  We study the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two
classical magnetic impurities in one-dimensional lattice models with flat
bands. As two representative examples, we pick the stub lattice and the diamond
lattice at half filling. We first calculate the exact RKKY interaction
numerically and then compare our data to results obtained via different
analytical techniques. In both our examples, we find that the RKKY interaction
exhibits peculiar features that can directly be traced back to the presence of
a flat band in the energy spectrum. Importantly, these features are not
captured by the conventional RKKY approximation based on non-degenerate
perturbation theory. Instead, we find that degenerate perturbation theory
correctly reproduces our exact results if there is an energy gap between the
flat and the dispersive bands, while a non-perturbative approach becomes
necessary in the absence of a gap.
