The $S=rac{1}{2}$ XY and XYZ models on the two or higher dimensional hypercubic lattice do not possess nontrivial local conserved quantities

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.

💡 Research Summary

The paper investigates the S = ½ quantum spin system defined on a d‑dimensional hypercubic lattice (d ≥ 2) with uniform nearest‑neighbor XY or XYZ exchange interactions and an arbitrary uniform magnetic field. The Hamiltonian is
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