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

We study a class of quantum spin systems in the mean-field setting of the complete graph. For spin $S=\frac{1}{2}$ the model is the Heisenberg ferromagnet, for general spin $S\in \frac{1}{2}\mathbb{N}$ it has a probabilistic representation as a cycle-weighted interchange process. We determine the free energy and the critical temperature (recovering results by T'oth and by Penrose when $S=\frac{1}{2}$). 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 $S\ge 1$.