Integrability from a Single Conservation Law in Quantum Spin Chains

Abstract: Identifying integrable systems has been one of the central problems in rigorous statistical mechanics. In this talk, I will discuss a recent result [1] on the mathematical structure of integrability in quantum spin chains with finite-range interactions. We prove that the existence of a specific conservation law, known as the Reshetikhin condition, implies the presence of infinitely many local conserved quantities—that is, integrability. This result shows that the entire hierarchy of conservation laws associated with solutions of the Yang–Baxter equation is already encoded in the lowest nontrivial conservation law. Combined with recent progress on the absence of integrability in generic systems [2], I will also discuss the sharp boundary between integrable and nonintegrable quantum spin chains.

[1] A.Hokkyo, "Integrability from a Single Conservation Law in Quantum Spin Chains", arXiv:2508.20713.

[2] A.Hokkyo, "Rigorous Test for Quantum Integrability and Nonintegrability", arXiv:2501.18400.

mathematical physicsdynamical systemsquantum algebrarepresentation theorysymplectic geometry

Audience: general audience

( video )

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BIMSA Integrable Systems Seminar

Series comments: The aim is to bring together experts in integrable systems and related areas of theoretical and mathematical physics and mathematics. There will be research presentations and overview talks.

Audience: Graduate students and researchers interested in integrable systems and related mathematical structures, such as symplectic and Poisson geometry and representation theory.

The zoom link will be distributed by mail, so please join the mailing list if you are interested in attending the seminar.

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