Simple counterexample for the ℒ2 classification of topological insulators based on the bulk–boundary correspondence

The so-called ℒ2 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 ℒ2 invariants imply the existence of gapless surface states, (ii) the ℒ2 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 ℒ2 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 ℒ2 invariant (at the trivial class of equivalence of the bulk Hamiltonian) for the 3D → 2D system. Furthermore, topologically stable surface states can exist at both trivial (Bi(111) surface) and nontrivial (Sb(111) surface) values of the bulk ℒ2 invariant. In view of these facts, the statement that the ℒ2 classification based on the bulk–boundary correspondence is responsible for the appearance and topological stability of surface states is doubtful.