Bulk-boundary correspondence in three-dimensional topological insulators

We discuss the relation between bulk topological invariants and the spectrum of surface states in three-dimensional noninteracting topological insulators. By studying particular models and considering general boundary conditions for the electron wave function on the crystal surface, we demonstrate that by use of experimental techniques that probe surface states, only strong topological and trivial insulating phases can be distinguished, the latter state being equivalent to a weak topological insulator. In a strong topological insulator, only the parity of the number of surface states, but not the number itself, is robust against time-reversal-invariant boundary perturbations. Our results suggest a Z(2) definition of the bulk-boundary correspondence, compatible with the Z(2) classification of topological insulators.