Long-Range Order in Antiferromagnetic Quantum Spin Systems

The existence of long-range order is proved under certain conditions for the antiferromagnetic XYZ model on the simple cubic or the square lattice. In particular, the spin-1/2 XXZ model on the square lattice is shown to have ground-state long-range order if the exchange anisotropy ∆ satisfies 0≦∆< <0.20 or ∆ > 1.72, which improves the result of Kubo and Kishi. The existence of long-range order of the z-component of the spin operator is proved for the XXZ model with XY-like anisotropy (0 ≦ ∆ ≦ 1) under certain conditions. A similar result is shown to hold for the long-range order in the x-direction for the Ising-like model (∆ ≧ 1). The XXZ model on the two-dimensional hexagonal lattice is proved to have finite ground-state long-range order for any value of ∆(≧0) if S ≧ l and for ∆ ?2.55 when S=1/2.