Anomalous currents on closed surfaces: extended proximity, partial quantization and qubits

Motivated by the surfaces of topological insulators, the Dirac anomaly's discontinuous dependence on the sign of the mass, m/|m|, is investigated on closed topologies when the mass terms are weak or only partially cover the surface. It is found that, unlike the massive Dirac theory on an infinite plane, there is a smoothly decreasing current when the mass region is not infinite; also, a massive finite region fails to exhibit a Hall current edge - exerting an extended proximity effect, which can, however, be uniformly small - and oppositely orientated Hall phases are fully quantized while accompanied by diffuse chiral modes. Examples are computed using Dirac energy eigenstates on a flat torus (genus one topology) and a closed cap cylinder (genus zero topology) for various mass-term geometries. Finally, from the resulting properties of the surface spectra, a potential application for a flux-charge qubit is presented.