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Dichotomy theorem separating complete integrability and non-integrability of isotropic spin chains
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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.
Forward citations
Cited by 2 Pith papers
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A Simple Necessary and Sufficient Condition for Yang--Baxter Integrability
The Reshetikhin condition on a nearest-neighbour spin-chain Hamiltonian is sufficient (and necessary) for the existence of a regular difference-form Yang–Baxter R-matrix, resolving a 1980s conjecture.
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Scalene Yang--Baxter triples as a source of hidden symmetries beyond the ordinary Yang--Baxter equation
A non-Hermitian spin chain that fails ordinary Yang-Baxter integrability still possesses a hidden staggered nilpotent symmetry generated by a cross-commuting scalene transfer matrix.
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