Vacuum degeneracy of chiral spin states in compactified space

Abstract

A chiral spin state is not only characterized by the T and P order parameter ${E}_{123}$${=\mathrm{S}}_{1}$\ensuremath{\cdot}${(\mathrm{S}}_{2}$\ifmmode\times\else\texttimes\fi{}${\mathrm{S}}_{3}$), it is also characterized by an integer k. In this paper we show that this integer k can be determined from the vacuum degeneracy of the chiral spin state on compactified spaces. On a Riemann surface with genus g the vacuum degeneracy of the chiral spin state is found to be 2${k}^{g}$. Among those vacuum states, some ${k}^{g}$ states have 〈${E}_{123}$〉>0, while other ${k}^{g}$ states have 〈${E}_{123}$〉0. The dependence of the vacuum degeneracy on the topology of the space reflects some sort of topological ordering in the chiral spin state. In general the topological ordering in a system is classified by topological theories.