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Bulk invariants and topological response in insulators and superconductors with nonsymmorphic symmetries

In this work we consider whether nonsymmorphic symmetries such as a glide plane can protect the existence of topological crystalline insulators and superconductors in three dimensions. In analogy to time-reversal symmetric insulators, we show that the presence of a glide gives rise to a quantized magnetoelectric polarizability, which we compute explicitly through the Chern-Simons 3-form of the bulk wave functions for a glide symmetric model. Our approach provides a measurable property for this insulator and naturally explains the connection with mirror symmetry protected insulators and the recently proposed $Z_2$ index for this phase. More generally, we prove that the magnetoelectric polarizability becomes quantized with any orientation-reversing space group symmetry. We also construct analogous examples of glide protected topological crystalline superconductors in classes D and C and discuss how bulk invariants are related to quantized surface thermal-Hall and spin-Hall responses.