We can also consider how the size of the system affects the low-frequency spectrum. For example, the lower part of the spectrum for a system with 492 sites is shown in Fig. 4a. In general the low-frequency modes found for the 180-site system are also seen in the 492-site system, but slightly shifted in frequency. The same sets of two-fold degeneracies are also found, which are again related to a symmetry in for these modes of the in-plane vortex. And, once again, there are modes higher up in the spectrum with downward cusps at .
Figure 4: Comparison of the lowest 19 modes in the spinwave spectrum for a square lattice circular system with 492 sites, with fixed boundary conditions, (a) containing an in-plane vortex at its center, and (b) no vortex (spinwave modes about ferromagnetically aligned state). The solid and dotted lines are used only to distinguish nearby modes.
In Fig. 4b, the low-frequency spectrum for the 492-site system without a vortex is shown. The sites were initially aligned in the xy-plane ferromagnetically. We find more or less the same sequence of nondegenerate and degenerate modes, which do not split at any . There is no mode that goes soft, as expected, since the ferromagnetic state is a stable energy minimum for , and there are no cusps.