Topological Crystalline Materials - General Formulation, Module Structure, and Wallpaper Groups -

Understand

We formulate topological crystalline materials on the basis of the twisted equivariant $K$-theory.

• Basic ideas of the twisted equivariant $K$-theory is explained with application to topological phases protected by crystalline symmetries in mind, and systematic methods of topological classification for crystalline materials are presented.
• Our formulation is applicable to bulk gapful topological crystalline insulators/superconductors and their gapless boundary and defect states, as well as bulk gapless topological materials such as Weyl and Dirac semimetals, and nodal superconductors.
• As an application of our formulation, we present a complete classification of topological crystalline surface states, in the absence of time-reversal invariance.