Magnon modes for a circular two-dimensional easy-plane ferromagnet in the cone state

We calculate the magnon modes in the presence of a vortex in a circular easy-plane ferromagnet with a magnetic field perpendicular to the plane of the magnet. We also determine the range of anisotropy and magnetic field for which the two vortex branches, known as light and heavy cone-state vortices, are stable. The analysis was done by combining analytical calculations in the continuum limit with numerical simulations of small discrete systems. For large enough systems the magnon modes are expressed by the S matrix for magnon-vortex scattering. For small systems the vortex structure and consequently the magnon scattering are affected by the finite size, for which a theory designed for isotropic magnets is extended here. The presence of magnetic field in combination with easy-plane anisotropy leads to a splitting of doublets both near a small magnetic field and when the magnetic field is comparable to the anisotropy field. Similar doublets with splitting determined by the magnetic field may be expected in the mode spectra of small magnetic particles.