Spin-$s$ Rational $Q$-system
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866911858527567872 |
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| author | Hou, Jue Jiang, Yunfeng Zhu, Rui-Dong |
| author_facet | Hou, Jue Jiang, Yunfeng Zhu, Rui-Dong |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_07640 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Spin-$s$ Rational $Q$-system Hou, Jue Jiang, Yunfeng Zhu, Rui-Dong High Energy Physics - Theory Mathematical Physics Exactly Solvable and Integrable Systems 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. |
| title | Spin-$s$ Rational $Q$-system |
| topic | High Energy Physics - Theory Mathematical Physics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2303.07640 |