Critical manifold of the Potts model: Exact results and homogeneity approximation

Abstract

The q-state Potts model has stood at the frontier of research in statistical mechanics for many years. In the absence of a closed-form solution, much of the past effort has focused on locating its critical manifold, trajectory in the parameter (q,e(J)) space where J is the reduced interaction, along which the free energy is singular. However, except in isolated cases, antiferromagnetic (AF) models with J<0 have been largely neglected. In this paper we consider the Potts model with AF interactions focusing on obtaining its critical manifold in exact and/or closed-form expressions. We first reexamine the known critical frontiers in light of AF interactions. For the square lattice we confirm the Potts self-dual point to be the sole critical frontier for J>0. We also locate its critical frontier for J<0 and find it to coincide with a solvability condition observed by Baxter in 1982 [R. J. Baxter, Proc. R. Soc. London Ser. A 388, 43 (1982)]. For the honeycomb lattice we show that the known critical frontier holds for all J, and determine its critical q(c) = 1/2(3 + sqrt[5]) = 2.61803 beyond which there is no transition. For the triangular lattice we confirm the known critical frontier to hold only for J>0. More generally we consider the centered-triangle (CT) and Union-Jack (UJ) lattices consisting of mixed J and K interactions, and deduce critical manifolds under homogeneity hypotheses. For K = 0 the CT lattice is the diced lattice, and we determine its critical manifold for all J and find q(c) = 3.32472. For K = 0 the UJ lattice is the square lattice and from this we deduce both the J > 0 and J < 0 critical manifolds and q(c) = 3. Our theoretical predictions are compared with recent numerical results.