Multiple-spin exchange in a two-dimensional Wigner crystal

Multiple-spin exchange constants in a two-dimensional Wigner crystal are calculated with a WKB approximation. Contributions of the Gaussian fluctuations around classical exchange paths are accurately calculated. In the dilute limit, the three-spin exchange interaction is found to be dominant in agreement with the previous study. Over a wide range of the electron density, however, n-spin exchange interactions with $n>~4$ are found to be of considerable magnitude, which makes the ground phase diagram nontrivial. Recent experimental findings by Okamoto and Kawaji [Phys. Rev. B 57, 9097 (1998)] are also discussed in the light of the present results.