Topological crystalline insulators on nonprimitive lattice

In this presentation, we focus on the topological crystalline insulators (TCIs) protected by mirror symmetry and those by glide symmetry. Such a mirror-symmetric TCI is characterized by an integer topological number known as mirror Chern number (MCN) and a glide-symmetric TCI is characterized by a Z2 topological number. Meanwhile, in nonprimitive lattices, a half of the reciprocal vectors may not be invariant under the mirror and glide operations, and the formulae of the topological numbers should be altered. In our previous work, we derived a new formula of the Z2 topological number for glide-symmetric TCI on the nonprimitive lattices. In the present work, we give a new formula of the MCN on the nonprimitive lattices. We then describe how the topological numbers for mirror- and glide-symmetric systems change in the nonprimitive lattices, in terms of the trajectory of the Weyl nodes within the intermediate Weyl semimetal phase between two bulk-insulating phases. We also show how the expressions of these topological numbers are reduced by adding additional symmetries.