The effects of competing magnetic interactions in stabilizing different spin configurations are drawing a renewed attention in order to both unveil emerging topological spin textures and to highlight microscopic mechanisms leading to their stabilization. The possible key role of the two-site exchange anisotropy in selecting specific helicity and vorticity of skyrmionic lattices has only recently been proposed. In this work we explore the phase diagram of a frustrated localized magnet on a two-dimensional centrosymmetric triangular lattice, focusing on the interplay between the two-ion anisotropy (TIA) and the single-ion anisotropy (SIA). The effects of an external magnetic field applied perpendicularly to the magnetic layer are also investigated. By means of Monte Carlo simulations, we find a profusion of different spin configurations, going from trivial to high-order Q skyrmionic and meronic lattices. In closer detail, we find that a dominant role is played by the two-ion over the single-ion anisotropy in determining the planar spin texture, whereas the strength and sign of SIA, together with the magnitude of the magnetic field, tune the perpendicular spin components, mostly affecting the polarity (and, in turn, the topology) of the spin-texture. Our analysis confirms the crucial role of anisotropic symmetric exchange in systems with dominant short-range interactions, at the same time predicting a rich variety of complex magnetic textures that may arise from a fine tuning of competing anisotropic mechanisms.
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