Relaxation schemes for normal modes of magnetic vortices: Application to the scattering matrix

Relaxation schemes for finding normal modes of nonlinear excitations are described and applied to the vortex-spinwave scattering problem in classical two-dimensional easy-plane Heisenberg models. The schemes employ the square of an effective Hamiltonian to ensure positive eigenvalues, together with an evolution in time or a self-consistent Gauss-Seidel iteration producing diffusive relaxation. We find some of the lowest frequency spinwave modes in a circular system with a single vortex present. The method is used to describe the vortex-spinwave scattering $(S$-matrix and phase shifts) and other dynamical properties of vortices in ferromagnets and antiferromagnets, for systems larger than that solvable by diagonalization methods. The lowest frequency modes associated with translation of the vortex center are used to estimate the vortex mass.