Minimal nonintegrable models with three-site interactions

TL;DR

This work provides a systematic framework to classify minimal nonintegrable spin-$1/2$ models with genuine three-site interactions. By mapping three-site terms to a composite-spin representation, the authors apply rigorous criteria—injectivity, the $2$-local conservation condition, the Reshetikhin condition, and frustration graphs—to certify nonintegrability of the deformed Fredkin chain and to classify a family of two-term composite-spin models into integrable and nonintegrable classes. They further extend these results to five three-site models, showing that only one (two decoupled FFD sectors) is integrable, while the others, involving either coupled FFD sectors or FFD plus Ising terms, are minimal nonintegrable. The frustration-graph analysis provides a complementary structural perspective on why integrability breaks down upon coupling integrable blocks. Overall, the paper sharpens our understanding of the boundary between integrability and nonintegrability beyond nearest-neighbor interactions and highlights how sparse, well-structured couplings can destroy local conservation laws.

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.

Figures (9)

• Figure 1: Forbidden induced subgraphs in the frustration-graph approach: a claw (left) and examples of even holes with four and six vertices (middle and right).
• Figure 2: Frustration graph of free-fermions-in-disguise model with the Hamiltonian defined in eq. \ref{['eq:FFD_h']}.
• Figure 3: Frustration graph of $\tilde{H}^{\mathrm{\@slowromancap ii@}}$ with Hamiltonian density eq. (\ref{['density_tilde_hB']}), in which edges induced by eq. (\ref{['anti_XXZ_YYZ']}) are marked with red and blue.
• Figure 4: Frustration graph of $\tilde{H}^{\mathrm{\@slowromancap iii@}}$ with Hamiltonian density given by eq. (\ref{['density_tilde_hGamma']}), where green and blue edges show edges generated by eq. (\ref{['anti_XXZ_ZYY']}).
• Figure 5: The frustration graph of $\tilde{H}^{\mathrm{\@slowromancap iv@}}$ with Hamiltonian density given by eq. (\ref{['density_tilde_hDelta']}).

Theorems & Definitions (18)

• Theorem 1
• proof
• Theorem 2
• Conjecture 1
• Theorem 3
• Theorem 4
• Theorem 5
• proof
• Theorem 6
• proof