Potts models with invisible states on general Bethe lattices

Abstract

The number of so-called invisible states which need to be added to the q-state Potts model to transmute its phase transition from continuous to first order has attracted recent attention. In the q = 2 case, a Bragg–Williams (mean-field) approach necessitates four such invisible states while a 3-regular random graph formalism requires seventeen. In both of these cases, the changeover from second- to first-order behaviour induced by the invisible states is identified through the tricritical point of an equivalent Blume–Emery–Griffiths model. Here we investigate the generalized Potts model on a Bethe lattice with z neighbours. We show that, in the q = 2 case, invisible states are required to manifest the equivalent Blume–Emery–Griffiths tricriticality. When z = 3, the 3-regular random graph result is recovered, while z → ∞ delivers the Bragg–Williams (mean-field) result.