Minimal model for the flat bands in copper-substituted lead phosphate apatite

Two recent preprints gave evidence that a copper-substituted lead apatite, denoted as CuPb$_9$(PO$_4$)$_6$OH$_2$ and also known as LK99, could be a room-temperature superconductor. While other research groups have not yet replicated the superconductivity in this material, a recent Density Functional Theory (DFT) calculation indicated the presence of two nearly flat bands near the Fermi level. Such flat bands are known to exhibit strongly correlated physics, which could potentially explain the reported high-$T_c$ superconductivity. In order to facilitate the theoretical study of the intriguing physics associated with these two flat bands, we propose a minimal tight-binding model which reproduces their main features. We also discuss implications for superconductivity