Dichotomy of quantum integrability and its breakdown
In quantum many-body systems, the presence or absence of local conserved quantities is a key factor that governs thermalization and other dynamical properties. Empirically, one expects the following “integrability dichotomy” for quantum chains with nearest-neighbour interactions: A system is either (a) completely integrable, with k-local conserved quantities for all integers k ≧ 3, or (b) nonintegrable, with no nontrivial local conserved quantities at all; there is no intermediate case. Although this dichotomy is an important statement both theoretically and for applications, a rigorous proof has long been lacking.
In this talk I present three recent results towards a rigorous understanding of this dichotomy.
(i) For parity-symmetric spin-1/2 chains, we prove the dichotomy and show that all completely integrable models in this class are exhausted by the previously known examples.
(ii) For general SU(2)-invariant spin chains, we again establish the dichotomy and demonstrate that complete integrability versus nonintegrability is decided by a simple check of the Reshetikhin condition.
(iii) For bosonic chains with symmetric hopping and general on-site terms including non-Hermitian contributions, we uncover a breakdown of the dichotomy in the non-Hermitian regime via a complete classification of local conserved quantities. We find models that are almost nonintegrable but possess a single 3-local conserved quantity, as well as models that are nearly completely integrable yet lack a 4-local conserved quantity.
These results raise broader questions: Where is the boundary between regimes where the integrability dichotomy holds and where it breaks down? And what principles govern quantum many-body systems beyond the dichotomy? Clarifying these issues is an important direction for future work.
[1] M. Yamaguchi, Y. Chiba, and N. Shiraishi, Complete Classification of Integrability and Non-integrability for Spin-1/2 Chain with Symmetric Nearest-Neighbor Interaction, arXiv:2411.02162. [2] N. Shiraishi and M. Yamaguchi, Dichotomy Theorem Separating Complete Integrability and Non-integrability of Isotropic Spin Chains, arXiv:2504.14315. [3] M. Yamaguchi and N. Shiraishi, in preparation.