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Stability of surface states of weak $\mathbb{Z}_2$ topological insulators and superconductors

We study the stability against disorder of surface states of weak $\mathbb{Z}_2$ topological insulators (superconductors) which are stacks of strong $\mathbb{Z}_2$ 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 $\mathbb{Z}_2$ 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 $\mathbb{Z}_2$ index. In other words the dimerized surfaces can be regarded as a strong $\mathbb{Z}_2$ topological insulator (superconductor). We argue that the surface states are not localized by disorder when the ensemble average of the Dirac mass term vanishes.