Dichotomy theorem separating complete integrability and non-integrability of isotropic spin chains

• URL: http://arxiv.org/abs/2504.14315v4
• Date: Fri, 25 Jul 2025 16:18:58 GMT
• Title: Dichotomy theorem separating complete integrability and non-integrability of isotropic spin chains
• Authors: Naoto Shiraishi, Mizuki Yamaguchi,
• Abstract summary: We prove a dichotomy theorem that whether a single quantity is zero or not sharply separates two scenarios.<n>This result excludes the possibility of an intermediate system with some but not all local conserved quantities.<n>This theorem also serves as a complete classification of integrability and non-integrability for $Sleq 13.5$, suggesting that all the integrable models are in the scope of the Yang-Baxter equation.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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