the planar and the axial spin components.[@pet98]

Motivated by these experiments, the properties of S = 1 anisotropic Heisenberg models (both on a tetragonal and on a hexagonal lattice) including nondiagonal spin exchange were investigated by Monte Carlo simulations. The quantum nature of the spins were taken into account in a semi–classical approximation, where $S^z$ was discretized and the spin length fixed to be $\left| \vec{S} \right| = \sqrt{\mbox{S} \left( \mbox{S} + 1 \right)}$. Applied external fields with axial and planar components were considered.

These simulations showed that, besides the disordering of the axial spin components at high temperatures, a second phase transition involving the planar spin components takes place at lower temperatures. For the tetragonal models, the effects of two different nondiagonal spin exchange terms were discussed. It was shown that the nondiagonal coupling involving only products of two spins could induce, for large values of the coupling constant, a discontinuous change of the total axial magnetization at the ordering temperature of the planar spin components, whereas $M_z$ always changed continuously when the second exchange term involving products of squares of spins was considered.

For the realistic antiferromagnetic model defined on a hexagonal lattice, which is supposed to describe 