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A 2-torsion topological phase exists for Hamiltonians symmetric under thewallpaper group with glide reflection symmetry, corresponding to theunorientable cycle of the Klein bottle fundamental domain. We prove a mod 2twisted Toeplitz index theorem, which implies a bulk-edge correspondencebetween this bulk phase and the exotic topological zero modes that it acquiresalong a boundary glide axis.

Crystallographic bulk-edge correspondence: glide reflections and twisted mod 2 indices