$T$-$Q$ relation and exact solution for the XYZ chain with general nondiagonal boundary terms

TL;DR

This paper derives an exact solution for the XYZ quantum spin chain with general nondiagonal boundary conditions using a novel approach to the Baxter's Q-operator and the fusion hierarchy, providing a complete spectral characterization.

Contribution

It introduces a new construction of the Baxter's Q-operator as a limit of transfer matrices with higher spin auxiliary spaces for the XYZ chain with open boundaries.

Findings

Derived the T-Q relation from the fusion hierarchy.

Obtained the Bethe Ansatz solution for eigenvalues.

Provided the complete spectrum of the Hamiltonian.

Abstract

We propose that the Baxter's $Q$-operator for the XYZ quantum spin chain with open boundary conditions is given by the $j\to \infty$ limit of the corresponding transfer matrix with spin-$j$ (i.e., $(2j+1)$-dimensional) auxiliary space. The associated $T$-$Q$ relation is derived from the fusion hierarchy of the model. We use this relation to determine the Bethe Ansatz solution of the eigenvalues of the fundamental transfer matrix. This solution yields the complete spectrum of the Hamiltonian.