Absence of nontrivial local conserved quantities in the spin-1 bilinear-biquadratic chain and its anisotropic extensions

TL;DR

This paper rigorously classifies the integrability of spin-1 bilinear-biquadratic models with anisotropy, proving nonintegrability for most cases and clarifying the origin of quantum many-body scars without local conserved quantities.

Contribution

It provides a complete classification of integrability in spin-1 models, establishing nonintegrability beyond Bethe ansatz solvable cases and extending proofs to anisotropic and symmetric models.

Findings

Only Bethe ansatz solvable models are integrable.

Most spin-1 models lack nontrivial local conserved quantities.

Quantum many-body scars occur independently of local conservation laws.

Abstract

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 classify the integrability of generic Hamiltonians that possess translational symmetry, $U(1)$ symmetry, time-reversal symmetry, and spin-flip symmetry. Our result accomplishes a breakthrough in nonintegrability proofs by expanding their scope to spin-1 systems.