Only two models in the class of S=1/2 zigzag spin chains are integrable (one classical, one Bethe-ansatz solvable); all others are non-integrable, with no missing integrable models and no intermediate cases having finitely many local conserved quantities.
The $S=\frac{1}{2}$ XY and XYZ models on the two or higher dimensional hypercubic lattice do not possess nontrivial local conserved quantities
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abstract
We study the $S=\frac{1}{2}$ quantum spin system on the $d$-dimensional hypercubic lattice with $d\ge2$ with uniform nearest-neighbor interaction of the XY or XYZ type and arbitrary uniform magnetic field. By extending the method recently developed for quantum spin chains, we prove that the model possesses no local conserved quantities except for the trivial ones, such as the Hamiltonian. This result strongly suggests that the model is non-integrable. We note that our result applies to the XX model without a magnetic field, which is one of the easiest solvable models in one dimension.
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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.
The quantum compass model on the square lattice possesses no nontrivial local conserved quantities besides the Hamiltonian.
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Complete classification of integrability and non-integrability of S=1/2 spin chains with symmetric next-nearest-neighbor interaction
Only two models in the class of S=1/2 zigzag spin chains are integrable (one classical, one Bethe-ansatz solvable); all others are non-integrable, with no missing integrable models and no intermediate cases having finitely many local conserved quantities.
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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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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.