The $Z_2$ Classification of Dimensional Reduced Hopf Insulators

The Hopf insulators are characterized by a topological invariant called Hopf index which classifies maps from three-sphere to two-sphere, instead of a Chern number or a Chern parity. In contrast to topological insulator, the Hopf insulator is not protected by any kind of symmetry. By dimensional reduction, we argue that there exists a new type of $\mathbb{Z}_2$ index for 2D Hamiltonian with vanishing Chern number. Specific model Hamiltonian with this nontrivial $\mathbb{Z}_2$ index is constructed. We also numerically calculate the topological protected edge modes of this dimensional reduced Hopf insulator and show that they are consistent with the $\mathbb{Z}_2$ classification.