Complete classification of integrability and non-integrability of S=1/2 spin chains with symmetric next-nearest-neighbor interaction

TL;DR

This paper provides a complete classification of integrable and non-integrable S=1/2 zigzag spin chains with symmetric next-nearest-neighbor interactions, confirming only two integrable models exist.

Contribution

It proves that within this class, only two models are integrable, and all others are non-integrable, establishing a comprehensive classification.

Findings

Only two integrable models identified in the class.

All other models are proven non-integrable.

No intermediate models with finite conserved quantities exist.

Abstract

We study S=1/2 quantum spin chains with shift-invariant and inversion-symmetric next-nearest-neighbor interaction, also known as zigzag spin chains. We completely classify the integrability and non-integrability of the above class of spin systems. We prove that in this class there are only two integrable models, a classical model and a model solvable by the Bethe ansatz, and all the remaining systems are non-integrable. Our classification theorem confirms that within this class of spin chains, there is no missing integrable model. This theorem also implies the absence of intermediate models with a finite number of local conserved quantities.