Figure 1: Profiles of vs. distance r from the vortex center, for single static out-of-plane vortices at the center of a circular system with 180 sites on a square lattice. The curves are labelled by the different values of . These results were obtained by the relaxation procedure described in the text (Sec. ii), using a fixed boundary condition as described in Sec. iv.
Now the approach is to make the spinwave perturbation calculation using the directions, , as the different local quantization axes for each site. This is similar in spirit to a spinwave calculation for an antiferromagnet,  in which the classical ground state can have sets of spins on the different sublattices aligned along different directions, and the perturbation analysis uses different coordinate systems for different sublattices.