and 
from the
static vortex spins were written in local coordinate frames for
each spin in terms of creation/annihilation operators, e.g.,
and replacing 
to give 
. Here
. The matrix equations of motion for the
coefficients was diagonalized numerically.
Some modes for a system of radius R=8a with a vortex at the
center are shown in Fig. 2; the 
's are
represented as arrows in the complex plane.
Using the semiclassical condition, 
, connections between the 
coefficients and 
can be established. Using the
formula 
for the first
for the second
one, with 
determined by (5), we have
and a similar equation for 
,
changing the signs before 
.
Then one can show that the connections are:
and similarly for 
,
changing 
and 
.
The equations (9) work well for low-energy modes.
For example, the arrows for 
in
Fig. 2a, b, are perpendicular to those for 
; in different sublattices they are
antiparallel for the mode with 
(Fig.
2a), and parallel for the 
mode (Fig.
2b). For the case 
(e.g., Fig.
2c), the ratio 
[Eq. (7)] can take
the values 
, and degenerate pairs of modes (
)
combine to form linear combinations with structure 
.
