# Locality of the anomalous Hall conductivity

## TL;DR

This paper introduces a local geometrical marker for the anomalous Hall conductivity, enabling analysis of inhomogeneous and bounded systems, extending beyond the traditional reciprocal-space integral approach.

## Contribution

It presents a novel local expression for the geometrical AHC, allowing for inhomogeneous and bounded sample analysis, including extrinsic geometrical contributions.

## Key findings

- The geometrical AHC can be expressed as a local property.
- The local marker probes AHC in inhomogeneous and bounded systems.
- Extrinsic geometrical contributions can be included in the marker.

## 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.