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

TL;DR

This paper provides the first rigorous proof that the general spin-1 bilinear-biquadratic chain model is nonintegrable, advancing understanding of quantum many-body dynamics and the complexity of spin-1 systems.

Contribution

It introduces a comprehensive proof of nonintegrability for the spin-1 bilinear-biquadratic chain, unifying methods and exploring implications for quantum scar systems.

Findings

Confirmed nonintegrability of the model

Unified proof using graph theoretical methods

Identified absence of local conserved quantities in certain systems

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, providing a rigorous proof of nonintegrability for the bilinear-biquadratic chain model remains an open challenge. While integrable solutions are known for specific parameter values, a comprehensive understanding of the model's general integrability has been elusive. In this paper, we present the first rigorous proof of nonintegrability for the general spin-$1$ bilinear-biquadratic chain models. Our proof not only confirms the nonintegrability of widely studied models but also extends to offer deeper insights into several areas. These include the unification of nonintegrability proofs using graph theoretical methods and 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 work marks a significant step toward understanding the complex dynamics of spin-$1$ systems and offers a framework that can be applied to a broader class of quantum many-body systems.