Classification of three-state Hamiltonians solvable by Coordinate Bethe Ansatz

TL;DR

This paper classifies all three-state spin chain Hamiltonians with rank 1 symmetry that are solvable via coordinate Bethe ansatz, extending known models and providing explicit spectra and eigenfunctions.

Contribution

It introduces four multi-parametric extensions of known 19-vertex Hamiltonians and identifies new 17- and 14-vertex Hamiltonians, solving them explicitly with Bethe ansatz.

Findings

Classified all rank 1 symmetric three-state Hamiltonians solvable by CBA.

Derived spectra, eigenfunctions, and Bethe equations for new Hamiltonians.

Extended known 19-vertex models to include 17- and 14-vertex cases.

Abstract

We classify all Hamiltonians with rank 1 symmetry, acting on a periodic three-state spin chain, and solvable through (generalisation of) the coordinate Bethe ansatz (CBA). We obtain in this way four multi-parametric extensions of the known 19-vertex Hamiltonians (such as Zamolodchikov-Fateev, Izergin-Korepin, Bariev Hamiltonians). Apart from the 19-vertex Hamiltonians, there exists 17-vertex and 14-vertex Hamiltonians that cannot be viewed as subcases of the 19-vertex ones. In the case of 17-vertex Hamiltonian, we get a generalization of the genus 5 special branch found by Martins, plus three new ones. We get also two 14-vertex Hamiltonians. We solve all these Hamiltonians using CBA, and provide their spectrum, eigenfunctions and Bethe equations. A special attention is made to provide the specifications of our multi-parametric Hamiltonians that give back known Hamiltonians.