Minimal nonintegrable models with three-site interactions

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.

💡 Research Summary

The paper tackles the long‑standing problem of characterizing integrability breaking in translationally invariant spin‑½ chains that contain genuine three‑site interactions. The authors introduce the notion of a “minimal non‑integrable model”: a periodic spin‑½ Hamiltonian that (i) includes true three‑site terms, (ii) saturates the injectivity condition with exactly two interaction terms, and (iii) loses either injectivity or regains integrability if any one of those terms is removed.

To certify non‑integrability they employ the recently formulated Hokkyo criterion, which under two structural assumptions—injectivity and a 2‑local conservation condition—reduces the problem to the existence (or absence) of a 3‑local conserved charge. If no 3‑local operator can be found whose commutator with the Hamiltonian reduces to length ≤ 2, then no non‑trivial local conserved charges exist for any range 3 ≤ k ≤ N/2. The authors also use the Reshetikhin condition as a complementary integrability test: any integrable nearest‑neighbour model must admit a 3‑local charge of the form Q^(3)=∑_j