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LK-99, with chemical formula Pb$_{10-x}$Cu$_x$(PO4)$_6$O, was recentlyreported to be a room-temperature superconductor. While this claim has met withlittle support in a flurry of ensuing work, a variety of calculations (mostlybased on density-functional theory) have demonstrated that the system possessessome unusual characteristics in the electronic structure, in particular flatbands. We have established previously that within DFT, the system is insulatingwith many characteristics resembling the classic cuprates, provided thestructure is not constrained to the $P$3(143) symmetry nominally assigned toit. Here we describe the basic electronic structure of LK-99 withinself-consistent many-body perturbative approach, quasiparticle self-consistentGW (QSGW) approximation and their diagrammatic extensions. QSGW predicts thatpristine LK-99 is indeed a Mott/charge transfer insulator, with a bandgap gapin excess of 3eV, whether or not constrained to the $P$3(143) symmetry. Thehighest valence bands occur as a pair, and look similar to DFT bands. Thelowest conduction band is an almost dispersionless state of largely Cu $d$character. When Pb$_9$Cu(PO$_4$)$_6$O} is hole-doped, the valence bands modifyonly slightly, and a hole pocket appears. However, two solutions emerge: ahigh-moment solution with the Cu local moment aligned parallel to neighbors,and a low-moment solution with Cu aligned antiparallel to its environment. Inthe electron-doped case the conduction band structure changes significantly:states of mostly Pb character merge with the formerly dispersionless Cu $d$state, and high-spin and low spin solutions once again appear. Thus we concludethat with suitable doping, the ground state of the system is not adequatelydescribed by a band picture, and that strong correlations are likely.

Multiple Slater determinants and strong spin-fluctuations as key  ingredients of the electronic structure of electron- and hole-doped  Pb$_{10-x}$Cu$_x$(PO4)$_6$O