Existence of Long-Range Order in the XXZ Model

The method of Kennedy, Lieb, and Shastry is applied to the antiferromagnetic 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≧1/2 and Δ≧0 at low temperature. On the 2D square lattice LRO exists in the ground state for any Δ≦0 if S≧1 and for 0≦Δ<0.13 and Δ=0 if S=1/2. The existence of LRO in the ground state of the S=1/2 XY model (Δ=0), therefore, has been confirmed, in agreement with the conjecture by Nishimori et al. The ground state of the S=1/2 antiferromagnetic Heisenberg model in two dimensions still remains to be clarified.