Crystalline Topological Insulators and Semimetals with $C_{nv}$ Symmetry

We explore a class of 3D materials with $C_{nv}$ symmetry. For $n =3,4$ and $6$, we find the first-known 3D topological insulators with robust surface modes, but $without$ spin-orbit coupling, and $not$ $needing$ time-reversal symmetry; the relevant symmetries are purely crystalline. To describe these $C_{nv}$ systems, we introduce the notion of a mirror chirality: an integer invariant which characterizes half-mirror-planes in the 3D Brillouin zone. In the evolution between two gapped phases with distinct mirror chiralities, we find that the intermediate gapless phase is a Weyl semimetal. Applications are discussed in the context of photonic crystals.