Braid group and anyons on a cylinder

In this paper we present a careful reexamination of anyons on a cylinder (or annulus), starting from the braid-group analysis. Proper attention is paid to the topological features arising from the existence of noncontractible loops. The rule for putting anyons on a square lattice has to be modified when the periodic boundary condition is imposed on one direction. In contrast to the annulus, one extra restriction is needed for the cylinder geometry to recover its symmetry between the two edges. We have performed some finite-system calculations. The consistency of our results has been checked by the agreement from two sets of seemingly different rules and also by that with free fermions. We have also calculated the spectral flow of the excited states with varying statistics or flux and have seen a lot of level crossings, which seem to be a general feature of anyon systems. The mean-field treatment is found to be good until level crossing occurs, and to be better if one starts with the hard-core boson rather than the fermion