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Physical review. B, Condensed matter

Spectrum of three anyons in a harmonic-oscillator potential

M. V. N. Murthy, J. Law, M. Brack, R. K. Bhaduri

Published Feb 15, 1992• 9 citations
Publisher

Abstract

The Hamiltonian of three anyons in a harmonic-oscillator potential is diagonalized using a set of harmonic-oscillator states, by treating these as bosons or fermions with an appropriate statistical interaction. The interpolation between the bosonic and fermionic limits is established, and the spectrum of the low-lying states is displayed.

DOI
10.1103/physrevb.45.4289

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The Hamiltonian of three anyons in a harmonic-oscillator potential is diagonalized using a set of harmonic-oscillator states, by treating these as bosons or fermions with an appropriate statistical interaction. The interpolation between the bosonic and fermionic limits is established, and the spectrum of the low-lying states is displayed.
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.
The ground-state energy and wave function of many anyons in a harmonic oscillator potential is studied using perturbation theory from the fermionic and bosonic ends. As the number of anyons increases to infinity, the ground-state energy scales with the power 3/2. As the statistics parameter changes, there are repeated level crossings between states having different angular momenta. Hence the ground state is only piecewise continuous, and the number of pieces increases with the number of anyons. This is to be contrasted with the problem of interpolating statistics in one dimension where there are no level crossings in the ground state for any number of particles.
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.