Complete Classification of Integrability and Non-integrability for Spin-1/2 Chain with Symmetric Nearest-Neighbor Interaction

TL;DR

This paper rigorously classifies all symmetric nearest-neighbor spin-1/2 chains, confirming that only known models are integrable and that no intermediate systems with finite conserved quantities exist, emphasizing the rarity of integrability.

Contribution

It provides a complete classification of integrability for symmetric spin-1/2 chains, proving non-integrability for all but known integrable models.

Findings

All non-integrable models lack nontrivial local conserved quantities.

Known integrable models are the only exceptions in this class.

No intermediate systems with finite conserved quantities exist.

Abstract

General spin-1/2 chains with symmetric nearest-neighbor interaction are studied. We rigorously prove that all spin models in this class, except for known integrable systems, are non-integrable in the sense that they possess no nontrivial local conserved quantities. This result confirms that there are no missing integrable systems, i.e., integrable systems in this class are exactly those that are already known. In addition, this result excludes the possibility of intermediate systems which have a finite number of nontrivial local conserved quantities. Our findings support the expectation that integrable systems are exceptional in quantum many-body systems and most systems are non-integrable.