SOME SUPERSYMMETRIC FEATURES IN THE SPECTRUM OF ANYONS IN A HARMONIC POTENTIAL

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-theta to one described by theta + pi. 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 theta. 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.