Proof of the absence of local conserved quantities in general spin-<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> chains with symmetric nearest-neighbor interaction

Mizuki Sanatani; Yuuya Chiba; Naoto Shiraishi

doi:10.1103/8l6f-z1jm

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Proof of the absence of local conserved quantities in general spin- $\frac{1}{2}$ chains with symmetric nearest-neighbor interaction

Mizuki Sanatani ^1,*, Yuuya Chiba ^2,†, and Naoto Shiraishi ^1,‡

¹Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan
²Nonequilibrium Quantum Statistical Mechanics RIKEN Hakubi Research Team, RIKEN Cluster for Pioneering Research (CPR), 2-1 Hirosawa, Wako, Saitama 351-0198, Japan

^*Contact author: yamaguchi-q@g.ecc.u-tokyo.ac.jp
^†Contact author: yuya.chiba@riken.jp
^‡Contact author: shiraishi@phys.c.u-tokyo.ac.jp

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Phys. Rev. B 114, 094437 – Published 28 August, 2026

DOI: https://doi.org/10.1103/8l6f-z1jm

Abstract

We provide a rigorous proof of the absence of nontrivial local conserved quantities in all spin- $\frac{1}{2}$ chains with symmetric nearest-neighbor interaction, except for known integrable systems. This result shows that there are no further integrable systems to be discovered. Our finding also implies that there are no intermediate systems with a finite number of nontrivial local conserved quantities. In addition, we clarify all short-support conserved quantities in nonintegrable systems, which we need to take into account in analyses of thermalization and level statistics.

Physical Review B

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Proof of the absence of local conserved quantities in general spin- $\frac{1}{2}$ chains with symmetric nearest-neighbor interaction

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Complete classification of integrability and nonintegrability for spin- $\frac{1}{2}$ chain with symmetric nearest-neighbor interaction

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Proof of the absence of local conserved quantities in general spin-12 chains with symmetric nearest-neighbor interaction

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Complete classification of integrability and nonintegrability for spin-12 chain with symmetric nearest-neighbor interaction

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Proof of the absence of local conserved quantities in general spin- $\frac{1}{2}$ chains with symmetric nearest-neighbor interaction

Complete classification of integrability and nonintegrability for spin- $\frac{1}{2}$ chain with symmetric nearest-neighbor interaction