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Physical review. B, Condensed matter

Correlation energy of the anyon gas

Charles B. Hanna, Alexander L. Fetter

Published Feb 1, 1993• 4 citations
Publisher

Abstract

The correlation energy of the anyon gas is expressed in terms of a coupling-constant integral involving two-body and three-body correlation functions. In the random-phase approximation, the dominant two-body contributions involve the energy difference between the collective modes of the interacting and noninteracting systems, which is evaluated numerically for the Bose gas and for semions.

DOI
10.1103/physrevb.47.3280

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The correlation energy of the anyon gas is expressed in terms of a coupling-constant integral involving two-body and three-body correlation functions. In the random-phase approximation, the dominant two-body contributions involve the energy difference between the collective modes of the interacting and noninteracting systems, which is evaluated numerically for the Bose gas and for semions.
The correlated random-phase-approximation (CRPA) method is applied to study the ground-state and excited-state properties of an anyon gas. We perform a systematic 1/N expansion to the calculation of elementary excitations, where N is the statistical parameter. We find that the leading-(1/N)-order contributions are the same as those obtained by calculations of the RPA response functions. We then consider corrections to the RPA by a perturbation of the exchange interactions, and show that these higher-order corrections are important at large wave vectors. Our calculation gives an excitation spectrum in good agreement with that obtained by the exact solution of a small system.
The correlation energy of an inhomogeneous electron gas is ruled by the density correlation function LLL. It obeys a Bethe-Salpeter equation (BSE), whose kernel is the functional derivative of the particle self-energy. It is solved on equal footing for two-particle and single-particle Green's functions within Hedin's GW approximation. According to the Hartree and exchange-correlation contributions to the kernel, the correlation expression decays into random-phase-approximation (RPA) -like and screened-exchange-like parts. The actual treatment of the screened potential WWW in the BSE for LLL leads to different second-order screened exchange corrections (SOSEX). Including correlation in the kernel even yields approximations beyond SOSEX. The progress beyond RPA is illustrated for spinless and spin-polarized homogeneous electron gases, indicating that the screening treatment is substantial.
In this paper, we first briefly introduce a two-component anyon model and present unusual commutation relations between the two components. In the Fermi (or Bose) representation, drawing an electrostatic-field analogy, we found that the three-body interaction between particles can be approximately reduced to a logarithmic interaction. In the random-phase approximation, we calculated the effective interaction and dielectric response function. The effect of screening is to make the effective interaction nonsingular at q=0 and to induce a current-current interaction that does not appear in the original interaction. Finally, we found that there exists a restricted class of plasma excitations in the long-wavelength case.
We study a tight-binding model for anyons on a square lattice using a self-consistent mean-field theory, where the anyon gauge field is approximated by its expectation value. For any anyon density per site ${\mathrm{\ensuremath{\rho}}}_{0}$=p/q and statistics \ensuremath{\nu}=1-1/n, the anyon system has a linear collective mode if and only if q is greater than twice the largest common factor of p and n. The collective mode dispersion, \ensuremath{\omega}(k), is calculated numerically for n=1 (hard-core bosons) and n=2 (semions) at certain densities.
The ground state energy of the free electron gas is calculated using the Rayleigh-Schrödinger variational method with the wave function ψ= D Π i<j f ( x ij ) where D is a determinant of plane waves and f ( x ij ) a correlation function. Consideration of the wave function in terms of the collective coordinates p k' the Fourier components of the density, suggests an accurate approximation for the energy integral which is then evaluated over the coordinates of the particles so that the use of subsidiary conditions is avoided. Effects omitted in the random phase approximation are included and the final results extend continuously over plasma and particle modes and should be valid in the range of densities encountered in real metals. The results agree closely with those of Nozières and Pines, and of Hubbard obtained by more elaborate methods.