Semiclassical theories of the anomalous Hall effect

TL;DR

This paper reviews the semiclassical theory of the anomalous Hall effect, emphasizing its gauge invariance, interpretability, and agreement with rigorous Green function methods, highlighting recent advances in the field.

Contribution

It provides a comprehensive review of the semiclassical approach to the anomalous Hall effect, including historical context and recent developments, demonstrating its effectiveness and consistency with rigorous methods.

Findings

Semiclassical theory aligns with Green function calculations.

Gauge invariant concepts simplify understanding of Hall current contributions.

Recent advances have clarified the semiclassical approach's validity.

Abstract

Recently, the semiclassical theory of the anomalous Hall effect induced by the Berry curvature in Bloch bands has been introduced. The theory operates only with gauge invariant concepts, that have a simple semiclassical interpretation and provides a clear distinction among various contributions to the Hall current. While the construction of such an approach to the anomalous Hall effect problem has been long sought, only the new semiclassical theory demonstrated the agreement with quantitative results of rigorous approaches based on the Green function techniques. The purpose of this work is to review the semiclassical approach including the early ideas and the recent achievements.