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Modern Physics Letters B

FEW-ANYON SPECTRA IN A HARMONIC OSCILLATOR POTENTIAL

Ranjan Ghosh, Sumathi Rao

Published May 10, 1996• 0 citations
Publisher

Abstract

We study the problem of four and five anyons in a harmonic oscillator potential and make some exact statements regarding its energy spectra. We show that the spectra exhibit some mirror symmetric (reflection symmetric about the semionic statistics point θ=π/2) features analogous to the mirror symmetry in the two and three anyon spectra.

DOI
10.1142/s0217984996000560

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We study the problem of four and five anyons in a harmonic oscillator potential and make some exact statements regarding its energy spectra. We show that the spectra exhibit some mirror symmetric (reflection symmetric about the semionic statistics point θ=π/2) features analogous to the mirror symmetry in the two and three anyon spectra.
We find that the energy spectra of four and five anyons in a harmonic potential exhibit some mirror symmetric (reflection symmetric about the semionic statistics point θ=π/2θ=π/2θ=π/2) features analogous to the mirror symmetry in the two and three anyon spectra. However, since the ℓ=0\ell=0ℓ=0 sector remains non-mirror symmetric, the fourth and fifth virial coeffients do not reflect this symmetry.
We use supersymmetric quantum mechanics to show that the spectrum of three anyons in a harmonic potential exhibits an almost complete mirror symmetry about semions. Barring a subset of exactly solved states, all states come in pairs such that the energy of one state with the statistics parameter \ensuremath{\theta} is equal to the energy of its partner state at \ensuremath{\pi}-\ensuremath{\theta}. Bosons, semions, and fermions have \ensuremath{\theta}=0,\ensuremath{\pi}/2, and \ensuremath{\pi}, respectively. From this, we show that the third virial coefficient of an ideal anyon gas is exactly mirror symmetric about semions.
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.