RKKY Interaction for the Spin Polarized Electron Gas

We extend the original work of Ruderman, Kittel, Kasuya, and Yosida (RKKY) on the interaction between two magnetic moments embedded in an electron gas to the case where the electron gas is spin polarized. The broken symmetry of a host material %(here we have a broken time reversal symmetry) introduces the Dzyaloshinsky-Moriya (DM) vector and tensor interaction terms, in addition to the standard RKKY term, so that the net interaction energy has the form: $ {\cal H} = J \vec S_1 \cdot \vec S_2 + \vec D \cdot \vec S_1 \times \vec S_2 + \vec S_1 \cdot \stackrel{\leftrightarrow}{\Gamma} \cdot \vec S_2 $. We find that for the spin-polarized electron gas, a non-zero tensor interaction $\stackrel{\leftrightarrow}{\Gamma}$ is present in addition to the scalar RKKY interaction $J$, while $\vec D$ is zero due to the presence of inversion symmetry. Explicit expressions for these are derived for the electron gas both in 2D and 3D. The RKKY interaction exhibits a beating pattern, caused by the presence of the two Fermi momenta $k_{F\uparrow}$ and $k_{F\downarrow}$, while the $R^{-3}$ distance dependence of the original RKKY result for the 3D electron gas is retained. This model serves as a simple example of the magnetic interaction in systems with broken symmetry, which goes beyond the RKKY interaction.