SYMMETRY-BREAKING SCHEMES FOR THE POTTS-MODEL ON A BETHE LATTICE

We prove that for Q-state ferromagnetic Potts models on Bethe lattices without an external field, the allowed spontaneous-broken-symmetry solutions are those in which the probability of a Potts spin being oriented in a particular one of the Q possible directions can take only two possible values. Hence the Potts spins may have probability p of occupying each of M of the Q states and probability 1-Mp/Q-M of occupying the remaining Q - M states. We further prove that the symmetry-breaking scheme with M = 1 has the lowest free energy and in consequence symmetry-breaking schemes with 1 < M less-than-or-equal-to Q/2 describe metastable structures.