Critical behaviour, scaling and universality in some two-dimensional spin models

Abstract

We report the results of a high-precision numerical study of certain two-dimensional spin models which are discrete versions of the O(2) and O(4) vector models. The models we studied were the Z(10) clock model and different versions of a Y model where Y is the doubly covered icosahedral group of 120 elements; the latter are discretized versions of O(4) vector models. We find strong evidence for two transitions in these models: one, at low temperature, is the well-known freezing transition where the discreteness of the spin space begins to be felt; the other one at much higher temperature is the so-called Kosterlitz-Thouless transition in the case of Z(10) and a similar transition for the case of Y which according to conventional wisdom should not exist. These transitions are well described by standard power-like singularities; we determine their critical indices which show universality for the different actions of the Y models and also some remarkable numerical regularities for the dependence of N of the different discretized O( N) models. All of these models seem to have a line of critical points connecting the two transitions.