Some typical spectra ( for ) for the two different boundary conditions are shown in Fig. 3 for a 180 site system. For both types of boundary conditions in Fig. 3, out-of-plane spin components are present in the static vortex for .
Figure 3: Comparison of the lowest 19 modes in the spinwave spectrum for a square lattice circular system with 180 sites, containing an in-plane vortex at its center, with (a) free boundary conditions, and (b) fixed boundary conditions. Degenerate modes are marked with solid circles. The solid and dotted lines are used only to distinguish nearby modes.
There are some striking features of these results, which are typical of all the eigenspectrum results for different sized systems and lattices. Consider the fixed boundary condition (Fig. 3b). One mode of the in-plane vortex (the lowest mode) comes close to zero frequency as approaches from below. This mode is still present for , with the out-of-plane vortex as static structure, and its frequency again rises away from zero. The spectrum suggests that this mode can be considered as a soft mode that is responsible for the energetic instability of an in-plane vortex to become an out-of-plane vortex, and vice-versa, as . The other obvious feature of the spectrum is that modes that are degenerate (solid circles) for the in-plane vortex become split for the out-of-plane vortex. This change also occurs for near . It is likely that the degeneracy for for the in-plane vortex comes about because these modes have an component that can be essentially ``up'' or ``'down''. On the other hand, for the out-of-plane vortex, this up-down symmetry is certainly violated, because the static spin configuration already possesses a nonzero profile. Also, one of the modes that comes out of the lowest degenerate pair goes very close to zero frequency for near 0.9 .
There are some differences for the free boundary condition, Fig. 3a. Here the ``soft mode'' that bears the most resemblance to that for the fixed boundary condition reaches a downward cusp at , but does not go below . On the other hand, this is the value of above which the lowest degenerate mode becomes nondegenerate, one component of which does go very close to zero frequency near , and then rises up again. For both boundary conditions, there are actually several modes higher up in the spectra that also come to downward cusps near , while other modes show no particular features near .