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

TL;DR

This paper establishes a mathematical framework linking glide reflection symmetries in certain topological phases to boundary zero modes via a mod 2 index theorem, expanding understanding of bulk-edge correspondence in unorientable systems.

Contribution

It proves a mod 2 twisted Toeplitz index theorem that connects bulk topological phases with boundary zero modes in glide-symmetric systems.

Findings

Established a bulk-edge correspondence for glide reflection symmetric phases.

Proved a mod 2 twisted Toeplitz index theorem.

Demonstrated the existence of exotic topological zero modes along glide boundaries.

Abstract

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