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
.