  Despite their dispersion-less character, flat bands (FBs) are often at the
heart of remarkable physical phenomena. Because, FBs may be responsible for
unconventional quantum electronic transport, our purpose is to investigate
their impact on the RKKY couplings (J(R)). As a good candidate, we choose the
The Lieb lattice which consists of one four-fold coordinated atom (A) and two
two-fold coordinated atoms (B,C) per unit cell. As in graphene, at the
neutrality point, J(R) is found to fall off as 1/R3. The coupling between a
pair of impurities on sublattice A are found ferromagnetic and isotropic. In
contrast, for impurities on the two-fold coordinated sites, J(R) reveals (i) a
strong angular anisotropy and (ii) can be both ferromagnetic (F) or
antiferromagnetic (AF). The AF character introduced by the FB can largely
dominate, and introduces frustration effects. The FB is found to have a drastic
impact on J(R) for impurities located on (B,C) sublattices. We have also
addressed the effect of tuning the carrier density that reveals remarkable
features as well.
