For comparison, we have also calculated the spectra for finite circular systems on triangular and hexagonal lattices. The same methods as described above for the square lattice were used, in which the primary physical difference is the coordination number z=6 for the triangular lattice, and z=3 for the hexagonal lattice. This leads to the different values of for the triangular lattice, and for the hexagonal lattice, as seen in the spectra shown in Fig. 12. These results are completely consistent with the Ansatz calculation for this mode.  There are substantial similarities in the spectra for the different lattices, including the splitting of the degeneracies for , the modes with downward cusps at , and the one component of the lowest degenerate pair coming close to zero frequency somewhat above . On the other hand, the symmetries of the lattices lead to small differences in the wavefunctions (not shown here).
Figure 12: Normal mode spectra (lowest 19 modes) of circular systems containing a vortex at the center, with fixed boundary conditions, for (a) triangular lattice with 174 sites; (b) hexagonal lattice with 192 sites. While many features of these results also appear in Fig. 2, the different values of are notable.