Stability of surface states of general weak Z2 topological insulators and superconductors

We study the stability against disorder of surface states of weak Z2 topological insulators (superconductors) which are stacks of strong Z2 topological insulators (superconductors), considering representative Dirac Hamiltonians in the Altland-Zirnbauer symmetry classes in various spatial dimensions. We show that, in the absence of disorder, surface Dirac fermions of weak Z2 topological insulators (superconductors) can be gapped out by a Dirac mass term which couples surface Dirac cones and leads to breaking of a translation symmetry (dimerization). The dimerization mass is a unique Dirac mass term in the surface Dirac Hamiltonian, and the two dimerized gapped phases which differ in the sign of the Dirac mass are distinguished by a Z2 index. In other words the dimerized surfaces can be regarded as a strong Z2 topological insulator (superconductor). We argue that the surface states are not localized by disorder when the ensemble average of the Dirac mass term vanishes.