Simple counterexample for the $\mathcal{Z}_2$ classification of topological insulators based on the bulk-boundary correspondence

The so-called $\mathcal{Z}_2$ classification of topological insulators has been previously proposed on the basis of the bulk-boundary correspondence. This classification is commonly accepted and involves the following statements [L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007)]: (i) nontrivial $\mathcal{Z}_2$ invariants imply the existence of gapless surface states, (ii) the $\mathcal{Z}_2$ invariants can be deduced from the topological structure of the Bloch wave functions of the bulk crystal in the Brillouin zone. In this work, a simple counterexample has been given for the $\mathcal{Z}_2$ classification. It has been shown that both topologically stable and topologically unstable surface states can exist on surfaces, at the same bulk, at the same space symmetry of a semi-infinite crystal, and, correspondingly, at a trivial value of the $\mathcal{Z}_2$ invariant (at the trivial class of equivalence of the bulk Hamiltonian) for the $3\mathrm{D}\to 2\mathrm{D}$ system. Furthermore, topologically stable surface states can exist at both trivial (Bi(111) surface) and nontrivial (Sb(111) surface) values of the bulk $\mathcal{Z}_2$ invariant. In view of these facts, the statement that the $\mathcal{Z}_2$ classification based on the bulk-boundary correspondence is responsible for the appearance and topological stability of surface states is doubtful.