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Rigorous Test for Quantum Integrability and Nonintegrability

Akihiro Hokkyo

TL;DR

This work provides a rigorously provable criterion to distinguish integrable from nonintegrable quantum spin chains with finite‑range interactions, based solely on the nonexistence of a $3$‑local conserved quantity when the length of the commutator with the Hamiltonian is bounded by $2$. Under injectivity of the two‑site interaction and a 1D dimensionality assumption, the absence of any $3$‑local obstruction $\hat{Q}$ with $\text{len}([\hat{Q},\hat{H}])\leq 2$ implies no $k$‑local conserved quantity for all $3\leq k\leq N/2$, yielding a size‑independent, algorithmic nonintegrability test. The paper demonstrates the framework on the XXZ chain with a nonuniform field and the $S=\tfrac{1}{2}$ XYZ model on the triangular lattice, proving nonintegrability by showing the $3$‑local obstruction cannot exist; it also develops a graphical proof strategy using string diagrams and constructs isomorphisms between locality spaces. Moreover, it extends the approach to higher dimensions and internal degrees of freedom, offering an internal‑degree‑of‑freedom nonintegrability test via reduction to two coupled chains. Overall, the work unifies prior nonintegrability proofs, provides a practical, system‑size‑independent algorithm, and lays groundwork for exploring spectrum‑generating algebras and quasi‑local conserved structures in many‑body systems.

Abstract

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 $S=1/2$ Heisenberg chain with nearest-and next-nearest-neighbor interactions, the $S=1$ bilinear-biquadratic chain and the $S=1/2$ XYZ model in two or higher dimensions. Moreover, our results also yield the first proof of nonintegrability for models such as the $S=1/2$ Heisenberg chain with a non-uniform magnetic field, the $S=1/2$ 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.

Rigorous Test for Quantum Integrability and Nonintegrability

TL;DR

This work provides a rigorously provable criterion to distinguish integrable from nonintegrable quantum spin chains with finite‑range interactions, based solely on the nonexistence of a ‑local conserved quantity when the length of the commutator with the Hamiltonian is bounded by . Under injectivity of the two‑site interaction and a 1D dimensionality assumption, the absence of any ‑local obstruction with implies no ‑local conserved quantity for all , yielding a size‑independent, algorithmic nonintegrability test. The paper demonstrates the framework on the XXZ chain with a nonuniform field and the XYZ model on the triangular lattice, proving nonintegrability by showing the ‑local obstruction cannot exist; it also develops a graphical proof strategy using string diagrams and constructs isomorphisms between locality spaces. Moreover, it extends the approach to higher dimensions and internal degrees of freedom, offering an internal‑degree‑of‑freedom nonintegrability test via reduction to two coupled chains. Overall, the work unifies prior nonintegrability proofs, provides a practical, system‑size‑independent algorithm, and lays groundwork for exploring spectrum‑generating algebras and quasi‑local conserved structures in many‑body systems.

Abstract

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 Heisenberg chain with nearest-and next-nearest-neighbor interactions, the bilinear-biquadratic chain and the XYZ model in two or higher dimensions. Moreover, our results also yield the first proof of nonintegrability for models such as the Heisenberg chain with a non-uniform magnetic field, the 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.

Paper Structure

This paper contains 21 sections, 15 theorems, 120 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setup and main result
  3. Models
  4. Main Result
  5. Examples
  6. The XXZ chain with a non-uniform magnetic field
  7. The $S=\frac{1}{2}$ XYZ model on the triangular lattice
  8. Injectivity of $\hat{H}_i^{(2)}$
  9. Confirmation of Assumption \ref{['eq:assumption_1dimensionality']}
  10. Absence of $3$-local quantity $\hat{Q}_{[3]}$ satisfying $\mathop{\mathrm{\mathrm{len}}}\nolimits([\hat{Q}_{[3]},\hat{H}])\leq 2$
  11. Outline of Proof
  12. String Diagram
  13. Isomorphisms between $\mathcal{B}^{(k)}_{\leq}$'s
  14. Isomorphisms between $\mathcal{B}^{(k)}_{<,b}$'s
  15. Absence of spectrum generating algebras
  16. ...and 6 more sections

Key Result

Theorem 1

Consider a Hamiltonian $\hat{H}$ with nearest-neighbor interactions and on-site potentials (see Eq. eq:Hamiltonian) satisfying the assumptions eq:assumption_injective and eq:assumption_1dimensionality. If there is no $3$-local quantity $\hat{Q}$ such that $\mathop{\mathrm{\mathrm{len}}}\nolimits([\h

Figures (2)

  • Figure 1: The triangular lattice can be regarded as the square lattice with additional edges. We define locality along the direction indicated by the horizontal lines.
  • Figure 2: The honeycomb lattice can be regarded as a subgraph of the square lattice. Therefore, a Hamiltonian with nearest-neighbor interactions on the honeycomb lattice takes the form of Eq. \ref{['eq:internal_Hamiltonian']}. We define locality along the direction indicated by the horizontal lines.

Theorems & Definitions (24)

  • Theorem 1: Main Result
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • Lemma 7
  • Proposition 8
  • ...and 14 more