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arXiv:0704.3457 · Physical Review B

Cooper channel and the singularities in the thermodynamics of a Fermi liquid

Andrey V. Chubukov, Dmitrii L. Maslov

Published Oct 9, 2007• 16 citations
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Abstract

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 $A(T)$ 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.

DOI
10.1103/physrevb.76.165111

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We analyze how the logarithmic renormalizations in the Cooper channel affect the nonanalytic temperature dependence of the specific heat coefficient \ensuremathγ(T)\ensuremath−\ensuremathγ(0)=A(T)T\ensuremath{\gamma}(T)\ensuremath{-}\ensuremath{\gamma}(0)=A(T)T\ensuremathγ(T)\ensuremath−\ensuremathγ(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 consider the nonanalytic terms in the spin susceptibility arising as a result of rescattering of pairs of quasiparticles. We emphasize the importance of rescattering in the Cooper channel for the analysis of the temperature dependences in the two-dimensional electron systems in the ballistic regime. In the calculation of the linear in TTT term we use angular harmonics in the Cooper channel, because for each harmonic the interaction amplitude is renormalized independently. We observe, that as a consequence of strong renormalizations in the Cooper ladder, the temperature derivative of the spin susceptibility may change its sign at low temperatures.
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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.
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 show that the singularities in the dynamical bosonic response functions of a two-dimensional Fermi liquid give rise to universal nonanalytic corrections to the results of the Fermi-liquid theory. In particular, we find a T2{T}^{2}T2 term in the specific heat, linear-in-TTT terms in the effective mass and in the uniform spin susceptibility ${\ensuremath{\chi}}_{s}(Q=0,T),$ and ∣Q∣|Q|∣Q∣ term in ${\ensuremath{\chi}}_{s}(Q,T=0).$ The existence of these terms has been the subject of recent controversy, which is resolved in this paper. We present exact expressions for all nonanalytic terms to second order in a generic interaction U(Q)U(Q)U(Q) and show that the nonanalytic terms originate exclusively from forward- and backward-scattering of particles with zero total momentum.
Using a recently suggested method of bosonization in an arbitrary dimension, we study the anomalous contribution of the low-energy spin and charge excitations to thermodynamic quantities of a two-dimensional (2D) Fermi liquid. The method is slightly modified for the present purpose such that the effective supersymmetric action no longer contains the high-energy degrees of freedom but still accounts for effects of the finite curvature of the Fermi surface. Calculating the anomalous contribution $\ensuremath{\delta}c(T)$ to the specific heat, we show that the leading logarithmic in temperature corrections to $\ensuremath{\delta}c(T)/{T}^{2}$ can be obtained in a scheme combining a summation of ladder diagrams and renormalization group equations. The final result is represented as the sum of two separate terms that can be interpreted as coming from singlet and triplet superconducting excitations. The latter may diverge in certain regions of the coupling constants, which should correspond to the formation of triplet Cooper pairs.