A method for solve integrable $A_2$ spin chains combining different representations

TL;DR

This paper develops a new method inspired by the nested Bethe ansatz to solve integrable A2 spin chains with mixed representations, providing eigenvalue solutions via coupled Bethe equations and proposing a conjecture for more general cases.

Contribution

It introduces a novel approach for solving non-homogeneous A2 spin chains with different representations, extending the applicability of Bethe ansatz techniques.

Findings

Derived coupled Bethe equations for mixed representation chains

Provided eigenvalue solutions for the trace of the monodromy matrix

Proposed a conjecture for solutions in more general non-homogeneous chains

Abstract

A non homogeneous spin chain in the representations $ \{3 \}$ and $ \{3^*\}$ of $A_2$ is analyzed. We find that the naive nested Bethe ansatz is not applicable to this case. A method inspired in the nested Bethe ansatz, that can be applied to more general cases, is developed for that chain. The solution for the eigenvalues of the trace of the monodromy matrix is given as two coupled Bethe equations different from that for a homogeneous chain. A conjecture about the form of the solutions for more general chains is presented. PACS: 75.10.Jm, 05.50+q 02.20 Sv