The above results show how there is one particular mode that goes soft for . To get a better idea of the physical structure of this mode or any of the modes, it is necessary to look at the associated wavefunctions.
Figure: Maps of the spin fluctuations in a square lattice system with 180 sites, for the six lowest frequency modes when Out-of-plane fluctuations (Eq. 3.13b) are proportional to the areas of the solid squares. In-plane fluctuations (Eq. 3.13a) are proportional to the white area within the larger squares. The total in-plane plus out-of-plane fluctuations are proportional to the areas of the larger squares. Part (a) shows the soft mode, while (b1) and (b2) are a degenerate pair, and (c), (d) and (e) are the next higher frequency modes.
Also, it is important to measure the spread of the wavefunctions, to understand whether a particular mode is localized on the vortex or extended throughout the system. For a system with 180 spins, the wavefunctions of the six lowest energy modes are shown in Figs. 7, 8, 9 , for , and , corresponding to well below , just below , and slightly above .
Figure 8: Maps of the spin fluctuations in the square lattice system as in Fig. 7, but for , just below the transition to an out-of-plane vortex. In (a), the soft mode's intensity concentrates itself more near the vortex core, and is strongly out-of-plane, while the higher modes in (b) through (e) have much smaller changes from their forms.
In these diagrams, two squares representing the squared wavefunction are plotted at each lattice site, in order to present both the in-plane and out-of-plane fluctuations for the selected mode on one diagram. The area of the inner solid square is proportional to the out-of-plane spin fluctuations for that site, as in Eq. (3.13b). The area of the larger open square is proportional to the total in-plane plus out-of-plane spin fluctuations, as in Eq. (3.13). The difference of the two areas (the white area outside the solid square, and inside the open square) is proportional to the in-plane spin fluctuations, as in Eq. (3.13a). For the soft mode (a), there is a substantial increase in the out-of-plane fluctuations as approaches , while the relative size of the in-plane fluctuations diminishes. For the other lowest modes, there are only minor changes in the fluctuations with . The mode labeled (b) is doubly degenerate, while (b1) and (b2) are its two components that are split above [Fig. 9]. Also note that the orientation of the two components of this mode is rather arbitrary, because there is an arbitrary phase between the two modes involved. This is the cause for the oblique angle of the line of nodes in mode (b1) in Figures 7 and 10.
For comparison, Fig. 10 shows the lowest modes on the 180-site system in the absence of the vortex, at , starting instead from a ferromagnetically aligned state. Some modes, including the one that most resembles the soft mode when the vortex is present, do not appear very different whether the vortex is present or absent (for this value of far from ). On the other hand, some modes, such as (b) and (d), clearly have amplitude at the vortex core that is not present when the vortex is removed.
Figure 9: Maps of the spin fluctuations in the square lattice system as in Fig. 7, but for , above . The soft mode in (a) is strongly out-of-plane, with intensity concentrated near the vortex core, while mode (b)'s degeneracy is now split.