Anyons on a torus: braid group, Aharonov-Bohm period, and numerical study

We present a careful construction of anyons on a torus starting with the braid-group analysis. The rules of Wen, Dagotto, and Fradkin for putting anyons on a torus are reproduced with some minor improvements. The existence of noncontractible loops leads to braid-group representations characterized not only by anyon statistics 0 but also by the magnetic fluxes фX and фy threading through the holes of the torus. The three parameters are tangled with each other. We explore the symmetries of the torus to separate the effects of фx and фy from those of Ө. It is shown that the anyon system always has a smaller period Ө/∏ in фx and фy than the natural period 1. We perform several numerical calculations to investigate the spectral flow and consistency of the method and find interesting features in the spectral flow, which are relevant in understanding the fractional quantum Hall effect.