Higher-order topological insulators and superconductors protected by inversion symmetry

We study surface states of topological crystalline insulators and superconductors protected by inversion symmetry. These fall into the category of "higher-order" topological insulators and superconductors which possess surface states that propagate along one-dimensional curves (hinges) or are localized at some points (corners) on the surface. We provide a complete classification of inversion-protected higher-order topological insulators and superconductors in any spatial dimension for the 10 symmetry classes by means of a layer construction. We discuss possible physical realizations of such states starting with a time-reversal-invariant topological insulator (class AII) in three dimensions or a time-reversal-invariant topological superconductor (class DIII) in two or three dimensions. The former exhibits one-dimensional chiral or helical modes propagating along opposite edges, whereas the latter hosts Majorana zero modes localized to two opposite corners. Being protected by inversion, such states are not pinned to a specific pair of edges or corners, thus offering the possibility of controlling their location by applying inversion-symmetric perturbations such as magnetic field.