The free energy in a class of quantum spin systems and interchange processes

Abstract

We study a class of quantum spin systems in the mean-field setting of the complete graph. For spin S=12, the model is the Heisenberg ferromagnet, and for general spin S∈12N, it has a probabilistic representation as a cycle-weighted interchange process. We determine the free energy and the critical temperature (recovering results by Tóth and by Penrose when S=12). The critical temperature is shown to coincide (as a function of S) with that of the q = 2S + 1 state classical Potts model, and the phase transition is discontinuous when S ≥ 1.