Dichotomy theorem separating complete integrability and non-integrability of isotropic spin chains
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| Format: | Preprint |
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2025
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| _version_ | 1866911076988223488 |
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| author | Shiraishi, Naoto Yamaguchi, Mizuki |
| author_facet | Shiraishi, Naoto Yamaguchi, Mizuki |
| contents | We investigate the integrability and non-integrability of isotropic spin chains with nearest-neighbor interaction with general spin $S$ in terms of the presence or absence of local conserved quantities. We prove a dichotomy theorem that whether a single quantity is zero or not sharply separates two scenarios: (i) this system has $k$-local conserved quantities for all $k$ (completely integrable), or (ii) this system has no nontrivial local conserved quantity (non-integrable). This result excludes the possibility of an intermediate system with some but not all local conserved quantities, which solves in the affirmative the Grabowski-Mathieu conjecture. This theorem also serves as a complete classification of integrability and non-integrability for $S\leq 13.5$, suggesting that all the integrable models are in the scope of the Yang-Baxter equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14315 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dichotomy theorem separating complete integrability and non-integrability of isotropic spin chains Shiraishi, Naoto Yamaguchi, Mizuki Statistical Mechanics Strongly Correlated Electrons Mathematical Physics Exactly Solvable and Integrable Systems Quantum Physics We investigate the integrability and non-integrability of isotropic spin chains with nearest-neighbor interaction with general spin $S$ in terms of the presence or absence of local conserved quantities. We prove a dichotomy theorem that whether a single quantity is zero or not sharply separates two scenarios: (i) this system has $k$-local conserved quantities for all $k$ (completely integrable), or (ii) this system has no nontrivial local conserved quantity (non-integrable). This result excludes the possibility of an intermediate system with some but not all local conserved quantities, which solves in the affirmative the Grabowski-Mathieu conjecture. This theorem also serves as a complete classification of integrability and non-integrability for $S\leq 13.5$, suggesting that all the integrable models are in the scope of the Yang-Baxter equation. |
| title | Dichotomy theorem separating complete integrability and non-integrability of isotropic spin chains |
| topic | Statistical Mechanics Strongly Correlated Electrons Mathematical Physics Exactly Solvable and Integrable Systems Quantum Physics |
| url | https://arxiv.org/abs/2504.14315 |