Teleportation of Berry curvature on the surface of a Hopf insulator

A Alexandradinata,Alexey A Soluyanov,Aleksandra Nelson

doi:10.1103/physrevb.103.045107

A Alexandradinata, Alexey A Soluyanov + Show 1 more

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https://doi.org/10.1103/physrevb.103.045107

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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.

Highlights

The paradigm of topological insulators asserts that an energy gap separates conduction and valence bands with opposite topological invariants
We propose that equal-energy bands with opposite Chern invariants can be spatially separated, onto opposite facets of a finite crystalline Hopf insulator
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

Summary

FACETED CHERN NUMBER AS A TOPOLOGICAL INVARIANT

We propose the faceted Chern number Cf as a topological invariant for any three-spatial-dimensional Hamiltonian that is a two-by-two matrix at each k ∈ BZ, with a spectral gap (at each k) separating a low-energy and high-energy bulk band with trivial first Chern class. For such category of Hamiltonians which include the Hopf insulator, we consider a surface termination whose reduced Brillouin zone (rBZ) is a 2D cut of the BZ.

BULK-TO-BOUNDARY FLOW OF BERRY CURVATURE

BULK-BOUNDARY CORRESPONDENCE

OUTLOOK