2D periodically ordered states of vortices-antivortices and skyrmion-like excitations in planar up to isotropic heisenberg classical spin arrays

In 2D classical spin arrays subject to planar Heisenberg interaction Jxy, periodic metastable states of vortex-antivortex pairs will be described. In the unit cell of 2nx2n spins with the symmetry of planar group P4, two vortices and two antivortices are arranged. Their pair energy increases nearly logarithmically with pair distance n. In contrast, for isotropic interaction J the excitations are half-skyrmion-like and their pair energy is ∼4πJ(1-1.02/n2), hence 4πJ in the continuous limit as for a single skyrmion. A collapse is prevented due to the increase of the mean energy per spin for decreasing distance n