Critical points in coupled Potts models and correlated percolation

TL;DR

This paper uses scale invariant scattering theory to exactly identify fixed points in coupled Potts models, revealing limitations on scale invariance and insights into correlated percolation and cluster properties.

Contribution

It provides the first exact determination of RG fixed points for coupled Potts models and clarifies the relationship between Potts spin clusters and Fortuin-Kasteleyn clusters.

Findings

Scale invariance in coupled Potts ferromagnets is limited to the Ashkin-Teller case.

Critical properties of Potts spin clusters differ from Fortuin-Kasteleyn clusters.

Results extend to continuous number of states, including the percolation limit.

Abstract

We use scale invariant scattering theory to exactly determine the renormalization group fixed points of a $q$-state Potts model coupled to an $r$-state Potts model in two dimensions. For integer values of $q$ and $r$ the fixed point equations are very constraining and show in particular that scale invariance in coupled Potts ferromagnets is limited to the Ashkin-Teller case ($q=r=2$). Since our results extend to continuous values of the number of states, we can access the limit $r\to 1$ corresponding to correlated percolation, and show that the critical properties of Potts spin clusters cannot in general be obtained from those of Fortuin-Kasteleyn clusters by analytical continuation.