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arXiv:cond-mat/9302036 · Physical Review Letters

Nonanalytic contributions to the self-energy and the thermodynamics of two-dimensional Fermi liquids

Dermot Coffey, Kevin S. Bedell

Published Aug 16, 1993• 50 citations
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Abstract

We calculate the entropy of a two-dimensional Fermi liquid using a model with a contact interaction between fermions. We find that there are ${\mathit{T}}^{2}$ contributions to the entropy from interactions separate from those due to the collective modes. These ${\mathit{T}}^{2}$ contributions arise fron nonanalytic corrections to the real part of the self-energy which may be calculated from the leading-log dependence of the imaginary part of the self-energy through the Kramers-Kronig relation. We find no evidence of a breakdown in Fermi liquid theory in 2D and conclude that Fermi liquids in 2D are similar to 3D Fermi liquids.

DOI
10.1103/physrevlett.71.1043

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We calculate the entropy of a two-dimensional Fermi liquid using a model with a contact interaction between fermions. We find that there are T2{\mathit{T}}^{2}T2 contributions to the entropy from interactions separate from those due to the collective modes. These T2{\mathit{T}}^{2}T2 contributions arise fron nonanalytic corrections to the real part of the self-energy which may be calculated from the leading-log dependence of the imaginary part of the self-energy through the Kramers-Kronig relation. We find no evidence of a breakdown in Fermi liquid theory in 2D and conclude that Fermi liquids in 2D are similar to 3D Fermi liquids.
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If the dynamical quasiparticle spectrum of a Fermi liquid (FL), ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{p}}^{\mathrm{dy}}$, given by the poles of the single-particle propagator, is used in the expression for the entropy of a noninteracting Fermi gas, one finds that the result is different from that calculated directly from the thermodynamic potential. In particular, the coefficient of the T3{\mathit{T}}^{3}T3lnT terms, the leading correction to the linear T dependence for a three-dimensional (3D) FL, is overestimated when calculated from ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{p}}^{\mathrm{dy}}$. The thermodynamic properties may be calculated correctly from a second quasiparticle spectrum, the statistical quasiparticle spectrum, ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{p}}^{\mathrm{st}}$. In this paper I calculate the corrections to leading linear energy and temperature dependences for a 2D FL. I compare the difference between the two spectra in the 2D and 3D cases. Although these differences are qualitatively the same, they are quantitatively larger in the 2D case.
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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 study a two-dimensional Fermi gas with an arbitrary short-range repulsive interaction. We show that the vertex part has an unusual singularity in the particle-particle channel for all momenta q2kF{\mathit{k}}_{\mathit{F}}kF​, which is not found in higher dimensions. We interpret this as a collective mode representing a bound excitation of two holes. This mode, however, does not lead to an instability, or to a breakdown of Fermi-liquid theory in the low-density limit. The resulting Fermi-surface phase shift vanishes, and a \ensuremath{\Vert}\ensuremath{\omega}${\mathrm{\ensuremath{\Vert}}}^{5/2}$ correction to \ensuremath{\Sigma}''(kF{\mathit{k}}_{\mathit{F}}kF​,\ensuremath{\omega}) is obtained. The 2D case is also contrasted with a similar calculation in 1D.