Geometric and nongeometric contributions to the surface anomalous Hall conductivity

TL;DR

This paper develops a formalism to calculate the full surface anomalous Hall conductivity in magnetoelectric insulators, distinguishing geometric and nongeometric contributions, and clarifies how surface preparation affects quantized surface currents.

Contribution

It introduces a comprehensive method to compute surface anomalous Hall conductivity, separating geometric and nongeometric parts, and addresses ambiguities related to surface conditions.

Findings

Berry-curvature term related to bulk anomalous Hall conductivity

Quantized changes in surface conductivity due to surface Hamiltonian adjustments

Existence of a nongeometric, surface-preparation independent component

Abstract

A static electric field generates circulating currents at the surfaces of a magnetoelectric insulator. The anomalous Hall part of the surface conductivity tensor describing such bound currents can change by multiples of $e^2/h$ depending on the insulating surface preparation, and a bulk calculation does not fix its quantized part. To resolve this ambiguity, we develop a formalism for calculating the full surface anomalous Hall conductivity in a slab geometry. We identify a Berry-curvature term, closely related to the expression for the bulk anomalous Hall conductivity, whose value can change by quantized amounts by adjusting the surface Hamiltonian. In addition, the surface anomalous Hall conductivity contains a nongeometric part that does not depend on the surface preparation.