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.
Chiba, Proof of absence of local conserved quantities in two- and hi gher-dimensional quantum Ising models , (preprint, 2024)
3 Pith papers cite this work. Polarity classification is still indexing.
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The S=1/2 XY and XYZ models on d≥2 hypercubic lattices possess no nontrivial local conserved quantities.
The quantum compass model on the square lattice possesses no nontrivial local conserved quantities besides the Hamiltonian.
citing papers explorer
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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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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.