R-matrices of three-state Hamiltonians solvable by Coordinate Bethe Ansatz

TL;DR

This paper reviews methods to derive R-matrices from Hamiltonians, classifies three-state models solvable by Coordinate Bethe Ansatz, and introduces a new R-matrix for 17-vertex Hamiltonians, expanding the understanding of integrable models.

Contribution

It provides a systematic approach to infer R-matrices from Hamiltonians and presents new solutions, including a novel R-matrix for 17-vertex models, advancing the classification of integrable three-state systems.

Findings

Recovered R-matrices for Zamolodchikov--Fateev and Izergin--Korepin models.

Related the generalized Bariev Hamiltonian to known branches, proving they generate the same Hamiltonian.

Produced a new R-matrix for 17-vertex Hamiltonians.

Abstract

We review some of the strategies that can be implemented to infer an $R$-matrix from the knowledge of its Hamiltonian. We apply them to the classification achieved in arXiv:1306.6303, on three state $U(1)$-invariant Hamiltonians solvable by CBA, focusing on models for which the $S$-matrix is not trivial. For the 19-vertex solutions, we recover the $R$-matrices of the well-known Zamolodchikov--Fateev and Izergin--Korepin models. We point out that the generalized Bariev Hamiltonian is related to both main and special branches studied by Martins in arXiv:1303.4010, that we prove to generate the same Hamiltonian. The 19-vertex SpR model still resists to the analysis, although we are able to state some no-go theorems on its $R$-matrix. For 17-vertex Hamiltonians, we produce a new $R$-matrix.