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We formulate topological crystalline materials on the basis of the twistedequivariant $K$-theory. Basic ideas of the twisted equivariant $K$-theory isexplained with application to topological phases protected by crystallinesymmetries in mind, and systematic methods of topological classification forcrystalline materials are presented. Our formulation is applicable to bulkgapful topological crystalline insulators/superconductors and their gaplessboundary and defect states, as well as bulk gapless topological materials suchas Weyl and Dirac semimetals, and nodal superconductors. As an application ofour formulation, we present a complete classification of topologicalcrystalline surface states, in the absence of time-reversal invariance. Theclassification works for gapless surface states of three-dimensionalinsulators, as well as full gapped two-dimensional insulators. Such surfacestates and two-dimensional insulators are classified in a unified way by 17wallpaper groups, together with the presence or the absence of (sublattice)chiral symmetry. We identify the topological numbers and their representationsunder the wallpaper group operation. We also exemplify the usefulness of ourformulation in the classification of bulk gapless phases. We present a newclass of Weyl semimetals and Weyl superconductors that are topologicallyprotected by inversion symmetry.

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