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

Abstract: 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 ($P6\text{}3/m$), 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 Pb${10}($PO$4)_6$O and Pb${10}($PO$_4)_6($OH$)_2$, the van der Waals ferromagnet Cr$_2$Ge$_2$Te$_6$, the semiconductor SiC, and the 2D dichalcogenide MoS$_2$.