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 $(s,t)\in [0,1{]}^2$ for a general ferroelectric six-vertex model. First, we show there is an open subset $\mathfrak{H}\subset [0,1{]}^2$, 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 $(s,t)\in \mathfrak{H}$. Second, we show that there is a unique pure state for this model of any slope $(s,t)$ on the boundary $\partial \mathfrak{H}$ of $\mathfrak{H}$. These results confirm predictions of Bukman-Shore from 1995.

arXiv comment35 pages, 9 figures, Version 2: Minor revisions

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• The genus-zero five-vertex model