On the transition between the disordered and antiferroelectric phases of the 6-vertex model

Abstract

The symmetric six-vertex model with parameters a,b,c>0 is expected to exhibit different behavior in the regimes a+b<c (antiferroelectric), |a−b|<c≤a+b (disordered) and |a−b|>c (ferroelectric). In this work, we study the way in which the transition between the regimes a+b=c and a+b<c manifests. When a+b<c, we show that the associated height function is localized and its extremal periodic Gibbs states can be parametrized by the integers in such a way that, in the n-th state, the heights n and n+1 percolate while the connected components of their complement have diameters with exponentially decaying tails. When a+b=c, the height function is delocalized. The proofs rely on the Baxter–Kelland–Wu coupling between the six-vertex and the random-cluster models and on recent results for the latter. An interpolation between free and wired boundary conditions is introduced by modifying cluster weights. Using triangular lattice contours (T-circuits), we describe another coupling for height functions that in particular leads to a novel proof of the delocalization at a=b=c. Finally, we highlight a spin representation of the six-vertex model and obtain a coupling of it to the Ashkin–Teller model on Z2 at its self-dual line sinh2J=e−2U. When J<U, we show that each of the two Ising configurations exhibits exponential decay of correlations while their product is ferromagnetically ordered.

Topics & Concepts