Wigner transformation, momentum space topology, and anomalous transport

TL;DR

This paper uses the Wigner transform and derivative expansion to analyze anomalous transport phenomena like the quantum Hall and chiral magnetic effects, linking them to momentum space topology and clarifying their presence in various materials.

Contribution

It provides a systematic analysis connecting topological invariants to anomalous currents and clarifies the conditions under which the chiral magnetic effect occurs or is absent.

Findings

Reproduces the conventional Hall conductivity in 2+1 D.

Explains the anomalous quantum Hall effect in topological insulators and Weyl semimetals.

Shows the absence of equilibrium chiral magnetic effect in certain solids and regularized quantum field theories.

Abstract

Using derivative expansion applied to the Wigner transform of the two - point Green function we analyse the anomalous quantum Hall effect (AQHE), and the chiral magnetic effect (CME). The corresponding currents are proportional to the momentum space topological invariants. We reproduce the conventional expression for the Hall conductivity in $2+1$ D. In $3+1$ D our analysis allows to explain systematically the AQHE in topological insulators and Weyl semimetals. At the same time using this method it may be proved, that the equilibrium CME is absent in the wide class of solids, as well as in the properly regularized relativistic quantum field theory.