Nontrivial surface topological physics from strong and weak topological insulators and superconductors

We investigate states on the surface of strong and weak topological insulators and superconductors that have been gapped by a symmetry breaking term. The surface of a strong 3D topological insulator gapped by a magnetic material is well known to possess a half quantum Hall effect. Furthermore, it has been known that the surface of a weak 3D topological insulator gapped by a charge density wave exhibits a half quantum spin Hall effect. To generalize these results to all Altland-Zirnbauer symmetry classes of topological insulators and superconductors, we reproduce the classification table for the ten symmetry classes by using the representation theory of Clifford algebras and construct minimal-size Dirac Hamiltonians. We find that if the surface dimension and symmetry class possesses a $\mathbb{Z}$ or $\mathbb{Z}_2$ topological invariant, then the resulting surface state with a gapped symmetry breaking term may have a nontrivial topological phase.