Violation of ensemble equivalence in the antiferromagnetic mean-field XY model

It is well known that long-range interactions pose serious problems for the formulation of statistical mechanics. We show in this paper that ensemble equivalence is violated in a simple mean-field model of N fully coupled cla ssical rotators with repulsive interaction (antiferromagnetic XY model). Wh ile in the canonical ensemble the rotators are randomly dispersed over all angles, in the microcanonical ensemble a bi-cluster of rotators separated b y angle pi, forms in the low energy limit. We attribute this behavior to th e extreme degeneracy of the ground state. We obtain empirically an analytic al formula for the probability density function for the angle made by the r otator, which compares extremely well with numerical data and should become exact in the zero energy limit. At low energy, in the presence of the bi-c luster, an extensive amount of energy is located in the single harmonic mod e, with the result that the energy temperature relation is modified. Althou gh still linear, T = alpha U, it has: the slope alpha approximate to 1.3, i nstead of the canonical value alpha = 2.