Topological crystalline materials: General formulation, module structure, and wallpaper groups

Topological crystalline materials are emergent topological phases due to crystalline space group symmetry. They are either gapful or gapless in the bulk, while hosting topological states at the boundary. Here, the authors define topological crystalline materials rigorously on the basis of a mathematical theory, known as twisted equivariant K-theory. Abstract mathematical ideas, such as the Mayer-Vietoris sequence and module structure, are explained in terms of band theory. The formulation is applicable to bulk gapful topological crystalline insulators/superconductors and their gapless boundary and defect states as well as to bulk gapless topological materials, such as Weyl and Dirac semimetals or nodal superconductors. The authors present a complete classification of topological crystalline surface states and band insulators protected by 17 wallpaper groups in the absence of time-reversal invariance, which may support topological states beyond simple Dirac fermions.