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

Alexander Glazman, Ron Peled

Published 2019-09-08Version 1

The symmetric six-vertex model with parameters $a,b,c>0$ is expected to exhibit different behavior in the regimes $a+b<c$ (antiferroelectric), $|a-b|<c\leq a+b$ (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 in the thermodynamic limit. It is shown that the height function of the six-vertex model delocalizes with logarithmic variance when $a+b=c$ while remaining localized when $a+b<c$. In the latter regime, the extremal translation-invariant Gibbs states of the height function are described. Qualitative differences between the two regimes are further exhibited for the Gibbs states of the six-vertex model itself. Via a coupling, our results further allow to study the self-dual Ashkin-Teller model on $\mathbb{Z}^2$. It is proved that on the portion of the self-dual curve $\sinh 2J = e^{-2U}$ where $J<U$ each of the two Ising configurations exhibits exponential decay of correlations while their product is ferromagnetically ordered. This is in contrast to the case $J=U$ (the critical 4-state Potts model) where it is known that all correlations have power-law decay. The proofs rely on the recently established order of the phase transition in the random-cluster model, which relates to the six-vertex model via the Baxter-Kelland-Wu coupling. Additional ingredients include the introduction of a random-cluster model with modified weight for boundary clusters, analysis of a spin (mixed Ashkin-Teller model) and bond representation (of the FK-Ising type) for the six-vertex model, both of which are proved to be strongly positively associated, and the introduction of triangular lattice contours and associated bijection for the analysis of the Gibbs states of the height function.