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Spin-s Rational Q-system
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Spin-s Rational Q-system
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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.
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Cited by 1 Pith paper
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Rational $Q$-systems for integrable spin chains without $U(1)$ symmetry
A rational Q-system with inhomogeneous QQ-relations is constructed for XXZ spin chains with anti-diagonal twist and non-diagonal boundary fields, and numerically shown to yield all physical solutions.
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