Spin-$s$ Rational $Q$-system

Abstract

Bethe ansatz equations for spin-$s$ Heisenberg spin chain with $s\ge 1$ are significantly more difficult to analyze than the spin-$\frac{1}{2}$ case, due to the presence of repeated roots. As a result, it is challenging to derive extra conditions for the Bethe roots to be physical and study the related completeness problem. In this paper, we propose the rational $Q$-system for the XXX${}_s$ spin chain. Solutions of the proposed $Q$-system give all and only physical solutions of the Bethe ansatz equations required by completeness. This is checked numerically and proved rigorously. The rational $Q$-system is equivalent to the requirement that the solution and the corresponding dual solution of the $TQ$-relation are both polynomials, which we prove rigorously. Based on this analysis, we propose the extra conditions for solutions of the XXX${}_s$ Bethe ansatz equations to be physical.

arXiv commenttypos corrected, some minor corrections

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• Exact Spin Correlators of Integrable Quantum Circuits from Algebraic Geometry
• Truncated string state space approach and its application to nonintegrable spin-$\frac{1}{2}$ Heisenberg chain
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• Spin-$s$ $Q$-systems: Twist and Open Boundaries