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

Yunfeng Jiang, Yi-Chao Liu, Yuan Miao, Zi-Xi Tan

TL;DR

This work extends the rational Q-system to integrable spin chains lacking $U(1)$ symmetry by deriving it from a combination of the $TQ$-relation and fusion relations. It develops explicit Q-systems for the XXZ chain with anti-diagonal twists and for the open XXZ chain with non-diagonal boundary fields, including inhomogeneous $TQ$-relations and corresponding inhomogeneous $QQ$-relations. Numerical tests against exact diagonalization confirm that the constructed Q-systems yield all and only physical solutions, providing a complete spectrum in these non-$U(1)$-preserving settings. The approach offers a general framework for building Q-systems in broad classes of models where $T$- and fusion relations are known, enabling systematic solutions beyond $U(1)$-symmetric cases.

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.

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

TL;DR

This work extends the rational Q-system to integrable spin chains lacking symmetry by deriving it from a combination of the -relation and fusion relations. It develops explicit Q-systems for the XXZ chain with anti-diagonal twists and for the open XXZ chain with non-diagonal boundary fields, including inhomogeneous -relations and corresponding inhomogeneous -relations. Numerical tests against exact diagonalization confirm that the constructed Q-systems yield all and only physical solutions, providing a complete spectrum in these non--preserving settings. The approach offers a general framework for building Q-systems in broad classes of models where - and fusion relations are known, enabling systematic solutions beyond -symmetric cases.

Abstract

The -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 symmetry. We extend the rational -system framework to integrable spin chains without 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 -system can be derived by combining -relation with fusion relations of higher-spin transfer matrices. This yields -relations analogous to the 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.

Paper Structure

This paper contains 29 sections, 138 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Q-system from TQ- and fusion
  3. TQ-relation
  4. Fusion relations
  5. The '$TQ$+fusion' system
  6. Q-system
  7. Minimal $Q$-system
  8. Equivalence of two systems
  9. Theorem 1
  10. XXZ chain with anti-diagonal twist
  11. Review of diagonal twist
  12. Anti-diagonal twist
  13. The '$TQ$ + Fusion' System
  14. Minimal Q-system
  15. Numerical results
  16. ...and 14 more sections

Figures (1)

  • Figure 1: The Young diagram corresponding to the rational $Q$-system with $L=9$ and $M=3$. The subscript of the notation $Q_{a,b}$ denotes its coordinate on the lattice: $(a,b)$ denotes the site which is located at the $a$-th row from the bottom to the top and the $b$-th column from the left to the right.