Existence of Long-Range Order in theXYModel

The existence of long-range order (LRO) at low temperature in the isotropic $\mathrm{XY}$ model with spin $S\geqq 1$ on simple cubic lattice in three or more dimensions is shown. This completes the proof of the existence of LRO at low temperature in the isotropic $\mathrm{XY}$ model with arbitrary spin size in three or more dimensions, as the proof for $S=\frac{1}{2}$ was given earlier by Dyson, Lieb, and Simon. The existence of LRO in the ground state is shown for $S\geqq \frac{3}{2}$ in two dimensions.