Q-operator and T-Q relation from the fusion hierarchy

TL;DR

This paper introduces a new way to derive the Baxter Q-operator for the XXZ spin chain using the fusion hierarchy, enabling the solution of the model for generic anisotropy parameters.

Contribution

It demonstrates that the Q-operator can be obtained as a limit of transfer matrices with higher spin auxiliary spaces, extending the Bethe Ansatz solution to generic anisotropy values.

Findings

Derived the T-Q relation for generic anisotropy parameters.

Solved the eigenvalues of the transfer matrix using the new T-Q relation.

Provided a complementary approach to existing solutions at roots of unity.

Abstract

We propose that the Baxter $Q$-operator for the spin-1/2 XXZ quantum spin chain is given by the $j\to \infty$ limit of the transfer matrix with spin-$j$ (i.e., $(2j+1)$-dimensional) auxiliary space. Applying this observation to the open chain with general (nondiagonal) integrable boundary terms, we obtain from the fusion hierarchy the $T$-$Q$ relation for {\it generic} values (i.e. not roots of unity) of the bulk anisotropy parameter. We use this relation to determine the Bethe Ansatz solution of the eigenvalues of the fundamental transfer matrix. This approach is complementary to the one used recently to solve the same model for the roots of unity case.