Monte Carlo simulation of a two dimensional anisotropic plane rotator model

Abstract

We have studied a classical system, consisting of two-component unit vectors (plane rotators) associated with a two dimensional square lattice, and interacting via the nearest neighbour pair potential(s)

where m is a positive integer and {φ _k } are the angles defining the orientation of the plane rotators in an arbitary reference frame. The two potential models W_m and – W_m possess essentially the same properties in the absence of an external field (spin-flip symmetry); moreover, for given values of a and b, all of the potential models W_m have the same partition function, and several mean values can be defined in a way which is independent of m. This model has been proven rigorously to possess a Kosterlitz-Thouless transition when a = b, and to possess a low temperature order-disorder transition when 0 ≤ |b| < a; when m = 2, this entails the existence of nematic- or antinematic- like order, depending on the sign of c. We have chosen b = 0, and characterized the system quntitatively by Monte Carlo simulation; calculations were carried out in the nematic representation (c = – 1, m = 2). Simulation results suggest a second order transition taking place at T*_c = kT _c/ε = 1·315 ± 0·015; the molecular field treatment over estimates this value by 50 per cent.