Motion of vortex pairs in the ferromagnetic and antiferromagnetic anisotropic Heisenberg model

We study the motion of a pair of unbound vortices in two-dimensional classical spin systems with easy-plane exchange symmetry (XY symmetry). Assuming a velocity-independent shape we derive an equation of motion for one vortex in the presence of the other. The results are compared with a direct molecular-dynamics simulation on a 50×50 square lattice at zero temperature. For both the ferromagnet and the antiferromagnet there exists a critical value ${\lambda }_{\mathit{c}}$ of the anisotropy parameter λ which separates two regimes with different stable vortex structures. For λ<${\lambda }_{\mathit{c}}$ the spins creating a vortex are essentially in the easy plane and the motion of a pair of vortices is mainly determined by repulsive or attractive forces for equal or different vorticities, respectively. For λ>${\lambda }_{\mathit{c}}$ the vortices have additional out-of-plane spin components which depend on λ. In the ferromagnetic case these z components are parallel to each other and act effectively on the other vortex like a magnetic field. Together with the attraction or repulsion discussed above, this effective field leads to rotation of the vortices around each other, or to a translation parallel to each other, depending on whether the products of the vorticity and the sign of the out-of-plane components are equal or different for the two vortices. For an antiferromagnetic out-of-plane vortex, however, the z components of the spins are antialigned and therefore do not give an effective magnetic field, here the vortices move essentially on straight lines. In our simulations we observe the above discussed trajectories. The fluctuations around these trajectories due to the discreteness of the lattice are more pronounced for λ<${\lambda }_{\mathit{c}}$ than for λ>${\lambda }_{\mathit{c}}$.