On different approaches to integrable lattice models

TL;DR

This paper explores integrable IRF lattice models based on affine Lie algebras, computes new Boltzmann weights for specific cases, and discusses the vertex-IRF correspondence to relate these weights to quantum R-matrices.

Contribution

It introduces new solutions for Boltzmann weights of IRF models based on rak{su}(2)_k and rak{su}(3)_k, expanding the understanding of integrable lattice models.

Findings

New Boltzmann weights for rak{su}(2)_k and rak{su}(3)_k IRF models

Explicit computation for fundamental and adjoint representations

Discussion of vertex-IRF correspondence and quantum R-matrix relations

Abstract

Interaction-Round the Face (IRF) models are two-dimensional lattice models of statistical mechanics defined by an affine Lie algebra and admissibility conditions depending on a choice of representation of that affine Lie algebra. Integrable IRF models, i.e., the models the Boltzmann weights of which satisfy the quantum Yang-Baxter equation, are of particular interest. In this paper, we investigate trigonometric Boltzmann weights of integrable IRF models. By using an ansatz proposed by one of the authors in some previous works, the Boltzmann weights of the restricted IRF models based on the affine Lie algebras $\mathfrak{su}(2)_k$ and $\mathfrak{su}(3)_k$ are computed for fundamental and adjoint representations for some fixed levels $k$. New solutions for the Boltzmann weights are obtained. We also study the vertex-IRF correspondence in the context of an unrestricted IRF model based on $\mathfrak {su}(3)_k$ (for general $k$) and discuss how it can be used to find Boltzmann weights in terms of the quantum $\hat{R}$ matrix when the adjoint representation defines the admissibility conditions.