Symmetry Protected Topological Hopf Insulator and its Generalizations

We study a class of $3d$ and $4d$ topological insulators whose topological nature is characterized by the Hopf map and its generalizations. We identify the symmetry ${\mathcal{C}}^{\mathrm{\prime }}$, a generalized particle-hole symmetry that gives the Hopf insulator a ${\mathbb{Z}}_2$ classification. The $4d$ analogue of the Hopf insulator with symmetry ${\mathcal{C}}^{\mathrm{\prime }}$ has the same ${\mathbb{Z}}_2$ classification. The minimal models for the $3d$ and $4d$ Hopf insulator can be heuristically viewed as “Chern-insulator$⋊S^1$” and “Chern-insulator$⋊T^2$” respectively. We also discuss the relation between the Hopf insulator and the Weyl semimetals, which points the direction for its possible experimental realization.