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

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.