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We discuss a topological classification of insulators and superconductors inthe presence of both (non-spatial) discrete symmetries in the Altland-Zirnbauerclassification and spatial reflection symmetry in any spatial dimensions. Byusing the structure of bulk Dirac Hamiltonians of minimal matrix dimensions andexplicit constructions of topological invariants, we provide the completeclassification, which still has the same dimensional periodicities with theoriginal Altland-Zirnbauer classification. The classification ofreflection-symmetry-protected topological insulators and superconductorsdepends crucially on the way reflection symmetry operation is realized. When aboundary is introduced, which is reflected into itself, these non-trivialtopological insulators and superconductors support gapless modes localized atthe boundary.

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