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

Abstract

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.