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arXiv:2602.07867

Minimal nonintegrable models with three-site interactions

Wen-Ming Fan, Kun Hao, Xiao-Hui Wang, Kun Zhang, Vladimir Korepin

quant-phmath-phnlin.SISubmitted Feb 8, 2026• Announced Feb 10, 2026• 0 citations
v1 · Feb 8, 2026v2 · Aug 20, 2026
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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.

Comments
41pages, 9 figures, 2 tables

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We prove that, for translationally invariant quantum spin chains with finite-range interactions, the existence of a specific conservation law implies the presence of infinitely many local conserved quantities, i.e., integrability. This shows that the standard hierarchy of local conserved quantities arising for nearest-neighbor Hamiltonians obtained via the canonical Yang-Baxter construction is already encoded in the lowest nontrivial conservation law, known as the Reshetikhin condition. Combined with recent rigorous results on nonintegrability, our theorem strongly restricts the possibility of partially integrable systems that admit only a finite but large number of local conserved quantities. Our work establishes a rigorous foundation for the systematic identification of new integrable models and deepens the algebraic understanding of conservation-law structures in quantum spin chains.
Conservation laws serve as the hallmark of integrability. The absence of conserved charges typically implies that the model is nonintegrable. The recently proposed Fredkin spin chain exhibits rich structures, and its ground state is analytically known. However, whether the Fredkin spin chain is integrable remains an open question. In this work, through rigorous analytical calculations, we demonstrate that the Fredkin spin chain, under both periodic and open boundary conditions, lacks local conserved charges, thereby confirming its nonintegrable nature. Furthermore, we find that when one or a portion of the Hamiltonian terms are removed (referred to as the truncated Fredkin spin chain), local conserved charges are still absent. Our findings suggest that in models involving three-site interactions, integrable models are generally rare.
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 nonintegrable 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 that 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 nonintegrable.
The integrability of a quantum many-body system, which is characterized by the presence or absence of local conserved quantities, drastically impacts the dynamics of isolated systems, including thermalization. Nevertheless, a rigorous and comprehensive method for determining integrability or nonintegrability has remained elusive. In this paper, we address this challenge by introducing rigorously provable tests for integrability and nonintegrability of quantum spin systems with finite-range interactions. Our results significantly simplify existing proofs of nonintegrability, such as those for the S=1/2S=1/2S=1/2 Heisenberg chain with nearest-and next-nearest-neighbor interactions, the S=1S=1S=1 bilinear-biquadratic chain and the S=1/2S=1/2S=1/2 XYZ model in two or higher dimensions. Moreover, our results also yield the first proof of nonintegrability for models such as the S=1/2S=1/2S=1/2 Heisenberg chain with a non-uniform magnetic field, the S=1/2S=1/2S=1/2 XYZ model on the triangular lattice, and the general spin XYZ model. This work also offers a partial resolution to the long-standing conjecture that integrability is governed by the existence of local conserved quantities with small support. Our framework ensures that the nonintegrability of one-dimensional spin systems with translational symmetry can be verified algorithmically, independently of system size.
We study integrable spin chains and quantum and classical cellular automata with interaction range ℓ≥3. This is a family of integrable models for which there was no general theory so far. We develop an algebraic framework for such models, generalizing known methods from nearest-neighbor interacting chains. This leads to a new integrability condition for medium-range Hamiltonians, which can be used to classify such models. A partial classification is performed in specific cases, including U(1)-symmetric three-site interacting models, and Hamiltonians that are relevant for interaction-round-a-face models. We find a number of models which appear to be new. As an application we consider quantum brickwork circuits of various types, including those that can accommodate the classical elementary cellular automata on light cone lattices. In this family we find that the so-called Rule150 and Rule105 models are Yang-Baxter integrable with three-site interactions. We present integrable quantum deformations of these models, and derive a set of local conserved charges for them. For the famous Rule54 model we find that it does not belong to the family of integrable three-site models, but we cannot exclude Yang-Baxter integrability with longer interaction ranges.