Time-reversal generalization of the Hopf insulator

The Hopf insulator is one of few examples of a topological insulator that can be a trivial insulator by adding trivial bands to either the valence band or the conduction band., i.e., delicate topological insulator. Here, we propose another example of the delicate topological insulator, which is obtained by applying Hopf map to time-reversal invariant systems. We show how to build these time-reversal generalized Hopf insulators mathematically and classify them using integer-valued Hopf invariants. The bulk invariants can be related to the non-trivial topological signature at the surface. Using lattice models, we demonstrate the relation between the bulk topology and rotation symmetry of the system.