Potts Models with (17) Invisible States on Thin Graphs

TL;DR

This paper investigates how adding 17 invisible states to a 2-state Potts model on thin graphs can change the phase transition from second to first order, using a mean-field approach and BEG model equivalence.

Contribution

It demonstrates that the transition order change occurs on thin graphs with a higher number of invisible states, extending previous mean-field results.

Findings

Transition order changes with invisible states on thin graphs.

17 invisible states are needed for transmutation in this model.

The BEG model is solved on thin graphs as a by-product.

Abstract

The order of a phase transition is usually determined by the nature of the symmetry breaking at the phase transition point and the dimension of the model under consideration. For instance, q-state Potts models in two dimensions display a second order, continuous transition for q = 2,3,4 and first order for higher q. Tamura et al recently introduced Potts models with "invisible" states which contribute to the entropy but not the internal energy and noted that adding such invisible states could transmute continuous transitions into first order transitions. This was observed both in a Bragg-Williams type mean-field calculation and 2D Monte-Carlo simulations. It was suggested that the invisible state mechanism for transmuting the order of a transition might play a role where transition orders inconsistent with the usual scheme had been observed. In this paper we note that an alternative mean-field approach employing 3-regular random ("thin") graphs also displays this change in the order of the transition as the number of invisible states is varied, although the number of states required to effect the transmutation, 17 invisible states when there are 2 visible states, is much higher than in the Bragg-Williams case. The calculation proceeds by using the equivalence of the Potts model with 2 visible and r invisible states to the Blume-Emery-Griffiths (BEG) model, so a by-product is the solution of the BEG model on thin random graphs.