Abstract
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; this originates from a bulk-to-boundary flow of Berry curvature which is not a type of Callan-Harvey anomaly inflow. In the continuum perspective, such nontrivial boundary states arise as nonchiral, Schrödinger-type modes on the domain wall of a generalized Weyl equation, describing a pair of opposite-chirality Weyl fermions acting as a 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 half an adiabatic cycle, but reverses the charge flow over the next half.
- Received 8 January 2020
- Revised 17 August 2020
- Accepted 17 December 2020
DOI:https://doi.org/10.1103/PhysRevB.103.045107
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