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Magnetic Exchange Interaction in the Spin Polarized Electron Gas

The exchange interaction between two magnetic moments embedded in a host metal is fundamental to the description of the magnetic behavior of solids. In the standard spin-degenerate electron gas, it leads to the well known Ruderman-Kittel-Kasuya-Yoshida (RKKY) interaction, which is of the Heisenberg form $J\vec S_1\cdot \vec S_2$, where the interaction strength $J$ oscillates and falls off with the distance between the moments in a characteristic way. Here, we study the more general case of the spin-polarized electron gas both in two and three dimensions, by evaluating the interaction strength as an integration over the product of the host Green's functions. We find that in addition to the Heisenberg term, an additional Ising-like term appears in the magnetic interaction, so that the net interaction for the spin-polarized gas is of the form $J_1 \vec S_1\cdot \vec S_2 + J_2 S_{1z} S_{2z}$. The interactions show a beating pattern as a function of distance, caused by the two different Fermi momenta for the two spins.