Surface states of topological crystalline insulators

We study topological crystalline insulator (TCI) phases hosting anomalous surface states. These include traditional surface states with two-dimensional Dirac dispersion as well as one-dimensional hinge modes characteristic of higher-order phases. Both of these can be captured by analyzing a general surface theory with multiple flavors of Dirac fermions and identifying global symmetry obstructions to achieving a fully gapped surface. This method is used to obtain a full classification of TCIs protected by time-reversal symmetry in the presence of strong spin-orbit coupling in the 230 space groups as well as TCIs protected by inversion in the ten different Altland-Zirnbauer symmetry classes. Furthermore, we establish a correspondence between the existence of surface states in a TCI and the existence of a stable (rather than fragile) obstruction to finding a basis of symmetric localized Wannier functions. Finally, we discuss possible extension of the results to magnetic space groups.