The packing of two species of polygons on the square lattice

TL;DR

This paper analyzes a fully packed double loop model on the square lattice, providing exact solutions for free energy and revealing phase transitions, thus advancing understanding of polygon packings and lattice models.

Contribution

It presents an exact solution for the fully packed double loop model using coordinate Bethe ansatz, connecting it to known models and revealing phase transition behavior.

Findings

Exact free energy expression obtained

Recovered entropy of the Ice model

Identified infinite order phase transition at specific fugacities

Abstract

We decorate the square lattice with two species of polygons under the constraint that every lattice edge is covered by only one polygon and every vertex is visited by both types of polygons. We end up with a 24 vertex model which is known in the literature as the fully packed double loop model. In the particular case in which the fugacities of the polygons are the same, the model admits an exact solution. The solution is obtained using coordinate Bethe ansatz and provides a closed expression for the free energy. In particular we find the free energy of the four colorings model and the double Hamiltonian walk and recover the known entropy of the Ice model. When both fugacities are set equal to two the model undergoes an infinite order phase transition.