Static and dynamic Friedel oscillations in an electron gas with fractional dispersion

We study the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in disordered systems, focusing on its static and dynamic behavior in one-, two-, and three-dimensional geometries. To model disorder, we introduce the fractional Laplacian, characterized by the Lévy index α<2, into the problem Hamiltonian. Our results demonstrate that disorder, modeled phenomenologically through the Lévy index α<2, significantly modifies the character of the RKKY interaction by suppressing and localizing Friedel oscillations, both statically and dynamically. As the disorder strength increases (i.e., as α decreases), the oscillations become increasingly localized, distorted, and rapidly damped. In the time domain, we show that dynamic spin-density excitations generated by magnetic impurities exhibit delayed and confined propagation in two dimensions, with significant attenuation at moderate distances. Our approach provides a unified framework for understanding indirect exchange interactions in disordered and low-dimensional systems such as diluted magnetic semiconductors like GaMnAs and InMnAs, which have significant implications for spintronic applications and magnetic quantum devices.