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arXiv:cond-mat/9301021 · International Journal of Modern Physics A

ASYMPTOTIC ANALYSIS AND SPECTRUM OF THREE ANYONS

Ghanashyam Date, M. Krishna, M. V. N. Murthy

Published Jun 20, 1994• 7 citations
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Abstract

The spectrum of anyons confined in a harmonic oscillator potential shows both linear and nonlinear dependence on the statistical parameter. While the existence of exact linear solutions has been shown analytically, the nonlinear dependence has been arrived at by numerical and/or perturbative methods. We develop a method which shows the possibility of a nonlinearly interpolating spectrum. To be specific we analyze the eigenvalue equation in various asymptotic regions for the three-anyon problem.

DOI
10.1142/s0217751x94001011

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The spectrum of anyons confined in a harmonic oscillator potential shows both linear and nonlinear dependence on the statistical parameter. While the existence of exact linear solutions has been shown analytically, the nonlinear dependence has been arrived at by numerical and/or perturbative methods. We develop a method which shows the possibility of a nonlinearly interpolating spectrum. To be specific we analyze the eigenvalue equation in various asymptotic regions for the three-anyon problem.
For three anyons confined in a harmonic oscillator, only the class of states that interpolates nonlinearly with the statistical parameter contributes to the third virial coefficient of a free anyon gas. Rather than evaluating the full three-body partition function as was done in an earlier publication [J. Law, A. Suzuki, and R. Bhaduri, Phys. Rev. A 46, 4693 (1992)], here only the nonlinear contribution is calculated, thus avoiding delicate cancellations between the irrelevant linear part and the two-body partition function. Our numerical results are consistent with the simple analytical form suggested recently by Myrheim and Olaussen [Phys. Lett. B 299, 267 (1993)].
We show that the low-lying spectrum of three noninteracting anyons in a harmonic potential can be qualitatively and quantitatively understood on the basis of a simple set of analytic wave functions. These wave functions are exact in both the boson and fermion limits and satisfy the anyone wave-function phase condition at all intermediate values of the statistical parameter 01. The resulting nonlinear spectrum agrees with those of exact numerical calculations to better than a few parts in a thousand.