Dichotomy theorem distinguishing non-integrability and the lowest-order Yang-Baxter equation for isotropic spin chains

• URL: http://arxiv.org/abs/2504.14315v1
• Date: Sat, 19 Apr 2025 14:35:49 GMT
• Title: Dichotomy theorem distinguishing non-integrability and the lowest-order Yang-Baxter equation for isotropic spin chains
• Authors: Naoto Shiraishi, Mizuki Yamaguchi,
• Abstract summary: We investigate the integrability and non-integrability of isotropic spin chains with nearest-neighbor interaction with general spin $S$.<n>We prove a dichotomy theorem that a single relation sharply separates two scenarios: (i) this system is non-integrable, or (ii) the lowest-order Yang-Baxter equation is satisfied.
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• 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 $S$. We prove a dichotomy theorem that a single relation sharply separates two scenarios: (i) this system is non-integrable, or (ii) the lowest-order Yang-Baxter equation is satisfied. This result solves in the affirmative the Grabowski-Mathieu conjecture stating that a model is integrable only if this model has a 3-local conserved quantity. 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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