3-state Hamiltonians associated to solvable 33-vertex models

TL;DR

This paper analyzes 33-vertex models with three internal states per site using the nested coordinate Bethe ansatz, identifying all solvable models and deriving their eigenvalues and Bethe equations.

Contribution

It provides a complete classification of solvable 33-vertex models with a nested Bethe ansatz approach and new constraints from Yang-Baxter solutions.

Findings

List of all solvable 33-vertex models.

Eigenvalues expressed in terms of rapidities.

New constraints from 4x4 R-matrix solutions.

Abstract

Using the nested coordinate Bethe ansatz, we study 33-vertex models, where only one global charge with degenerate eigenvalues exists and each site possesses three internal degrees of freedom. In the context of Markovian processes, they correspond to diffusing particles with two possible internal states which may be exchanged during the diffusion (transmutation). The first step of the nested coordinate Bethe ansatz is performed providing the eigenvalues in terms of rapidities. We give the constraints ensuring the consistency of the computations. These rapidities also satisfy Bethe equations involving $4\times 4$ R-matrices, solutions of the Yang--Baxter equation which implies new constraints on the models. We solve them allowing us to list all the solvable 33-vertex models.