Classification of topological crystalline insulators based on representation theory

Topological crystalline insulators define a new class of topological insulator phases with gapless surface states protected by crystalline symmetries. In this work, we present a general theory to classify topological crystalline insulator phases based on the representation theory of space groups. Our approach is to directly identify possible nontrivial surface states in a semi-infinite system with a specific surface, of which the symmetry property can be described by 17 two-dimensional space groups. We reproduce the existing results of topological crystalline insulators, such as mirror Chern insulators in the pm or pmm groups, Cnv topological insulators in the p4m,p31m, and p6m groups, and topological nonsymmorphic crystalline insulators in the pg and pmg groups. Aside from these existing results, we also obtain the following results: (1) there are two integer mirror Chern numbers (Z2) in the pm group but only one (Z) in the cm or p3m1 group for both the spinless and spinful cases; (2) for the pmm (cmm) groups, there is no topological classification in the spinless case but Z4 (Z2) classifications in the spinful case; (3) we show how topological crystalline insulator phase in the pg group is related to that in the pm group; (4) we identify topological classification of the p4m,p31m, and p6m for the spinful case; (5) we find topological nonsymmorphic crystalline insulators also existing in pgg and p4g groups, which exhibit new features compared to those in pg and pmg groups. We emphasize the importance of the irreducible representations for the states at some specific high-symmetry momenta in the classification of topological crystalline phases. Our theory can serve as a guide for the search of topological crystalline insulator phases in realistic materials.