Color symmetry breaking in the Potts spin glass

The Potts spin glass is an analogue of the Sherrington-Kirkpatrick model in which each spin can take one of $\kappa$ possible values, which we interpret as colors. It was suggested in arXiv:2310.06745 that the order parameter for this model is always invariant with respect to permutations of the colors. We show here that this is false whenever $\kappa \ge 58$.