  In this paper, we investigate the Kane-Mele model and endeavor to
demonstrate, through analytical calculations, how the presence of topological
edge states influences the RKKY interaction. We illustrate that the effect
diminishes as one moves away from the edges. To facilitate our analytical
approach, we initially utilize a one-dimensional wire exhibiting linear
dispersion for each spin as an approximation to the Kane-Mele model. We examine
its impact on the RKKY interaction. Subsequently, we establish a correspondence
between the edge states of the Kane-Mele model and a one-dimensional quantum
wire model, wherein the coupling strength diminishes with increasing distance
from the edges. Finally, we compare the analytical results with numerical
findings obtained using the Landauer-Buttiker formulation.
