Teleportation of Berry curvature on the surface of a Hopf insulator

The paradigm of topological insulators asserts that an energy gap separates conduction and valence bands with opposite topological invariants. Here, we propose that equal-energy bands with opposite Chern invariants can be spatially separated, onto opposite facets of a finite crystalline Hopf insulator. On a single facet, the number of Berry-curvature quanta is in one-to-one correspondence with the bulk homotopy invariant of the Hopf insulator;...