Spin-$s$ Rational $Q$-system

TL;DR

This paper introduces a rational $Q$-system for the spin-$s$ XXX chain that precisely characterizes physical solutions of Bethe ansatz equations, overcoming previous difficulties with repeated roots.

Contribution

It proposes and rigorously proves a new rational $Q$-system that ensures completeness and physicality of solutions for higher spin chains.

Findings

The $Q$-system characterizes all physical solutions.

Solutions are equivalent to polynomial solutions of the $TQ$-relation.

Extra conditions for physical solutions are proposed and validated.

Abstract

Bethe ansatz equations for spin-$s$ Heisenberg spin chain with $s\ge1$ are significantly more difficult to analyze than the spin-$\tfrac{1}{2}$ 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.