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Physical review. B, Condensed matter

Quasiparticle spectra and the calculation of thermodynamics for a two-dimensional Fermi liquid

Dermot Coffey

Published May 15, 1995• 1 citation
Publisher

Abstract

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 ${\mathit{T}}^{3}$lnT 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.

DOI
10.1103/physrevb.51.14069

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If the dynamical quasiparticle spectrum of a Fermi liquid (FL), \ensuremathϵpdy{\mathrm{\ensuremath{\epsilon}}}_{\mathit{p}}^{\mathrm{dy}}\ensuremathϵpdy​, 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 \ensuremathϵpdy{\mathrm{\ensuremath{\epsilon}}}_{\mathit{p}}^{\mathrm{dy}}\ensuremathϵpdy​. The thermodynamic properties may be calculated correctly from a second quasiparticle spectrum, the statistical quasiparticle spectrum, \ensuremathϵpst{\mathrm{\ensuremath{\epsilon}}}_{\mathit{p}}^{\mathrm{st}}\ensuremathϵpst​. 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.
The two-dimensional (2D) nature of the high-Tc materials has lead to renewed interest in the properties of fermions in two dimensions. Calculations of the corrections to the leading Fermi-liquid (FL) behaviour of the single-particle self-energy Σ(p, E), and the specific heat have shown that the differences between 2D and three-dimensional FLs arise from the difference in phase space between two and three dimensions. Here we investigate the imaginary part Σ″(p, ξp) of Σ(p, E) on shell at a low density which has a (p - pF)2 In (p - pf) dependence in two dimensions where pF is the Fermi wavenumber. We find that both particle-hole (ph) and particle-particle (pp) contributions are important in this limit so that in principle no class of diagrams can be neglected in the low-density limit in two dimensions, and one is effectively limited to second order in the interaction for reliable quantitative results. This is in contrast with three dimensions where the ph contribution becomes negligible compared with the pp contribution at low densities and only pp ladder diagrams survive. We argue, however, that the density dependence of Σ(p, E) is unchanged by high orders in perturbation theory and that in the limit pF → 0 the form of the dependence of Σ(p, E) on p - pF is given by the second-order contribution. We find that the (p - pF)2 In (p - pf) behaviour is limited to a values of p - pf Λ; pF so that the region over which FL behaviour is evident shrinks with decreasing density.
The spin susceptibility of a two-dimensional (2D) Fermi liquid has been recently shown, by Hirashima and Takahashi [J. Phys. Soc. Jpn. 67 (1998) 3816], to vary as χ( T )=χ 0 +α T as a function of temperature T , where χ 0 and α are constants. From the analytic form of the free energy which is compatible with this χ and the third law of thermodynamics, we can deduce that the entropy S , and hence the heat capacity, of the system varies as T 2 . This T 2 -term of S is also compatible with the theoretical prediction for the self-energy part of the 2D quasiparticles.
We review the leading momentum, frequency, and temperature dependences of the single-particle self-energy and the corresponding term in the entropy of a two-dimensional Fermi liquid (FL) with a free-particle spectrum. We calculate the corrections to these leading dependences for the paramagnon model and the electron gas and find that the leading dependences are limited to regions of energy and temperature which decrease with decreasing number density of fermions. This can make it difficult to identify the frequency- and temperature-dependent characteristics of a FL ground state in experimental quantities in low-density systems even when complications of band structure and other degrees of freedom are absent. This is an important consideration when the normal-state properties of the undoped cuprate superconductors are analyzed.
Asymptotic expansions for the thermodynamic functions of an ideal Fermi gas are obtained about the point z=1z=1z=1, where zzz is the fugacity. We do this in three dimensions for the three different types of single-particle energy spectra, namely nonrelativistic, relativistic, and massless. To relate to liquid-3He^{3}\mathrm{He}3He films, we also study the nonrelativistic spectrum in two dimensions.
The Fermi liquid theory may provide a good description of the thermodynamic properties of an interacting particle system when the interaction between the particles contributes to the total energy of the system with a quantity which may depend on the total particle number, but does not depend on the temperature. In such a situation, the ideal part of the Hamiltonian, i.e. the energy of the system without the interaction energy, also provides a good description of the system's thermodynamics. If the total interaction energy of the system, being a complicated function of the particle populations, is temperature dependent, then the Landau's quasiparticle gas cannot describe accurately the thermodynamics of the system. A general solution to this problem is presented in this paper, in which the quasiparticle energies are redefined in such a way that the total energy of the system is identical to the sum of the energies of the quasiparticles. This implies also that the thermodynamic properties of the system and those of the quasiparticle gas are identical. By choosing a perspective in which the quasiparticle energies are fixed while the density of states along the quasiparticle axis vary, we transform our quasiparticle system into an ideal gas which obey fractional exclusion statistics.
Fermi liquid theory is the basic paradigm within which we understand the normal behavior of interacting electron systems, but quantitative values for the parameters that occur in this theory are currently unknown in many important cases. One such case is the two-dimensional homogeneous electron gas (2D HEG), which is realized in a wide variety of semiconductor devices. We have used quantum Monte Carlo (QMC) methods to calculate the Landau interaction functions between pairs of quasiparticles. We use these to study the Fermi liquid parameters, finding that finite-size effects represent a serious obstacle to the direct determination of Fermi liquid parameters in QMC calculations. We have used QMC data in the literature for other properties of the 2D HEG to assemble a set of ``best available'' values for the Fermi liquid parameters.