Some supersymmetric features in the spectrum of anyons in a harmonic potential

Abstract

The energy spectrum of two or more anyons in a harmonic oscillator potential exhibits some supersymmetric features. The supersymmetry operators map a theory with statistics parameter θ to one described by θ+π. Thus fermions and bosons are mapped into each other, whereas semions remain invariant under a combined action of supersymmetry and parity. All the states of a given theory fall into families. The energy levels of states within the same family differ by multiples of the harmonic frequency. There is a bosonic SO(2,1) symmetry for any value of θ. Finally, supersymmetry explains why most energy levels for three anyons have a mirror symmetry about semions. Consequently, the third virial coefficient is exactly mirror symmetric.