Integrable vertex and loop models on the square lattice with open boundaries via reflection matrices

TL;DR

This paper reviews methods for constructing integrable vertex models with open boundaries on the square lattice using reflection matrices, explicitly solving several models and linking them to integrable loop models relevant for polymer surface behavior.

Contribution

It explicitly derives integrable vertex models with open boundaries for multiple cases and connects them to integrable loop models, expanding the understanding of boundary effects in these systems.

Findings

Eigen spectra obtained via Bethe ansatz methods.

Explicit solutions for six-vertex, 15-vertex, and 19-vertex models.

Connection established between vertex models and loop models with open boundaries.

Abstract

The procedure for obtaining integrable vertex models via reflection matrices on the square lattice with open boundaries is reviewed and explicitly carried out for a number of two- and three-state vertex models. These models include the six-vertex model, the 15-vertex $A_2^{(1)}$ model and the 19-vertex models of Izergin-Korepin and Zamolodchikov-Fateev. In each case the eigenspectra is determined by application of either the algebraic or the analytic Bethe ansatz with inhomeogeneities. With suitable choices of reflection matrices, these vertex models can be associated with integrable loop models on the same lattice. In general, the required choices {\em do not} coincide with those which lead to quantum group-invariant spin chains. The exact solution of the integrable loop models -- including an $O(n)$ model on the square lattice with open boundaries -- is of relevance to the surface critical behaviour of two-dimensional polymers.