Nonexistence and uniqueness for pure states of ferroelectric six‐vertex models

Abstract

In this paper, we consider the existence and uniqueness of pure states with some fixed slope (s,t)∈false[0,1false]2$(s, t) \in [0, 1]^2$ for a general ferroelectric six‐vertex model. First, we show there is an open subset frakturH⊂false[0,1false]2$\mathfrak {H} \subset [0, 1]^2$, which is parameterized by the region between two explicit hyperbolas, such that there is no pure state for the ferroelectric six‐vertex model of any slope false(s,tfalse)∈frakturH$(s, t) \in \mathfrak {H}$. Second, we show that there is a unique pure state for this model of any slope false(s,tfalse)$(s, t)$ on the boundary ∂frakturH$\partial \mathfrak {H}$ of H$\mathfrak {H}$. These results confirm predictions of Bukman–Shore from 1995.

The genus-zero five-vertex model

“…Existence of a unique limit is still open in general, but not essential for the calculations in this paper. Aggarwal [1] showed (in the context of the six-vertex model) that in certain cases, which we call "coexistence" cases below, there is no ergodic limit. He also showed, however, that on the coexistence phase boundary, that is, when (s, t) is on the upper right boundary of N (see Figure 11), there is a unique limit measure µ s,t (by showing that there is a unique ergodic Gibbs measure of slope (s, t)).…”

“…The measures ν N (X, Y ) have subsequential limits which are Gibbs measures of slope (s, t). Conjecturally (see [1]) in this case there is a unique limit which is equal to the µ s,t defined above, and is the unique ergodic Gibbs measure of this slope.…”