Weyl points and spin-orbit coupling in copper-substituted lead phosphate apatite

We study the impact of spin-orbit coupling on the topological band properties of copper-substituted lead phosphate apatite using a combination of group-theoretical analysis and full-relativistic density-functional theory calculations. We characterize Weyl points at time-reversal invariant momenta and find that a band inversion due to spin-orbit coupling leads to additional Weyl points close to the Fermi edge at general momenta. To determine the position of the altogether 66 Weyl points in the Brillouin zone, we develop an algorithm that follows a Berry-curvature-derived vector field to its monopole: the Weyl point. The emerging surface Fermi arcs and their spin polarization reveal avoided crossings and a Fermi loop detached from the Weyl points.