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arXiv:2410.23286 · Physical review. B./Physical review. B

Proof of nonintegrability of the spin-1 bilinear-biquadratic chain model

HaRu K. Park, SungBin Lee

Published Apr 28, 2025• 9 citations
arXivHTMLPDF

Abstract

Spin-1 chain models have been extensively studied in condensed matter physics, significantly advancing our understanding of quantum magnetism and low-dimensional systems, which exhibit unique properties compared to their spin-$1/2$ counterparts. Despite substantial progress in this area, a rigorous proof of nonintegrability for the bilinear-biquadratic chain model has been lacking. While integrable solutions are known for specific parameter values, a comprehensive understanding of the model's general integrability has remained elusive. In this paper, we present a rigorous proof of nonintegrability for general spin-1 bilinear-biquadratic chain models, by extending the proofs of nonintegrability on spin-$1/2$ models. Our proof not only confirms the nonintegrability of widely studied models but also provides offer deeper insights into several areas. These include the identification of the absence of local conserved quantities in quantum many-body scar systems with perfect fidelity revivals, such as the AKLT model. This paper marks a significant step toward understanding the complex dynamics of spin-1 systems.

DOI
10.1103/physrevb.111.134444

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Spin-1 chain models have been extensively studied in condensed matter physics, significantly advancing our understanding of quantum magnetism and low-dimensional systems, which exhibit unique properties compared to their spin-1/21/21/2 counterparts. Despite substantial progress in this area, a rigorous proof of nonintegrability for the bilinear-biquadratic chain model has been lacking. While integrable solutions are known for specific parameter values, a comprehensive understanding of the model's general integrability has remained elusive. In this paper, we present a rigorous proof of nonintegrability for general spin-1 bilinear-biquadratic chain models, by extending the proofs of nonintegrability on spin-1/21/21/2 models. Our proof not only confirms the nonintegrability of widely studied models but also provides offer deeper insights into several areas. These include the identification of the absence of local conserved quantities in quantum many-body scar systems with perfect fidelity revivals, such as the AKLT model. This paper marks a significant step toward understanding the complex dynamics of spin-1 systems.
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.
Establishing a rigorous proof of integrability or nonintegrability in quantum many-body systems remains one of the most challenging tasks, as it requires demonstrating the presence or absence of local conserved quantities and deciphering the complex dynamics of the system. In this paper, we introduce a graph-theoretical approach to address this issue. This framework not only provides a comprehensive method for analyzing the nonintegrability of quantum systems but also brings advantages for integrable systems by enabling the classification of local conserved quantities through a straightforward count of isolated loops in the associated graphs. As an illustrative example, we focus on the PXP model, a system widely believed to be nonintegrable, and rigorously confirm its nonintegrability by demonstrating the absence of local conserved quantities through graph-theoretical analysis. This proof for the PXP model conveys several key insights: It not only confirms the model's nonintegrability but also indicates that the quantum many-body scarring properties of the model is not associated with local conserved quantities. Our innovative approach is broadly applicable for proving (non-)integrability in other quantum many-body systems, significantly simplifying the process and offering numerous potential applications.
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 phenomenon of quantum many-body scars has received widespread attention both in theoretical and experimental physics in recent years due to its unique physical properties. In this paper, based on the su(2) algebraic relations, we propose a general method for constructing scar models by combining simple modules. This allows us to investigate many-body scar phenomena in high-spin systems. We numerically verify the thermalization and nonintegrability of this model and demonstrate the dynamical properties of the scar states. We also provide a theoretical analysis of the properties of these scar states. For spin-1 case, we find that our 1D chain model reduces to the famous PXP model [C. J. Turner et al., Phys. Rev. B 98, 155134 (2018)] under special parameter conditions. In addition, due to the continuous tunability of the parameters, our model also enables us to investigate the transitions of quantum many-body scar phenomena from nonintegrable to integrable system.
Absence of local conserved quantities, or nonintegrability, is often assumed when discussing various phenomena in quantum many-body systems, such as thermalization and transport. However, no concrete proof of this property is known in electron--phonon coupled systems, a typical setting for condensed matter physics. In this paper, we show that the one-dimensional Holstein model has no nontrivial local conserved quantities other than the Hamiltonian itself and the total fermion number operator. We further show that the absence of nontrivial local conserved quantities also holds for the more general Holstein--Hubbard model. We also establish the absence of nontrivial local conserved quantities in these models on higher-dimensional hypercubic lattices. Our result has accomplished an advance in nonintegrability proofs by expanding their scope to systems in which particles with different statistical properties are mixed.