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.
@article{arxiv.2512.01551,
title = {Rational $Q$-systems for integrable spin chains without $U(1)$ symmetry},
author = {Yunfeng Jiang and Yi-Chao Liu and Yuan Miao and Zi-Xi Tan},
journal= {arXiv preprint arXiv:2512.01551},
year = {2025}
}