Rational $Q$-systems for integrable spin chains without $U(1)$ symmetry

TL;DR

This paper extends the rational $Q$-system method to integrable spin chains lacking $U(1)$ symmetry, enabling complete solutions of Bethe ansatz equations in more general models.

Contribution

The authors develop a generalized $Q$-system framework for non-$U(1)$ symmetric integrable models, including derivation from fusion relations and validation through numerical solutions.

Findings

The extended $Q$-system reproduces all physical solutions.

It incorporates inhomogeneous terms in $QQ$-relations.

Numerical solutions match exact diagonalization results.

Abstract

The $Q$-system is an efficient method for finding complete physical solutions of Bethe ansatz equations, but so far its application has been confined to systems possessing $U(1)$ symmetry. We extend the rational $Q$-system framework to integrable spin chains without $U(1)$ symmetry, exemplified by the closed XXZ model with anti-diagonal twists and the open XXZ model with non-diagonal boundary fields. We demonstrate that the $Q$-system can be derived by combining $TQ$-relation with fusion relations of higher-spin transfer matrices. This yields $QQ$-relations analogous to the $U(1)$ symmetric case but incorporating additional inhomogeneous terms. We present numerical solutions that are validated against exact diagonalization, confirming that it generates all and exclusively physical solutions.