Vertex Models and Random Labyrinths: Phase Diagrams for Ice-type Vertex Models

TL;DR

This paper introduces a geometric method to construct phase diagrams for ice-rule vertex models, linking disordered phases to loop models and providing insights into ordered phases through polymer loop representations.

Contribution

It presents a simple geometric recipe for phase diagram construction and establishes a novel mapping between vertex models and loop/polymer models.

Findings

Disordered phase maps onto intersecting loop model

Ordered phases correspond to polymer loop models with cross-links

Provides a unified framework for understanding phase behavior in vertex models

Abstract

We propose a simple geometric recipe for constructing phase diagrams for a general class of vertex models obeying the ice rule. The disordered phase maps onto the intersecting loop model which is interesting in its own right and is related to several other statistical mechanical models. This mapping is also useful in understanding some ordered phases of these vertex models as they correspond to the polymer loop models with cross-links in their vulcanised phase.