The antiferromagnetic transition for the square-lattice Potts model

TL;DR

This paper provides an exact solution for the antiferromagnetic transition of the Q-state Potts model on the square lattice, revealing complex critical behaviors, including first- and second-order transitions, depending on Q and lattice parameters.

Contribution

It offers a detailed analysis of the Bethe ansatz equations and numerical methods to characterize the transition, including continuum limits and parafermionic truncations, advancing understanding of critical phenomena in the Potts model.

Findings

Critical point exhibits first-order transition for generic Q.

For Beraha numbers, the transition becomes second order with specific critical exponents.

The model's continuum limit involves two bosons, with implications for conformal field theory.

Abstract

We solve the antiferromagnetic transition for the Q-state Potts model (defined geometrically for Q generic) on the square lattice. The solution is based on a detailed analysis of the Bethe ansatz equations (which involve staggered source terms) and on extensive numerical diagonalization of transfer matrices. It involves subtle distinctions between the loop/cluster version of the model, and the associated RSOS and (twisted) vertex models. The latter's continuum limit involves two bosons, one which is compact and twisted, and the other which is not, with a total central charge c=2-6/t, for sqrt(Q)=2cos(pi/t). The non-compact boson contributes a continuum component to the spectrum of critical exponents. For Q generic, these properties are shared by the Potts model. For Q a Beraha number [Q = 4 cos^2(pi/n) with n integer] the two-boson theory is truncated and becomes essentially Z\_{n-2} parafermions. Moreover, the vertex model, and, for Q generic, the Potts model, exhibit a first-order critical point on the transition line, i.e., the critical point is also the locus of level crossings where the derivatives of the free energy are discontinuous. In that sense, the thermal exponent of the Potts model is generically nu=1/2. Things are profoundly different for Q a Beraha number, where the transition is second order, with nu=(t-2)/2 determined by the psi\_1 parafermion. As one enters the adjacant Berker-Kadanoff phase, the model flows, for t odd, to a minimal model of CFT with c=1-6/t(t-1), while for t even it becomes massive. This provides a physical realization of a flow conjectured by Fateev and Zamolodchikov in the context of Z\_N integrable perturbations. Finally, we argue that the antiferromagnetic transition occurs as well on other two-dimensional lattices.