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 C', a generalized particle-hole symmetry that gives the Hopf insulator a Z(2) classification. The 4d analog of the Hopf insulator with symmetry C' has the same Z(2) classification. The minimal models for the 3d and 4d Hopf insulator can be heuristically viewed as "Chern insulator x S-1" and "Chern insulator x T-2" respectively. We also discuss the relation between the Hopf insulator and the Weyl semimetals, which points toward a direction for its possible experimental realization.