This representation has the great advantage that it contains the physical explanation of how the symmetry is broken; for , once there are static out-of-plane spin components, the two different senses of change in the spinwave phase are not equivalent. Then mode (b1), with its phase changing in the negative sense around the vortex center, falls lower than mode (b2). The two wavefunctions are shown in Fig. 14, for , just above . Once there are nonzero spin components in the static vortex structure, this lack of equivalence for the two senses of rotation of the phase is very reasonable. It can also explain the higher degenerate pairs, because they also have the spinwave phase changing smoothly as one moves around the vortex center, some with higher winding numbers---the phase change of the spinwave in these cases changes by , where n is an integer [See Fig. 15]. Furthermore, this viewpoint shows why all of these degeneracies are split at , because the plus and minus senses of rotation of the phase are not equivalent, no matter what the winding number. Conversely, the modes that do not occur as degenerate pairs do not have a slow change in the phase around the vortex center. This new effect might be described as a coupling of the vorticity of the original vortex to the winding number of the phase of its spinwave excitations. Further details on the forms and characteristics of these wavefunctions will be published elsewhere.
Figure 15: Wavefunctions for the third lowest degenerate pair of modes of the 180 site system with a vortex at it center, as in Fig. 13, at . The frequency of these modes is , and they have winding numbers -3 and +3.