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Despite their dispersion-less character, flat bands (FBs) are often at theheart of remarkable physical phenomena. Because, FBs may be responsible forunconventional quantum electronic transport, our purpose is to investigatetheir impact on the RKKY couplings (J(R)). As a good candidate, we choose theThe Lieb lattice which consists of one four-fold coordinated atom (A) and twotwo-fold coordinated atoms (B,C) per unit cell. As in graphene, at theneutrality point, J(R) is found to fall off as 1/R3. The coupling between apair of impurities on sublattice A are found ferromagnetic and isotropic. Incontrast, for impurities on the two-fold coordinated sites, J(R) reveals (i) astrong angular anisotropy and (ii) can be both ferromagnetic (F) orantiferromagnetic (AF). The AF character introduced by the FB can largelydominate, and introduces frustration effects. The FB is found to have a drasticimpact on J(R) for impurities located on (B,C) sublattices. We have alsoaddressed the effect of tuning the carrier density that reveals remarkablefeatures as well.

RKKY couplings in the Lieb lattice: flat-band induced frustration