Topological Hopf-Chern Insulators and the Hopf Superconductor

We introduce new three-dimensional topological phases of two-band models using the Pontryagin-Thom construction. In symmetry class A these are the infinitely many Hopf-Chern topological insulators, which are hybrids of layered Chern insulators and the Hopf insulator. In symmetry class C, there is a $\mathbb{Z}_2$-classification with the non-trivial topological phase, the Hopf superconductor, being realized by a construction that doubles the usual Hopf insulator in momentum space. For these new topological phases we investigate the energy spectrum in the presence of a boundary and find gapless surface modes, confirming the validity of the bulk-boundary correspondence.