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 .

Wed Sep 6 18:51:57 CDT 1995