Generalization of the Ruderman-Kittel-Kasuya-Yosida Interaction for Nonspherical Fermi Surfaces

The Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction $\Phi \left(\mathrm{R}\right)$ is generalized for the case of nonspherical Fermi surfaces. In general $\Phi \left(\mathrm{R}\right)$ is found to fall off as $\frac{1}{R^3}$, and to oscillate with a period corresponding to a calipering of the Fermi surface in the R direction. For the special cases of parallel or cylindrical regions of the Fermi surface, a slower falloff of $\Phi \left(\mathrm{R}\right)$ ($\frac{1}{R}$ and $\frac{1}{R^2}$, respectively) is obtained. The general expression for the Fourier transform $\phi \left(\mathrm{q}\right)$ is also considered and displays Kohn anomalies whose form depends on the shape of the Fermi surface in the vicinity of calipering pairs of points. The usual infinite slope in $\phi \left(\mathrm{q}\right)$ for the spherical Fermi surfaces changes in the more general case and becomes a discontinuous slope for a "waist" and a logarithmic singularity in $\phi \left(\mathrm{q}\right)$ for parallel regions of the Fermi surface. A number of possible applications of the results are discussed.