Existence of Long-Range Order in the $\mathrm{XXZ}$ Model

The method of Kennedy, Lieb, and Shastry is applied to the antiferromagnetic $\mathrm{XXZ}$ model to examine the existence of long-range order (LRO). On the 3D simple-cubic lattice LRO is proved to exist for any $S\geqq \frac{1}{2}$ and $\Delta \geqq 0$ at low temperature. On the 2D square lattice LRO exists in the ground state for any $\Delta \leqq 0$ if $S\geqq 1$ and for $0\leqq \Delta <0.13\mathrm{}$ and $\Delta =0\mathrm{}$ if $S=\frac{1}{2}$. The existence of LRO in the ground state of the $S=\frac{1}{2} \mathrm{XY}$ model ($\Delta =0\mathrm{}$), therefore, has been confirmed, in agreement with the conjecture by Nishimori et al. The ground state of the $S=\frac{1}{2}$ antiferromagnetic Heisenberg model in two dimensions still remains to be clarified.