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

2025/04/19 by Shiraishi, Naoto, Yamaguchi, Mizuki · 2 citations
#Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Physics (quant-ph) #Statistical Mechanics (cond-mat.stat-mech) #Strongly Correlated Electrons (cond-mat.str-el)

paper · doi:10.48550/arxiv.2504.14315

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≤ 13.5, suggesting that all the integrable models are in the scope of the Yang-Baxter equation.

Cited by

• Absence of local conserved charges of the Fredkin spin chain and its truncated versions
• Integrability from a Single Conservation Law in Quantum Spin Chains

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