Influence of fractional dispersion on finite-temperature indirect exchange interactions

We study how disorder and finite temperature modify the indirect exchange interaction between magnetic impurities in one, two, and three dimensions by replacing the standard quadratic dispersion with a fractional dispersion ε_{k}∼k^{α} (Lévy index α<2 plays a role of the phenomenological disorder descriptor) in the thermal Green's function. In this fractional framework, increasing disorder suppresses and spatially localizes Friedel oscillations, while finite temperature further damps their amplitude and reduces the spatial and temporal coherence of the spin response, leading to an aperiodic, strongly attenuated behavior at large distances and long times. The resulting dynamic spin density exhibits a shortened interaction range, enhanced retardation, and features reminiscent of subdiffusive spin transport. We derive explicit relations between the generalized diffusion coefficient D_{α}, the Lévy index α, and microscopic parameters such as Fermi energy and carrier density, opening a route to experimentally determine these phenomenological quantities. Our results provide a compact description of disordered, thermally driven RKKY interactions and suggest practical strategies for tuning spin coupling in low-dimensional spintronic devices via temperature and controlled disorder.