Fractional statistics, Z p spin system, and XY model

Abstract

The Zp spin system in two dimensions from the point of view of fractional statistics is studied herein. It is found that, for p≤4, the system represents a gas of particles having a specific fermion number when p=2 corresponds to fermion number 1 and p=3 and 4 corresponds to a fractional fermion number. However, for p≥5, the corresponding system does not represent particles having any specific statistics; rather, it represents a hedgehog behavior. The XY model represented by a Zp spin system at p→∞ limit corresponds to bosonic kinks composed of vortex–antivortex pairs below a certain temperature. The two massive phases observed in a Zp system for 2≤p≤4 is related to their intrinsic fermionic character, and the three phases observed for p≥5 when a massless phase occurs in between two massive phases is related to the hedgehog behavior of the corresponding particles so far as their statistics is concerned. The XY model corresponding to a Zp spin system at p→∞ represents massive bosonic kinks as bound states of vortex–antivortex pairs below a certain temperature when the spin waves are massless and, as the temperature goes up, the bound states dissociate and massless kinks and massive spin waves appear. The nonexistence of an ordered phase for a continuous symmetry model is related to the inherent bosonic character of the corresponding particles.