Classification of integrability and non-integrability for some quantum spin chains

Abstract: Quantum non-integrability, or the absence of local conserved quantity, is a necessary condition for various empirical laws observed in macroscopic systems. Examples are thermal equilibration, the Kubo formula in linear response theory, and the Fourier law in heat conduction, all of which require non-integrability. From these facts, it is widely believed that integrable systems are highly exceptional, and non-integrability is ubiquitous in generic quantum many-body systems. Many numerical simulations also support this expectation. Nevertheless, conventional approaches in mathematical physics cannot address this belief, and establishing non-integrability of vast majority of generic quantum many-body systems is left as an open problem.

In this study, we address this problem and provide an affirmative result for a wide class of quantum many-body systems. Precisely, we rigorously classify the integrability and non-integrability of all spin-1/2 chains with symmetric nearest-neighbor interactions [1]. Our classification demonstrates that except for the known integrable models, all systems are indeed non-integrable. This result provides a rigorous proof of the ubiquitousness of non-integrability, as well as the absence of undiscovered integrable systems which remains to be discovered. Moreover, it is proved that there is no partially integrable systems with finite number of local conserved quantities.

In addition, recent extensions of non-integrability proofs to spin-1 systems [2] and others will be presented.

[1] M. Yamaguchi, Y. Chiba, N. Shiraishi, ``Complete Classification of Integrability and Non-integrability for Spin-1/2 Chain with Symmetric Nearest-Neighbor Interaction,'' arXiv:2411.02162

[2] A. Hokkyo, M. Yamaguchi, Y. Chiba, ``Proof of the absence of local conserved quantities in the spin-1 bilinear-biquadratic chain and its anisotropic extensions,'' arXiv:2411.04945

mathematical physicsdynamical systemsquantum algebrarepresentation theorysymplectic geometry

Audience: general audience

( slides | 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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