγ(T)\ensuremath−\ensuremathγ(0)\ensuremath∝T3lnT comes from two physically distinct processes. The first process involves a thermal excitation of a single particle-hole pair, whose components interact via a nonanalytic dynamic vertex. The second process involves an excitation of three particle-hole pairs which interact via the analytic static fixed-point vertex. We show that the single-pair contribution is expressed via the backscattering amplitude of quasiparticles at the Fermi surface. The three-pair contribution does not have a simple expression in terms of scattering in particular directions. We clarify the relation between these results and previous literature on both 3D and 2D systems, and discuss the relation between the nonanalyticities in \ensuremathγ\ensuremath{\gamma}\ensuremathγ and those in spin susceptibilities.
We calculate, using perturbative calculations and Ward identities, the basic parameters of the Fermi liquid: the scattering vertex, the Landau interaction function, the effective mass, specific heat, and physical susceptibilities for a model of two-dimensional (2D) fermions with a short-ranged interaction at nonzero temperature. The leading temperature dependence of the spin components of the scattering vertex, the Landau function, and the spin susceptibility is found to be linear. We find that the standard T=0T=0T=0 relationships for a Galilean-invariant Fermi liquid are violated by finite-temperature terms. The coefficients in the temperature corrections to these relationships involve a subtle interplay between contributions from small and large $(\ensuremath{\sim}{2k}_{F})$ momentum processes. A connection with previous studies of the 2D Fermi-liquid parameters is discussed. We conclude that the linear leading temperature dependence of the parameters is a generic feature of the 2D Fermi liquid.
We consider the nonanalytic temperature dependences of the specific heat coefficient, C(T)/T, and spin susceptibility, chi(s)(T), of 2D interacting fermions beyond the weak-coupling limit. We demonstrate within the Luttinger-Ward formalism that the leading temperature dependences of C(T)/T and chi(s)(T) are linear in T, and are described by the Fermi liquid theory. We show that these temperature dependences are universally determined by the states near the Fermi level and, for a generic interaction, are expressed via the spin and charge components of the exact backscattering amplitude of quasiparticles. We compare our theory to recent experiments on monolayers of He3.
We analyze how the logarithmic renormalizations in the Cooper channel affect the nonanalytic temperature dependence of the specific heat coefficient $\ensuremath{\gamma}(T)\ensuremath{-}\ensuremath{\gamma}(0)=A(T)T$ in a two-dimensional Fermi liquid. We show that A(T)A(T)A(T) is expressed exactly in terms of the fully renormalized backscattering amplitude, which includes the renormalization in the Cooper channel. In contrast to the one-dimensional case, both charge and spin components of the backscattering amplitudes are subject to this renormalization. We show that the logarithmic renormalization of the charge amplitude vanishes for a flat Fermi surface when the system becomes effectively one dimensional.
We revisit the issue of the temperature dependence of the specific heat C(T)C(T)C(T) for interacting fermions in one dimension. The charge component Cc(T){C}_{c}(T)Cc​(T) scales linearly with TTT, but the spin component Cs(T){C}_{s}(T)Cs​(T) displays a more complex behavior with TTT as it depends on the backscattering amplitude, g1{g}_{1}g1​, which scales down under renormalization group transformation and eventually behaves as ${g}_{1}(T)\ensuremath{\sim}1∕\mathrm{log}\phantom{\rule{0.2em}{0ex}}T$. We show, however, by direct perturbative calculations that Cs(T){C}_{s}(T)Cs​(T) is strictly linear in TTT to order g12{g}_{1}^{2}g12​ as it contains the renormalized backscattering amplitude not on the scale of TTT, but at the cutoff scale set by the momentum dependence of the interaction around 2kF2{k}_{F}2kF​. The running amplitude g1(T){g}_{1}(T)g1​(T) appears only at third order and gives rise to an extra T∕log3TT∕{\mathrm{log}}^{3}\phantom{\rule{0.2em}{0ex}}TT∕log3T term in Cs(T){C}_{s}(T)Cs​(T). This agrees with the results obtained by a variety of bosonization techniques. We also show how to obtain the same expansion in g1{g}_{1}g1​ within the sine-Gordon model.
We extend previous calculations of the nonanalytic terms in the spin susceptibility ${\ensuremath{\chi}}_{s}(T)$ and the specific heat C(T)C(T)C(T) to systems in a magnetic field. Without a field, ${\ensuremath{\chi}}_{s}(T)$ and C(T)∕TC(T)∕TC(T)∕T are linear in TTT in two dimensions (2D), while in 3D, ${\ensuremath{\chi}}_{s}(T)\ensuremath{\propto}{T}^{2}$ and $C(T)∕T\ensuremath{\propto}{T}^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}T$. We show that in a magnetic field, the linear in TTT terms in 2D become scaling functions of ${\ensuremath{\mu}}_{B}H∕T$. We present explicit expressions for these functions and show that at high fields ${\ensuremath{\mu}}_{B}H⪢T$, ${\ensuremath{\chi}}_{s}(T,H)$ scales as $\ensuremath{\mid}H\ensuremath{\mid}$. We also show that in 3D, ${\ensuremath{\chi}}_{s}(T,H)$ becomes nonanalytic in a field and at high fields scales as ${H}^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\ensuremath{\mid}H\ensuremath{\mid}$.
We calculate the basic parameters of the Fermi liquid: the scattering vertex, the Landau interaction function, the effective mass, and physical susceptibilities for a model of two-dimensional (2D) fermions with a short-ranged interaction at nonzero temperature. The leading temperature dependences of the spin components of the scattering vertex, the Landau function, and the spin susceptibility are found to be linear. T-linear terms in the effective mass and in the "charge-sector" quantities are found to cancel to second order in the interaction, but the cancellation is argued not to be generic. The connection with previous studies of the 2D Fermi-liquid parameters is discussed.