Teleportation of Berry curvature on the surface of a Hopf insulator

Abstract

The existing paradigm for topological insulators asserts that an energy gap separates conduction and valence bands with opposite topological invariants. Here, we propose that \textit{equal}-energy bands with opposite Chern invariants can be \textit{spatially} separated -- onto opposite facets of a finite crystalline Hopf insulator. On a single facet, the number of curvature quanta is in one-to-one correspondence with the bulk homotopy invariant of the Hopf insulator -- this originates from a novel bulk-to-boundary flow of Berry curvature which is \textit{not} a type of Callan-Harvey anomaly inflow. In the continuum perspective, such nontrivial surface states arise as \textit{non}-chiral, Schr\"odinger-type modes on the domain wall of a generalized Weyl equation -- describing a pair of opposite-chirality Weyl fermions acting as a \textit{dipolar} source of Berry curvature. A rotation-invariant lattice regularization of the generalized Weyl equation manifests a generalized Thouless pump -- which translates charge by one lattice period over \textit{half} an adiabatic cycle, but reverses the charge flow over the next half.