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

TL;DR

This paper proves that the $S=1/2$ XY and XYZ models on higher-dimensional hypercubic lattices lack nontrivial local conserved quantities, indicating their non-integrability.

Contribution

It extends a method from quantum spin chains to higher dimensions, showing these models have no local conserved quantities besides trivial ones.

Findings

Models lack nontrivial local conserved quantities.

Result applies to the XX model without magnetic field.

Supports the non-integrability of these models.

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.