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In higher-order topological insulators, bulk and surface electronic statesare gapped, while there appear gapless hinge states protected by spatialsymmetry. Here we show by ab initio calculations that the La apatite electrideis a higher-order topological crystalline insulator. It is a one-dimensionalelectride, in which the one-dimensional interstitial hollows along the $c$ axissupport anionic electrons, and the electronic states in these one-dimensionalchannels are well approximated by the one-dimensional Su-Schrieffer-Heegermodel. When the crystal is cleaved into a hexagonal prism, the 120$^\circ$hinges support gapless hinge states, with their filling quantized to be 2/3.This quantization of the filling comes from a topological origin. We find thatthe quantized value of the filling depends on the fundamental blocks thatconstitute the crystal. The apatite consists of the triangular blocks, which iscrucial for giving nontrivial fractional charge at the hinge.

Higher-Order Topological Crystalline Insulating Phase and Quantized Hinge Charge in Topological Electride Apatite