Collective-variable approach to the dynamics of nonlinear magnetic excitations with application to vortices

We propose a collective-variable ansatz for a system of N extended, but finite, nonlinear excitations in magnetic systems. In contrast to earlier approaches, where the interactions between the different excitations have been treated only through an external force term, we explicitly consider a dependence of the microscopic spin field on all the coordinates and velocities of the localized objects. This leads to N coupled equations of motion with parameters (mass and gyro tensors) which explicitly depend on the mutual distances of the excitations. We apply this ansatz to vortices in two-dimensional Heisenberg ferromagnets with weak easy-plane anisotropy. For vortex pairs we find either rotational or translational motion, with an additional cyclotronlike oscillation on top of the main trajectories. Due to the interactions between the two vortices we obtain two different eigenvalues of the mass and gyro tensors with values depending on the distance between the vortices, their vorticities, and the sign of their out-of-plane structures. In contrast to the single-vortex mass, which depends logarithmically on the system size L, we find two-vortex masses, which are independent of L, but depend on their mutual distance. These predictions are in good qualitative agreement with numerical simulations of the complete spin systems. However, since these simulations are performed at zero temperature, where the vortex pairs extend throughout the whole system, we observe a strong influence of the boundaries on the absolute values of the masses.