Integrable quantum chains combining site states in different representations of $su(3)$

TL;DR

This paper derives a general form of the local matrix for $su(3)$ quantum chains with sites in various representations, and solves a non-homogeneous chain using a nested Bethe ansatz approach, proposing a broader conjecture.

Contribution

It generalizes the $L$-operator for $su(3)$ to arbitrary representations and solves a non-homogeneous chain, introducing a conjecture for $su(n)$ cases.

Findings

Derived the general $L$-matrix for $su(3)$ in any representation.

Solved a non-homogeneous $su(3)$ chain using nested Bethe ansatz.

Proposed a conjecture for $su(n)$ chains with mixed representations.

Abstract

The general expression for the local matrix $L(\theta)$ of a quantum chain with the site space in any representation of $su(3)$ is obtained. This is made by generalizing $L(\theta)$ from the fundamental representation and imposing the fulfilment of the Yang-Baxter equation. Then, a non-homogeneous spin chain combining different representations of $su(3)$ is solved by a method inspired in the nested Bethe ansatz. The solution for the eigenvalues of the trace of the monodromy matrix is given as two coupled Bethe equations. A conjecture about the solution of a chain with the site states in different representations of $su(n)$ is presented.