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arXiv:2411.02162 · Physical Review Research

Complete classification of integrability and nonintegrability for spin- 1 2 chain with symmetric nearest-neighbor interaction

Mizuki Sanatani, Yuuya Chiba, Naoto Shiraishi

Published Jun 9, 2026• 1 citation
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Abstract

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.

DOI
10.1103/k8xr-brs7

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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.
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.
We provide a rigorous proof of the absence of nontrivial local conserved quantities in all spin-1/2 chains with symmetric nearest-neighbor interaction, except for known integrable systems. This result shows that there are no further integrable system that awaits to be discovered. Our finding also implies that there is no intermediate systems with a finite number of nontrivial local conserved quantities. In addition, we clarify all short-support conserved quantities in non-integrable systems, which we need to take into account in analyses of thermalization and level statistics.
We investigate the integrability and non-integrability of isotropic spin chains with nearest-neighbor interaction with general spin SSS in terms of the presence or absence of local conserved quantities. We prove a dichotomy theorem that whether a single quantity is zero or not sharply separates two scenarios: (i) this system has kkk-local conserved quantities for all kkk (completely integrable), or (ii) this system has no nontrivial local conserved quantity (non-integrable). This result excludes the possibility of an intermediate system with some but not all local conserved quantities, which solves in the affirmative the Grabowski-Mathieu conjecture. This theorem also serves as a complete classification of integrability and non-integrability for S≤13.5S\leq 13.5S≤13.5, suggesting that all the integrable models are in the scope of the Yang-Baxter equation.
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 provide a complete classification of the integrability and nonintegrability of the spin-1 bilinear-biquadratic model with a uniaxial anisotropic field, which includes the Heisenberg model and the Affleck-Kennedy-Lieb-Tasaki model. It is rigorously shown that, within this class, the only integrable systems are those that have been solved by the Bethe ansatz method, and that all other systems are nonintegrable, in the sense that they do not have nontrivial local conserved quantities. Here, “nontrivial” excludes quantities like the Hamiltonian or the total magnetization, and “local” refers to sums of operators that act only on sites within a finite distance. This result establishes the nonintegrability of the Affleck-Kennedy-Lieb-Tasaki model and, consequently, demonstrates that the quantum many-body scars observed in this model emerge independently of any conservation laws of local quantities. Furthermore, we extend the proof of nonintegrability to more general spin-1 models that encompass anisotropic extensions of the bilinear-biquadratic Hamiltonian and completely classified the integrability of generic Hamiltonians that possess translational symmetry, U ( 1 ) symmetry, time-reversal symmetry, and spin-flip symmetry. Our result has accomplished a breakthrough in nonintegrability proofs by expanding their scope to spin-1 systems.
Abstract We rigorously prove that the S=1/2S=1/2S=1/2 S = 1 / 2 anisotropic Heisenberg chain (XYZ chain) with next-nearest-neighbor interaction, which is anticipated to be non-integrable, is indeed non-integrable in the sense that this system has no nontrivial local conserved quantity. Our result covers some important models including the Majumdar–Ghosh model, the Shastry–Sutherland model, and many other zigzag spin chains as special cases. These models are shown to be non-integrable while they have some solvable energy eigenstates. In addition to this result, we provide a pedagogical review of the proof of non-integrability of the S=1/2S=1/2S=1/2 S = 1 / 2 XYZ chain with Z magnetic field, whose proof technique is employed in our result.