Localized representation and surface signature of Hopf insulators.

The Hopf insulator is a 3D topological insulator that can’t be described in terms of a 10-fold classification. It also differs from fragile topological insulators. The main requirement for its existence is a two-rank Hamiltonian. In case when Hopf Hamiltonian has a trivial first Chern class, it obeys Z classification.

In our work, we address two questions. Firstly, we study the possibility for the Hopf insulator to possess localized Wannier representation and the existence of its topological obstructions. We propose that Wannier functions are exponentially localized and preserve the symmetries of the system. What obstructs the equivalence of the Hopf insulator to the atomic limit is the finiteness of the Wannier function width.

The second part of our work is related to the surface signature of the Hopf insulator. We claim that the surface states can be gapped out by surface potential without violating the symmetry or closing the bulk gap. However, surface states have a nontrivial first Chern number that equals to the bulk Hopf invariant. This bulk-edge correspondence can be explained by a new type of bulk-to-boundary Berry curvature flow originating from gauge-invariant magnetoelectric polarizability.