Spin-$s$ $Q$-systems: Twist and Open Boundaries

TL;DR

This paper systematically studies the spin-$s$ XXX chain with twisted and open boundaries using the rational $Q$-system, revealing new phenomena like hidden symmetries and magnetic strings, and confirming the method's completeness.

Contribution

It establishes the rational $Q$-system framework for spin-$s$ chains with boundary conditions and uncovers novel boundary-induced phenomena supported by analytical and numerical analysis.

Findings

Confirmed completeness of the rational $Q$-system for these models.

Discovered hidden symmetries leading to degeneracies.

Identified magnetic strings dependent on boundary magnetic fields.

Abstract

In integrable spin chains, the spectral problem can be solved by the method of Bethe ansatz, which transforms the problem of diagonalization of the Hamiltonian into the problem of solving a set of algebraic equations named Bethe equations. In this work, we systematically investigate the spin-$s$ XXX chain with twisted and open boundary conditions using the rational $Q$-system, which is a powerful tool to solve Bethe equations. We establish basic frameworks of the rational $Q$-system and confirm its completeness numerically in both cases. For twisted boundaries, we investigate the polynomiality conditions of the rational $Q$-system and derive physical conditions for singular solutions of Bethe equations. For open boundaries, we uncover novel phenomena such as hidden symmetries and magnetic strings under specific boundary parameters. Hidden symmetries lead to the appearance of extra degeneracies in the Hilbert space, while the magnetic string is a novel type of exact string configuration, whose length depends on the boundary magnetic fields. These findings, supported by both analytical and numerical evidences, offer new insights into the interplay between symmetries and boundary conditions.