To study the tendency of the vortex to concentrate a mode near its core, we define the rms spread of a wavefunction using the total in-plane plus out-of-plane fluctuations as a weighting factor, as follows;
In a similar way, the rms spread of the system itself can be evaluated by using a constant weighting factor. For the system with 180 sites that is discussed in Figs. 7, 8, and 9, the rms spread of the system is 5.355 lattice constants. The rms spreads of the lowest frequency wavefunctions are shown in Fig. 11. There is a substantial reduction in for the soft mode as , while for the other modes there tends to be less drastic changes. Only mode (b1), which crosses the soft mode (a) slightly above , shows a similar sized change. Because the soft mode has an rms spread much smaller than the rms spread of the system for near , we interpret this to mean that the soft mode is a mode localized on the vortex, while the other lowest modes are more extended over the whole system. It is possible that there could be other modes higher up in the spectrum which are also localized in this sense, but it could be difficult to detect them because of the limited size of the systems that can be easily solved numerically.
Figure 11: The rms spreads of the 6 lowest frequency wavefunctions for the square lattice system with 180 sites, versus anisotropy parameter . The letters refer to the modes shown in Figs. 4--6. For the lattice itself, , while the radius of the system is about 8 lattice constants.