Teleportation of Berry curvature on the surface of a Hopf insulator

TL;DR

This paper introduces a novel form of topological surface states in Hopf insulators where Berry curvature is spatially separated and exhibits a bulk-to-boundary flow unrelated to anomaly inflow, revealing new topological phenomena.

Contribution

It demonstrates that equal-energy bands with opposite Chern numbers can be spatially separated on different surfaces of a Hopf insulator, with a novel bulk-to-boundary Berry curvature flow and a generalized Thouless pump.

Findings

Surface Berry curvature corresponds to bulk homotopy invariant.

Non-chiral, Schrödinger-type surface modes arise from a generalized Weyl equation.

A lattice regularization reveals a generalized Thouless pump with reversed charge flow.

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.