Spectrum of the transfer matrices of the spin chains associated with the $A^{(2)}_3$ Lie algebra

TL;DR

This paper derives the eigenvalues and Bethe ansatz equations for the $A^{(2)}_3$ integrable quantum spin chain, including non-diagonal boundary conditions, using fusion techniques and recursive relations.

Contribution

It introduces a universal method to solve $A^{(2)}_n$-related models with non-diagonal boundaries, extending previous solutions to more general boundary conditions.

Findings

Derived recursive relations for fused transfer matrices.

Obtained eigenvalues and Bethe ansatz equations.

Validated method for both open and periodic boundary conditions.

Abstract

We study the exact solution of quantum integrable system associated with the $A^{(2)}_3$ twist Lie algebra, where the boundary reflection matrices have non-diagonal elements thus the $U(1)$ symmetry is broken. With the help of the fusion technique, we obtain the closed recursive relations of the fused transfer matrices. Based on them, together with the asymptotic behaviors and the values at special points, we obtain the eigenvalues and Bethe ansatz equations of the system. We also show that the method is universal and valid for the periodic boundary condition where the $U(1)$ symmetry is reserved. The results in this paper can be applied to studying the exact solution of the $A^{(2)}_n$-related integrable models with arbitrary $n$.