Symmetry-enforced double Weyl points, multiband quantum geometry, and singular flat bands of doping-induced states at the Fermi level

Two common difficulties in the design of topological quantum materials are that the desired features lie too far from the Fermi level and are spread over a too-large energy range. Doping-induced states at the Fermi level provide a solution, where nontrivial topological properties are enforced by the doping-reduced symmetry. To show this, we consider a regular placement of dopants in a lattice of space group (SG) 176 (${P6}_{3}/m$), which reduces the symmetry to SG 143 ($P3$). Our two- and four-band models feature double Weyl points, Chern bands, Van Hove singularities, nontrivial multiband quantum geometry due to mixed orbital character, and singular flat bands. We relate these features to density-functional theory (DFT) calculations for dopant and vacancy bands of lead apatite ${\mathrm{Pb}}_{10}{({\mathrm{PO}}_{4})}_{6}\mathrm{O}$ and ${\mathrm{Pb}}_{10}{({\mathrm{PO}}_{4})}_{6}{(\mathrm{OH})}_{2}$, the van der Waals ferromagnet ${\mathrm{Cr}}_{2}{\mathrm{Ge}}_{2}{\mathrm{Te}}_{6}$, the semiconductor SiC, and the 2D dichalcogenide ${\mathrm{MoS}}_{2}$.