  In this paper, we revisit the effect of flat bands on the
Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction by using a coordinate
transformation that detangles flat-band states from dispersive ones. Under this
transformation, original flat-band systems containing magnetic impurities are
mapped onto a generalized Fano-Anderson model, where flat-band states act as
Fano defects. From this perspective, several features of exact RKKY couplings
calculated numerically can be understood easily. As an illustrative example, we
analyze a dimerized diamond chain model, which can exhibit either gapped or
gapless spectra depending on the ratio of hopping integrals. We find that
anomalous decay in the RKKY couplings arises exclusively in the gapless case
and with specific magnetic coupling configurations. Furthermore, the
conventional wisdom regarding the signs of RKKY interactions breaks down under
certain conditions. These subtleties arising from flat bands find explanation
within our present approach. Our investigation offers deeper insights into how
flat bands influence carrier-mediated exchange interactions, with implications
extending to broader contexts.
