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

Akihiro Hokkyo, Mizuki Yamaguchi, Yuuya Chiba

TL;DR

The authors prove a rigorous nonintegrability classification for the spin-1 BLBQ chain with a uniaxial field and its anisotropic extensions, showing that, apart from Bethe-ansatz integrable points, the model lacks nontrivial local conserved quantities of length k≥3. Using an operator-basis built from spherical-harmonic-like elements and Shiraishi's framework, they demonstrate that 3≤k≤N/2 local conserved quantities do not exist, while 1- and 2-local conservations reduce to known normals (e.g., total magnetization) or are absent, respectively. They further extend the analysis to generic translationally invariant, U(1)-symmetric, time-reversal, and spin-flip symmetric spin-1 models, and derive necessary integrability conditions for extended anisotropic Hamiltonians, confirming integrability only at celebrated Bethe-ansatz points (including Fateev–Zamolodchikov cases). The results establish AKLT nonintegrability and clarify the relation between QMBS and local conservation laws in spin-1 systems, offering a blueprint for exploring nonintegrability in higher-spin chains.

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.

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

TL;DR

The authors prove a rigorous nonintegrability classification for the spin-1 BLBQ chain with a uniaxial field and its anisotropic extensions, showing that, apart from Bethe-ansatz integrable points, the model lacks nontrivial local conserved quantities of length k≥3. Using an operator-basis built from spherical-harmonic-like elements and Shiraishi's framework, they demonstrate that 3≤k≤N/2 local conserved quantities do not exist, while 1- and 2-local conservations reduce to known normals (e.g., total magnetization) or are absent, respectively. They further extend the analysis to generic translationally invariant, U(1)-symmetric, time-reversal, and spin-flip symmetric spin-1 models, and derive necessary integrability conditions for extended anisotropic Hamiltonians, confirming integrability only at celebrated Bethe-ansatz points (including Fateev–Zamolodchikov cases). The results establish AKLT nonintegrability and clarify the relation between QMBS and local conservation laws in spin-1 systems, offering a blueprint for exploring nonintegrability in higher-spin chains.

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, symmetry, time-reversal symmetry, and spin-flip symmetry. Our result accomplishes a breakthrough in nonintegrability proofs by expanding their scope to spin-1 systems.

Paper Structure

This paper contains 20 sections, 13 theorems, 119 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Setup and Main Result
  3. Proof strategy
  4. Basis
  5. Proof outline
  6. Proof
  7. Step 1 (analysis of $r_{{\bf B}^{k+1}_i} = 0$) for $k=3$ case
  8. Step 2 (analysis of $r_{{\bf B}^{k}_i} = 0$) for $k=3$ case
  9. Step 1 (analysis of $r_{{\bf B}^{k+1}_i} = 0$) for $3 \leq k \leq N/2$
  10. Step 2 (analysis of $r_{{\bf B}^{k}_i} = 0$) for $3 \leq k \leq N/2$
  11. $k=2$ case
  12. $k=1$ case
  13. Nonintegrability of systems with anisotropic interactions
  14. Conclusion and Discussion
  15. For $J_2=0$ and $J_2=2J_1$
  16. ...and 5 more sections

Key Result

Theorem 1

In the BLBQ model eq:H_BLBQ not satisfying Eq. eq:except, $k$-local conserved quantities with $3 \leq k \leq N/2$ are absent.

Figures (1)

  • Figure 1: We perform a complete classification of the BLBQ model into integrable and nonintegrable systems. In the case without anisotropic field $D$, shown on the left, there are three known integrable systems, as indicated by three lines. In the case with nonzero $D$, shown on the right, there is one known integrable system. We prove the nonintegrability of the complement of these integrable systems.

Theorems & Definitions (23)

  • Theorem 1: Main Result
  • Theorem 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • ...and 13 more