Minimal nonintegrable models with three-site interactions

TL;DR

This paper classifies minimal nonintegrable spin-1/2 models with three-site interactions, establishing a clear boundary between integrability and nonintegrability beyond nearest-neighbor systems.

Contribution

It introduces and rigorously proves the nonintegrability of a deformed Fredkin spin chain and constructs five classes of minimal three-site interaction models, four of which are nonintegrable.

Findings

The deformed Fredkin chain is nonintegrable with no nontrivial local conserved charges.

Four classes of minimal three-site models are identified as nonintegrable.

One class of the constructed models remains integrable.

Abstract

A systematic understanding of integrability breaking in translationally invariant spin chains with genuine three-site interactions remains lacking. In this work, we introduce and classify minimal nonintegrable spin-$1/2$ Hamiltonians, defined as models that saturate injectivity while admitting no nontrivial local conserved charges beyond the Hamiltonian. We first rigorously establish the nonintegrability of the deformed Fredkin spin chain with periodic boundary conditions by mapping it to a nearest-neighbor composite-spin representation and excluding all admissible $3$-local conserved charges. Guided by its structure, we then construct five classes of spin-$1/2$ models with genuine three-site interactions. One class is integrable, while the remaining four contain exactly two interaction terms and constitute the minimal nonintegrable three-site models. Our results delineate a sharp boundary between integrability and nonintegrability beyond the nearest-neighbor paradigm.