Comment on "Origin of correlated isolated flat bands in copper-substituted lead phosphate apatite"

In this comment on "Origin of correlated isolated flat bands in copper-substituted lead phosphate apatite" (arXiv:2307.16892 [cond-mat.supr-con]), we discuss the flat half-occupied two-band manifold that appears in $\mathrm{Pb}_9\mathrm{Cu}(\mathrm{PO}_4)_6(\mathrm{OH})_2$ when using a semilocal DFT functional. We argue that the flat band is an artifact of the functional's overestimation of the energy of the oxygen p states in the valence band. When using the HSE hybrid functional, the energy of the oxygen p states is reduced, and the copper-derived manifold splits into one fully occupied and one empty band. While these results do not rule out the possibility of superconductivity in doped LK-99, they do predict that stoichiometric $\mathrm{Pb}_9\mathrm{Cu}(\mathrm{PO}_4)_6(\mathrm{OH})_2$ is an insulator, not a superconductor. Furthermore, we have shown that future first-principles studies of these materials should employ hybrid functionals or other advanced methods to ensure that the oxygen-derived valence-band energies are correctly described.