Ground-State Symmetry inXYModel of Magnetism

Numerical studies by Betts and Oitmaa have led those authors to conjecture that in the $\mathrm{XY}$ model the ground-state magnetization $M_z$ is zero. This is a model of spins on a lattice with interactions $-J({S_n}^x{S_m}^x+{S_n}^y{S_m}^y)$, that can also describe a hard-sphere boson fluid. In the present note I prove the conjecture for arbitrary spins $\mathcal{S}=\frac{1}{2}, 1, \frac{3}{2}, \dots$ for positive $J$, and treat various generalizations.