  We study a class of quantum spin systems in the mean-field setting of the
complete graph. For spin $S=\tfrac12$ the model is the Heisenberg ferromagnet,
for general spin $S\in\tfrac12\mathbb{N}$ 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
$S=\tfrac12$). 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 $S\geq1$.
