Chiralities of nodal points along high-symmetry lines with screw rotation symmetry

Abstract

Screw rotations in nonsymmorphic space group symmetries induce the presence of hourglass and accordion-shape band structures along screw invariant lines whenever spin-orbit coupling is non-negligible. These structures induce topological enforced Weyl points on the band intersections. In this work we show that the chirality of each Weyl point is related to the representations of the cyclic group on the bands that form the intersection. To achieve this, we calculate the Picard group of isomorphism classes of complex line bundles over the two-dimensional sphere with cyclic group action, and we show how the chirality (Chern number) relates to the eigenvalues of the rotation action on the rotation invariant points. Then we write an explicit Hamiltonian endowed with a cyclic action whose eigenfunctions restricted to a sphere realize the equivariant line bundles described before. As a consequence of this relation, we determine the chiralities of the nodal points appearing on the hourglass and accordion shape structures on screw invariant lines of the nonsymmorphic materials ${\mathrm{PI}}_{3}$ (SG: $\mathrm{P}{6}_{3}$), ${\mathrm{Pd}}_{3}\mathrm{N}$ (SG: $\mathrm{P}{6}_{3}22$), ${\mathrm{AgF}}_{3}$ (SG: $\mathrm{P}{6}_{1}22$), and ${\mathrm{AuF}}_{3}$ (SG: $\mathrm{P}{6}_{1}22$), and we corroborate these results with the Berry curvature and symmetry eigenvalues calculations for the electronic wave function.