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Physical review. B, Condensed matter

Ground state of many anyons in a harmonic potential

R. Chitra, Diptiman Sen

Published Nov 1, 1992• 28 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.46.10923

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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.
Using an anyon-fermion mapping method, we investigate the ground-state properties of hard-core anyons confined in a one-dimensional harmonic trap. The concise analytical formula of the reduced one-body density matrix is obtained. Based on this formula, we evaluated the momentum distribution, the natural orbitals, and their occupation distributions for different statistical parameters. The occupation and occupation fraction of the lowest natural orbital versus anyon number are also displayed. It is shown that the ground-state properties of anyons interplay between bosons and fermions continuously. We can expect that the hard-core anyons of larger statistical parameters exhibit similar properties to the hard-core bosons although anyon systems satisfy specific fractional statistics.
In this thesis, the case of two anyons in a harmonic potential in two dimensions is studied by using Feynman path integration. Following Wilczek's ideas, in the concept of the anyon the extraordinary spin and statistics are fractional. Significant applications of anyons are to the fractional quantum Hall effect and anyon superconductivity. Moreover, the configuration space of anyonic systems is multiply connected, and then the path integral is very useful to solve such problems. Nevertheless, there is the alternative approach that can be applied to the same problem, namely, the Chern-Simons construction. By adding the Chern-Simons term into the ordinary Lagrangian, normal particles, particles with the usual statistics, cna become anyons. In this thesis we show that the ground-state energy of two anyons in a harmonic potential can be evaluated both by applying the path integral technique and by the use of the Chern-Simons theory. We show that the complexity of the topology of the anyonic system can be handled readily. It is shown that the ground-state energy of two anyons in a harmonic potential depends on the statistical parameter, alpha, representing the statistics of the anyon.