Proof of the absence of local conserved quantities in general spin-1/2 chains with symmetric nearest-neighbor interaction

TL;DR

This paper rigorously proves that generic spin-1/2 chains with symmetric nearest-neighbor interactions lack additional local conserved quantities beyond known integrable models, clarifying the structure of conserved quantities and implications for thermalization.

Contribution

It provides a comprehensive proof that no unknown local conserved quantities exist in these chains, confirming the uniqueness of known integrable systems.

Findings

No additional local conserved quantities in non-integrable chains

Clarification of short-support conserved quantities in non-integrable systems

Implications for thermalization and level statistics analyses

Abstract

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