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arXiv:hep-th/9505020

MIRROR SYMMETRY IN THE FEW ANYON SPECTRA IN A HARMONIC OSCILLATOR POTENTIAL

Ranjan Ghosh, Sumathi Rao

Published May 3, 1995• 0 citations
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Abstract

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 $θ=π/2$) features analogous to the mirror symmetry in the two and three anyon spectra. However, since the $\ell=0$ sector remains non-mirror symmetric, the fourth and fifth virial coeffients do not reflect this symmetry.

DOI
10.48550/arxiv.hep-th/9505020

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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.
We propose a new generalized statistical mechanics for anyons based on a simple ansatz that has the correct familiar limiting cases. We derive the anyon equation of state from the grand partition function. We develop the virial expansion and obtain all the virial coefficients. Furthermore, we demonstrate how thermodynamic quantities such as the molar specific heat can readily be calculated. We discuss the properties of semions and also demonstrate that ideal non-relativistic anyons do not exhibit finite temperature condensation.
We present new family of exact analytic solutions for three anyons in a harmonic potential (or in free space) in terms of generalized harmonics on S3S^3S3, which supplement the known solutions. The new solutions satisfy the hard-core condition when α=13,1α={1\over 3},1α=31​,1 (ααα being the statistical parameter) but otherwise, have finite non-vanishing two-particle colliding probability density, which is consistent with self-adjointness of the Hamiltonian. These solutions, however, do not have one-to-one mapping property between bosonic and fermionic spectra.
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)].