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arXiv:0707.4422 · Physical Review B

Specific heat of a one-dimensional interacting Fermi system: Role of anomalies

Andrey V. Chubukov, Dmitrii L. Maslov, Ronojoy Saha

Published Feb 12, 2008• 10 citations
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Abstract

We revisit the issue of the temperature dependence of the specific heat $C(T)$ for interacting fermions in one dimension. The charge component ${C}_{c}(T)$ scales linearly with $T$, but the spin component ${C}_{s}(T)$ displays a more complex behavior with $T$ as it depends on the backscattering amplitude, ${g}_{1}$, 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 ${C}_{s}(T)$ is strictly linear in $T$ to order ${g}_{1}^{2}$ as it contains the renormalized backscattering amplitude not on the scale of $T$, but at the cutoff scale set by the momentum dependence of the interaction around $2{k}_{F}$. The running amplitude ${g}_{1}(T)$ appears only at third order and gives rise to an extra $T∕{\mathrm{log}}^{3}\phantom{\rule{0.2em}{0ex}}T$ term in ${C}_{s}(T)$. This agrees with the results obtained by a variety of bosonization techniques. We also show how to obtain the same expansion in ${g}_{1}$ within the sine-Gordon model.

DOI
10.1103/physrevb.77.085109

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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 g1(T)\ensuremath∼1∕logT{g}_{1}(T)\ensuremath{\sim}1∕\mathrm{log}\phantom{\rule{0.2em}{0ex}}Tg1​(T)\ensuremath∼1∕logT. 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.
Bosonization predicts that the specific heat C(T)C(T)C(T) of a one-dimensional interacting Fermi system is a sum of the specific heats of free collective charge and spin excitations, plus the term with the running backscattering amplitude which flows to zero logarithmically with decreasing TTT. We verify whether this result is reproduced in the ggg-ology model. Of specific interest are the anomalous terms in C(T)C(T)C(T) that depend on the bare backscattering amplitude. We show that these terms can be incorporated into a renormalized spin velocity. We do this by proving the equivalence of the results for C(T)C(T)C(T) obtained within the ggg-ology model and by bosonization with velocities obtained by the numerical solution of the Bethe-ansatz equations for the Hubbard model.
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 dynamic spin susceptibility $\ensuremath{\chi}(q,\ensuremath{\omega})$ of one-dimensional interacting fermions. To second order in the interaction, backscattering results in a logarithmic correction to $\ensuremath{\chi}(q,\ensuremath{\omega})$ at $q\ensuremath{\ll}{k}_{F}$, even if the single-particle spectrum is linearized near the Fermi points. Consequently, the dynamic spin structure factor $\mathrm{Im}\ensuremath{\chi}(q,\ensuremath{\omega})$ is nonzero at frequencies above the single-particle continuum. In the boson language, this effect results from the marginally irrelevant backscattering operator of the sine-Gordon model. Away from the threshold, the high-frequency tail of $\mathrm{Im}\ensuremath{\chi}(q,\ensuremath{\omega})$ due to backscattering is larger than that due to finite mass by a factor of kF/q{k}_{F}/qkF​/q. We derive the renormalization group equations for the coupling constants of the $g\ensuremath{-}\text{ology}$ model at finite $\ensuremath{\omega}$ and qqq and find the corresponding expression for $\ensuremath{\chi}(q,\ensuremath{\omega})$, valid to all orders in the interaction but not in the immediate vicinity of the continuum boundary, where the finite-mass effects become dominant.
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.
A model of one-dimensional fermions characterized by a large momentum-transfer interaction g1{g}_{1}g1​ and a small momentum-transfer interaction g2{g}_{2}g2​ is considered. Results obtained previously by the renormalization-group method are extended by studying the triplet Cooper-pair response function and uniform charge and spin response functions. The sign of g1{g}_{1}g1​ dominates the behavior of the system. For g1>0{g}_{1}>0g1​>0, ${g}_{1}\ensuremath{-}2{g}_{2}>0$ the triplet Cooper response diverges as $T\ensuremath{\rightarrow}0$ and is seen to dominate the singlet Cooper response which is also divergent. The uniform static spin susceptibility is nondivergent for g1>0{g}_{1}>0g1​>0 and diverges to zero for g1<0{g}_{1}<0g1​<0. The specific heat is found to be linear in temperature in all regions. These results, obtained by the second-order renormalization group, yield the correct qualitative results in regions where exact solutions are available.
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.