To obtain one more way to understand the degeneracies, we can plot the complete wavefunctions, including the phase information. This can be done by drawing an arrow in the complex plane for each lattice site, where the length of the arrow is proportional to , and the direction of the arrow is determined by the phase of . Similar arrows can be drawn for the other component of the wavefunction, . For in-plane vortices, represents the in-plane spin fluctuations, while represents the out-of-plane spin fluctuations. For out-of-plane vortices, represents only a part of the in-plane spin fluctuations, while represents a combination of out-of-plane and in-plane spin fluctuations, depending on the static out-of-plane spin structure. This interesting representation of the two modes (b1) and (b2) is shown in Fig. 13, for , well below . Here we clearly see the distinction between these degenenerate modes, which is made in terms of the phase of the spinwave and the way it changes around the center position of the vortex. Mode (b2) has the phase of both components and changing in the positive sense around the vortex center, while mode (b1) has the phase of both components changing in the negative sense around the vortex center. Of course, this representation is not unique; we could make linear combinations of these two modes and produce equivalent wavefunctions that do not have this ``vortexlike'' and ``antivortex-like'' appearance, but which would produce squared wavefunctions more like those already shown for in Fig. 7b1 and Fig. 7b2.
Figure 13: Wavefunctions for the lowest degenerate modes (b1) and (b2) in the 180 site square lattice system with a vortex at the center, at , well below . The line-head arrows are the complex amplitudes , and the hollow-head arrows are the complex amplitudes . The relative sizes and phases of these amplitudes are preserved in these diagrams. relates to the out-of-plane spin fluctuations, and relates to the in-plane spin fluctuations. The frequency of these modes is =0.6776 .
Figure 14: Wavefunctions for the modes (b1) and (b2) in the 180 site system with a vortex at the center, as in Fig. 13, but for , just above , where they are now nondegenerate. Mode (b1) has frequency , and mode (b2) has frequency .