# Integrability from a single conservation law in quantum spin chains

## TL;DR

This paper proves that in translationally invariant quantum spin chains, the existence of a specific conservation law guarantees the presence of infinitely many conserved quantities, establishing a fundamental link between a single law and integrability.

## Contribution

It demonstrates that the hierarchy of conservation laws in quantum spin chains is determined by the lowest nontrivial conservation law, providing a rigorous foundation for identifying integrable models.

## Key findings

- Existence of a conservation law implies infinite conserved quantities.
- The hierarchy of conservation laws is encoded in the lowest law.
- Partially integrable systems with finite conserved quantities are strongly restricted.

## Abstract

We prove that for translationally invariant quantum spin chains with finite-range interactions, the existence of a specific conservation law known as the Reshetikhin condition implies the presence of infinitely many local conserved quantities, i.e., integrability. This 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 rigorous results on nonintegrability, our theorem strongly restricts the possibility of partially integrable systems that admit only a finite but large number of local conserved quantities. Our work establishes a rigorous foundation for the systematic identification of new integrable models and deepens the algebraic understanding of conservation-law structures in quantum spin chains.