Higher-order topological insulators protected by inversion and rotoinversion symmetries

We provide the bulk topological invariant for chiral higher-order topological insulators in (i) fourfold rotoinversion invariant bulk crystals, and (ii) inversion-symmetric systems with or without an additional threefold rotation symmetry. These states of matter are characterized by a nontrivial Z₂ index, which we define in terms of symmetric hybrid Wannier functions of the filled bands, and can be readily calculated from the knowledge of the crystalline symmetry labels of the bulk band structure. The topological invariant determines the generic presence or absence of protected chiral gapless one-dimensional modes localized at the hinges between conventional gapped surfaces.