Classification of topological insulators and superconductors in the presence of reflection symmetry

TL;DR

This paper extends the classification of topological insulators and superconductors to include reflection symmetry, revealing how boundary modes depend on symmetry realization and maintaining the periodicity of the original AZ classification.

Contribution

It provides a complete classification framework for topological phases with reflection symmetry using Dirac Hamiltonians and explicit invariants, expanding the AZ scheme.

Findings

Classification depends on how reflection symmetry is realized.

Boundary modes are gapless and localized at boundaries invariant under reflection.

The classification retains the same dimensional periodicities as AZ classification.

Abstract

We discuss a topological classification of insulators and superconductors in the presence of both (non-spatial) discrete symmetries in the Altland-Zirnbauer classification and spatial reflection symmetry in any spatial dimensions. By using the structure of bulk Dirac Hamiltonians of minimal matrix dimensions and explicit constructions of topological invariants, we provide the complete classification, which still has the same dimensional periodicities with the original Altland-Zirnbauer classification. The classification of reflection-symmetry-protected topological insulators and superconductors depends crucially on the way reflection symmetry operation is realized. When a boundary is introduced, which is reflected into itself, these non-trivial topological insulators and superconductors support gapless modes localized at the boundary.