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Physical review. B, Condensed matter

Quantum mechanics of the fractional-statistics gas: Particle-hole interaction

Charles B. Hanna, R. B. Laughlin, Alexander L. Fetter

Published Jan 1, 1991• 24 citations
Publisher

Abstract

We compute the collective-mode spectrum of the fractional-statistics gas using the set of single particle-hole pairs of the Hartree-Fock ground state as a variational basis. We find that the particle-hole interaction reduces the energy of this mode to a finite value but does not cause it to disperse linearly to zero at long wavelengths, as would be expected of a superfluid. We attribute this to a flaw in the variational ground state, implicitly rectified by computing the response functions in the random-phase approximation. The formalism introduced in this paper provides the machinery for performing precise versions of such calculations.

DOI
10.1103/physrevb.43.309

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We develop a time-dependent Hartree-Fock approximation that is appropriate for Bose-condensed systems. Defining a depletion Green's function allows the construction of condensate and depletion particle densities from eigenstates of a single time-dependent (oscillating) Hamiltonian, guaranteeing that our approach is a number-conserving approximation. The poles of this Green's function yield the energies of number-changing excitations for which the condensate particle number is held fixed, which we show has a gapped spectrum in the superfluid state. The linearized time-dependent version of this has poles at the collective frequencies of the system, yielding the expected zero sound mode for a uniform infinite system. We show how the approximations may be expressed in a general linear response formalism.
We develop a general variational theory for inhomogeneous Fermi systems such as the electron gas in a metal surface, the surface of liquid 3He^{3}\mathrm{He}3He, or simple models of heavy nuclei. The ground-state wave function is expressed in terms of two-body correlations, a one-body attenuation factor, and a model-system Slater determinant. Massive partial summations of cluster expansions are performed by means of Born-Green-Yvon and hypernetted-chain techniques. An optimal single-particle basis is generated by a generalized Hartree-Fock equation in which the two-body correlations screen the bare interparticle interaction. The optimization of the pair correlations leads to a state-averaged random-phase-approximation equation and a strictly microscopic determination of the particle-hole interaction.