The free energies of six-vertex models and the n-equivalence relation

TL;DR

This paper investigates the free energies of six-vertex models with various boundary conditions, establishing their uniqueness in the thermodynamic limit and linking boundary arrow densities to free energy classification.

Contribution

It introduces the n-equivalence relation to classify thermodynamic limit properties and connects boundary arrow densities to free energy, extending prior transfer matrix results.

Findings

Free energy on rectangles is unique in the infinite limit.

Boundary arrow densities determine free energy classification.

Free energy matches Lieb and Sutherland's results for symmetric boundary conditions.

Abstract

The free energies of six-vertex models on general domain D with various boundary conditions are investigated with the use of the n-equivalence relation which classifies the thermodynamic limit properties. It is derived that the free energy of the six-vertex model on the rectangle is unique in the limit in which both the height and the width goes to infinity. It is derived that the free energies of the model on D are classified through the densities of left/down arrows on the boundary. Specifically the free energy is identical to that obtained by Lieb and Sutherland with the cyclic boundary condition when the densities are both equal to 1/2. This fact explains several results already obtained through the transfer matrix calculations. The relation to the domino tiling (or dimer, or matching) problems is also noted.