Computer simulation study of a one dimensional plane rotator system with long-range interactions

Abstract

We consider a classical system of particles, consisting of two-dimensional unit vectors associated with a one-dimensional lattice u _k|k ∊ Z and interacting via translationally invariant pair potential(s)

here m is a positive integer and T _m is a Tchebyshev polynomial of the first kind

where ϕ are the angles defining the orientations of the plane rotators in an arbitrary reference frame. For the case m = 1, Fröhlich et al. have proved rigorously the existence of a ferromagnetically ordered phase at low but finite temperature; moreover, all the potential models W _m give the same partition function, and several mean values can be defined in an m-independent way. For example, when m = 2, this entails the existence of nematic-like order. The system was characterized quantitatively by Monte Carlo simulation, and calculations were performed in the nematic representation (m = 2); simulation results suggest a second-order transition at T∗_c = (≡ kT∗_c/ε) = 2.16 ± 0.01. Comparison with molecular field and spherical model treatments is also reported: the former, but not the latter, agrees reasonably well with the simulation results.