Topological phase transition in a two-dimensional nematic n-vector model: A numerical study

A Monte Carlo study, based on a cluster algorithm, is made of the two-dimensional ${\mathrm{RP}}^{\mathit{n}\mathrm{-}1}$ model. Introducing various types of order parameters, we find strong evidence for a topological phase transition, driven by a condensation of defects. For n=3, the model describes a nematic liquid crystal. In this case, the transition is associated to a divergence of the correlation length and of the susceptibility, and a cusp in the specific heat. The results for n=40 are compared with the analytical ones for n=∞. The critical temperature is correctly reproduced, but the detailed nature of the transition is not. Crossover phenomena are detected, which reflect the noncommutativity of the n=∞ and the thermodynamic limit for some observables.