RKKY and Dzyaloshinsky-Moriya Interaction in the Electron Gas

We illustrate the origin of the Dzyaloshinsky-Moriya interaction $ \vec S_1 \times \vec S_2$ between two localized spins embedded in a solid by considering the example of the electron gas with spin-split bands, which serves as a simple pedagogical model for the understanding of this interaction. In this case, where symmetry is broken, the magnetic interaction acquires the Dzyaloshinsky-Moriya term in addition to the well-known RKKY interaction term $\vec S_1 \cdot \vec S_2 $, so that the net interaction has the form $H = J \vec S_1 \cdot \vec S_2 + \vec D \cdot \vec S_1 \times \vec S_2$. For the standard electron gas with spin degenerate states, the DM term vanishes yielding the well known RKKY interaction results. Explicit expressions for the magnitudes of the interactions are obtained for the electron gas in two and three dimensions. This simple model serves as a pedagogical example for the origin of the Dzialoshinski-Moriya interaction in a system with broken time-reversal symmetry.