Topological classification with additional symmetries from Clifford algebras

Understand

We classify topological insulators and superconductors in the presence of additional symmetries such as reflection or mirror symmetries.

• For each member of the 10 Altland-Zirnbauer symmetry classes, we have a Clifford algebra defined by operators of the generic (time-reversal, particle-hole, or chiral) symmetries and additional symmetries, together with gamma matrices in Dirac Hamiltonians representing topological insulators and superconductors.
• Following Kitaev's approach, we classify gapped phases of non-interacting fermions under additional symmetries by examining all possible distinct Dirac mass terms which can be added to the set of generators of the Clifford algebra.
• We find that imposing additional symmetries in effect changes symmetry classes and causes shifts in the periodic table of topological insulators and superconductors.