Locality of the anomalous Hall conductivity

Abstract

The geometrical intrinsic contribution to the anomalous Hall conductivity (AHC) of a metal is commonly expressed as a reciprocal-space integral: as such, it only addresses unbounded and macroscopically homogeneous samples. Here we show that the geometrical AHC has an equivalent expression as a local property. We define a ``geometrical marker'' which actually probes the AHC in inhomogeneous systems (e.g., heterojunctions), as well as in bounded samples. The marker may even include extrinsic contributions of geometrical nature.