Short-range magnetic order in a three-dimensional XXZ model with competing exchange interactions

Within the framework of the Tyablikov approximation for two-time temperature Green's functions, we have studied transverse spin correlations in a three-dimensional XXZ model with a competing ferromagnetic exchange between the X and Y projections of spins and antiferromagnetic exchange between the Z projections of spins, which allows the possibility of the appearance of either a ferromagnetic order of the easy-plane type or a two-sublattice antiferromagnetism of the easy-axis type (on bipartition lattices). The temperature and spatial behaviors of transverse correlators both in the paramagnetic temperature range and in the magnetically ordered state directly below the temperature of the phase transition have been studied. It is shown that the stronger exchange interaction is responsible for the appearance of a long-range magnetic order of a specific type for the spin projections along the appropriate direction of exchange, whereas the weaker exchange interaction at the same time establishes a short-range magnetic order of a competing type for the spin projections that are perpendicular to the direction of the long-range magnetic order.