Localized magnon mode of in-plane magnetic vortices in easy-plane magnets

We consider localized magnon modes of magnetic vortices in two-dimensional classical magnets, with exchange or single-ion easy-plane anisotropy stronger than the critical value required to stabilize in-plane vortices. A discrete lattice ansatz for the structure of a magnon mode on one vortex is analyzed. Its lowest eigenmodes are found to be identical with modes obtained from numerical diagonalization for ferro- (FM) and antiferromagnets (AFM), showing that the ansatz is exact. For the AFM model, one mode is found to be localized, with frequency reaching a size-independent asymptotic limit. A continuum treatment leads to an effective Schrödinger problem that requires a small-radius cutoff due to the singularity of the vortex core. We find, however, that it is not possible to choose this cutoff consistently to recover the exact local mode frequency from the ansatz. This suggests that strong discrete lattice effects not well described by the usual continuum approximation always appear from the core of in-plane vortices.