| We study a topological classification of insulators and superconductors in the presence of non-spatial discrete symmetries in the Altland-Zirnbauer classification and spatial symmetries in any spatial dimension. We provide a unified method, the construction of bulk Dirac Hamiltonians in minimal matrix dimension, to classify topological phases. Using this method, we first reproduce the classification of non-spatial symmetric topological insulators and superconductors in the Altland-Zirnbauer symmetry classes. Such non-trivial topological insulators and superconductors possess protected gapless modes in each boundary. Furthermore, we extend the classification to spatial symmetric systems, such as re ection symmetry. When a specific boundary that does not break system's symmetries is introduced in these non-trivial topological systems, gapless modes are present at this boundary. |
