Instability of planar vortices in two-dimensional easy-plane Heisenberg model with distance-dependent interactions

It is known that magnetic vortices in two dimensional Heisenberg models with easy-plane anisotropy exhibit an instability depending on the anisotropy strength. In this paper, we study the statistic behavior of the two-dimensional easy-plane Heisenberg models with distance-dependent interactions, $J_{xy}(r)$ and $J_z(r)$ for in-plane and out-of-plane components. We develop analytical and numerical methods for accurate determination of critical anisotropy, above which out-of-plane vortices are stable. In particular, we explore the vortex formation of the Gaussian-type interaction model and determine the critical anisotropy accurately for square, hexagonal and triangular lattices.