Potts Models with Invisible States on General Bethe Lattices

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 generalised Potts model on a Bethe lattice with z neighbours. We show that, in the q=2 case, r_c(z)=[4 z / 3(z-1)] [(z-1)/(z-2)]^z invisible states are required to manifest the equivalent Blume-Emery-Griffiths tricriticality. When z=3, the 3-regular, random-graph result is recovered, while the infinite z limit delivers the Bragg-Williams, mean-field result.