Potts Models with Invisible States on General Bethe Lattices

TL;DR

This paper explores how adding invisible states to the q=2 Potts model on Bethe lattices induces a change from second- to first-order phase transitions, generalizing previous mean-field and random-graph results.

Contribution

It derives a formula for the number of invisible states needed on Bethe lattices with arbitrary coordination number to induce tricriticality, unifying previous results.

Findings

Number of invisible states depends on lattice coordination number z.

For z=3, the result matches the random-graph case.

As z approaches infinity, the mean-field result is recovered.

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 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.