We have found that some of the higher modes also have strong changes associated with the vortex instability; partial evidence is the downward cusps in the spectrum. Additional evidence appears in the wavefunctions themselves. For example, mode (c) (third lowest in Fig. 3b) is one of the modes with a cusp, and for , there is not too much difference in this mode's structure, regardless of the presence or absence of the vortex [compare Fig. 7c and Fig. 10c], except that the mode is more concentrated at the system center when the vortex is present. However, this mode also concentrates itself additionally onto the vortex core for near , just as mode (a) does, as seen in Fig. 8c. Presumably, the other modes higher up in the spectrum that possess downward cusps in their -dependences are also strongly affected by the vortex instability.
The crossover from in-plane to out-of-plane vortices exhibits itself in an even more obvious way. A significant fraction of the modes are degenerate for , but all of these degeneracies split for . These degeneracies must be associated with a symmetry of the in-plane vortex, that is broken in the out-of-plane vortex. For example, consider the lowest degenerate mode, (b), in Fig. 3b, for . We might expect that a vortex in an infinite, continuum limit system would have a degenerate pair of zero-frequency modes associated with translation of its position in the two lattice directions. This pair would then be shifted to finite frequency on the discrete lattice, and when the vortex is additionally confined to a finite system as we have here, they would correspond to the two different directions along which the vortex center position could oscillate, rather than translate. But clearly, with this interpretation there is some problem to understand how this spatial symmetry could be broken in the out-of-plane vortex, or why the out-of-plane vortex wouldn't have a degenerate pair of translation modes. On the other hand, it is known that the dynamical response of the out-of-plane vortex to an external force is substantially different from that for the in-plane vortex. This is because the the gyrovector (vorticity times at vortex core) of the in-plane vortex is zero, but for the out-of-plane vortex it is nonzero . However, one would still need to explain the additional symmetries associated with the other degeneracies as well.
A simpler way to view the degeneracies is that they are most closely associated with symmetries of the in-plane vortex in spin space, the most important of which is that it is invariant under reversal of the out-of-plane component, because that component is zero. Then, all of the degeneracies must somehow be associated with the symmetry of those modes under reversal of their out-of-plane spin components, . For the in-plane vortex, this is equivalent to . Once we have an out-of-plane vortex for , the static vortex structure has all either greater than 0 or all less than 0. Then it is clear that the perturbations (specifically, ) about that static structure cost different energies depending on whether they increase or decrease each , leading to a breaking of the up-down symmetry that was present in the in-plane vortex. However, thinking this way, there is still a problem to understand which of the modes would be in degenerate pairs for .