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Rigorous Test for Quantum Integrability and Nonintegrability
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Rigorous Test for Quantum Integrability and Nonintegrability
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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.
Forward citations
Cited by 6 Pith papers
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Proof of the absence of local conserved quantities in the Holstein model
The one-dimensional Holstein model and Holstein-Hubbard model have no nontrivial local conserved quantities other than the Hamiltonian and total fermion number.
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Absence of thermalization after a local quench and strong violation of the eigenstate thermalization hypothesis
In XX spin chains with open boundaries, a local quench via a single-spin impurity prevents thermalization and produces a strong violation of the eigenstate thermalization hypothesis, including its weak version.
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Violating the All-or-Nothing Picture of Local Charges in Non-Hermitian Bosonic Chains
Non-Hermitian bosonic chains with symmetric hopping can host k-local charges for selected k only, providing counterexamples to all-or-nothing integrability and showing the Grabowski-Mathieu 3-local test is not universal.
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Solving models with generalized free fermions II: Path-product expansion and conserved charges
Derives path-product expansion for free-fermion modes and local conserved charges in generalized free-fermion models from Krylov basis generating function.
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The $S=\frac{1}{2}$ XY and XYZ models on the two or higher dimensional hypercubic lattice do not possess nontrivial local conserved quantities
The S=1/2 XY and XYZ models on d≥2 hypercubic lattices possess no nontrivial local conserved quantities.
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Absence of nontrivial local conserved quantities in the quantum compass model on the square lattice
The quantum compass model on the square lattice possesses no nontrivial local conserved quantities besides the Hamiltonian.
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