Normal modes of vortices in easy-plane ferromagnets

We investigate the linear spin-wave spectrum of two-dimensional easy-plane classical Heisenberg ferromagnets in the presence of a vortex, using numerical diagonalization on small systems. The spectra of normal modes for both in-plane and out-of-plane vortices are determined, for square, triangular, and hexagonal lattices. Some of the modes show a strong localization of their amplitudes near the center of the vortex. Moreover, we investigate a particular mode that drives the crossover in the static vortex structure from purely in-plane to a vortex with well-localized out-of-plane component as the easy-plane character of the system is reduced below a certain threshold.