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 as well, 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 non-contractible 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.