Berry-phase description of topological crystalline insulators

We study a class of translational-invariant insulators with discrete rotational symmetry. These insulators have no spin-orbit coupling, and in some cases have no time-reversal symmetry either; i.e., the relevant symmetries are purely crystalline. Nevertheless, topological phases exist which are distinguished by their robust surface modes. Like many well-known topological phases, their band topology is unveiled by the crystalline analog of Berry phases, i.e., parallel transport across certain noncontractible loops in the Brillouin zone. We also identify certain topological phases without any robust surface modes; they are uniquely distinguished by parallel transport along bent loops, whose shapes are determined by the symmetry group. Our findings have experimental implications in cold-atom systems, where the crystalline Berry phase has been directly measured.