New invariants, complete classification and fast diagnosis for topological crystalline insulators

Topological crystalline insulators have nontrivial band topology jointly protected by onsite symmetries of time reversal and charge conservation, as well as by spatial symmetries of lattice. Different types of lattice symmetries correspond to different topological invariants in the bulk and distinct characteristic surface states. We discuss newly discovered Z₂ invariants in 3D protected by rotation (screw) symmetry, inversion symmetry and roto-reflection symmetry, respectively. Each of the new invariants in the bulk is related to the (d-2)-dimensional edge states on the boundary. With the new invariants, we establish a complete classification of topological crystalline insulators for each of the 230 space groups in 3D, along with explicit, microscopic constructions for each class. To search for corresponding materials for the topological states in these new classes, we have established quantitative mappings from symmetry eigenvalues of valence bands to topological invariants all space groups. Applying these mappings, we have developed a fully automated diagnosis algorithm for all nonmagnetic crystals. The algorithm is applied to diagnosing 39519 materials, in which 8056 are found topological.