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International Journal of Modern Physics A

EXACT SOLUTIONS OF A MANY-ANYON PROBLEM

Stefan V. Mashkevich

Published Dec 30, 1992• 13 citations
Publisher

Abstract

A scheme for obtaining exact analytic solutions of the problem of an arbitrary number of anyons in a harmonic well is developed. Its essence consists in establishing a set of wave functions with the demanded interchange properties followed by finding stationary state functions within this set by the method of successive approximations. The energy of the corresponding states depends linearly on the statistical parameter. The discussion of the obtained states is carried out.

DOI
10.1142/s0217751x92003586

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International Journal of Modern Physics B•May 20, 1992•10 citations
A scheme for obtaining exact analytic solutions of the problem of an arbitrary number of anyons in a harmonic well is developed. Its essence consists in establishing a set of wave functions with the demanded interchange properties followed by finding stationary state functions within this set by the method of successive approximations. The energy of the corresponding states depends linearly on the statistical parameter. The discussion of the obtained states is carried out.
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.
We study a many-body system consisting of NNN identical anyons in an external harmonic-oscillator potential. A partial set of exact solutions is obtained by using Jacobi coordinates. For completing the full investigation, we employ a perturbative method to compute those spectra, which have not been analytically obtained. Near the boson limit, we point out that the requirement of the boson being hard core is needed for performing the perturbative calculations, and the first-order NNN-anyon ground-state energies are explicitly presented by using a proper regulation procedure. Up to third perturbative order from the fermion limit, the three-anyon ground-state energy corrections are explicitly presented. Finally we compute the first-order three-anyon ground-state wave function near the fermion limit.
We consider N anyons, with statistical parameter ν, in a homogeneous magnetic fieldand discuss exact and perturbative solutions. We show that the energy, E(ν), obeys[Formula: see text] for any bose or fermi state and that the extremal cases correspondto the states where the relative angular momenta of all pairs of particles are negativeand non-negative, respectively. The anyonic continuations of these extremal states aregiven, their energy is linear in ν. We conjecture that the above bounds on [Formula: see text] hold forany ν. This implies that the energy of any many-anyon state decreases monotonicallywith the statistical parameter ν (ν taking any real value). For small enough ν, E=const. [Formula: see text] and the wave function is known. Increasing ν, the wave functionbecomes singular at some bose point. In the following intermediate region (which maybe absent), E(ν) decreases more slowly; the exact form of E(ν) and the wave functionare unknown. Increasing ν further, above some bose point the energy becomes constantand the wave function is again known.