Nontrivial topological effects on the surface of strong and weak topological insulators and superconductors

We investigate states on the surface of strong or weak topological insulators and superconductors that have been gapped by a magnetic material or by a charge density wave. The surface of a strong 3D topological insulator gapped by a magnetic material is well known to possess a half quantum Hall effect. Furthermore, a recent paper (arXiv:1110.3420) showed that the surface of a weak 3D topological insulator gapped by charge density wave has a half quantum spin Hall effect. We generalize these results to all classes of topological insulator and superconductors. We find that if, in d-1 dimension and that symmetry class, there is a $Z$ or $Z_2$ topological invariant, then the resulting surface state may have a nontrivial topological phase.