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 ${\mathrm{Pb}}_{10}{\left({\mathrm{PO}}_4\right)}_6\mathrm{O}$ and ${\mathrm{Pb}}_{10}{\left({\mathrm{PO}}_4\right)}_6{\left(\mathrm{OH}\right)}_2$, the van der Waals ferromagnet ${\mathrm{Cr}}_2{\mathrm{Ge}}_2{\mathrm{Te}}_6$, the semiconductor SiC, and the 2D dichalcogenide ${\mathrm{MoS}}_2$.

Figure 1

Two-band model for SG $P3$. [(a),(b)] Top and side view of the trigonal lattice with nearest-neighbor hopping terms, where symmetry-related directions are shown in the same color. (c) Brillouin zone with TRIMs (orange), high-symmetry points (purple), and k-path (red) that is used for the band structures with parameter sets A and B in (e) and (h), respectively. The parameters are summarized in Table 1. (d) Symmetry-enforced Chern numbers on planes of constant $k_z$. Berry curvature ${\mathrm{\Omega }}_2^{xy}$ [(f),(i)] and quantum metric $g_2^{xx}$ [(g),(j)] at $k_z=0$ corresponding to the models to the left.

Figure 2

Four-band model for SG $P3$. [(a),(b)] Top and side view of the trigonal lattice with nearest- and next-nearest-neighbor hopping terms, where symmetry-related directions are shown in the same color. [(e),(g)] Resulting band structures with the corresponding (orbital-resolved) density of states (DOS) for parameter sets C and D, see Table 1. [(c),(d)] Quantum metric $g^{xx}$ for band 2 and bands (12) with parameter set C. [(f),(h)] Berry curvature ${\mathrm{\Omega }}^{xy}$ for bands 2 and (12) with parameter set C.

Figure 3

Visualization of the compact localized state (CLS) of the singular flat band and the destructive interference for triangles (a) and lines (b). The orbital weights (green) and hoppings (red, purple) involve $w=(1+i\sqrt{3})/2$.

Figure 4

Berry curvature ${\mathrm{\Omega }}_{-}^{xy}$ [(a),(b)] and quantum metric $g_{-}^{xx}$ [(c),(d)] at $k_z=\pi /2$ for the parameter sets A and B of the two-band model.

Figure 5

Band structure fulfilling the flat-band conditions.