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Discrete Approach.

Numerical diagonalization for magnon modes on a vortex was performed for circular square lattice systems with radius R<20a and fixed boundary conditions, using the method of Ref. [11]. Spin deviations 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 .

Gary M Wysin
Wed Sep 6 18:51:57 CDT 1995