Hall conductivity as topological invariant in phase space

TL;DR

This paper extends the topological invariant description of quantum Hall conductivity to systems with non-uniform magnetic fields using phase space methods, applicable to complex and interacting models.

Contribution

It introduces a phase space topological invariant based on Weyl symbols for quantum Hall systems with non-uniform magnetic fields, generalizing previous momentum space formulations.

Findings

Topological invariant in phase space for non-uniform magnetic fields

Applicable to interacting and complex tight-binding models

Generalizes TKNN invariant to broader conditions

Abstract

It is well known that the quantum Hall conductivity in the presence of constant magnetic field is expressed through the topological TKNN invariant. The same invariant is responsible for the intrinsic anomalous quantum Hall effect (AQHE), which, in addition, may be represented as one in momentum space composed of the two point Green's functions. We propose the generalization of this expression to the QHE in the presence of non-uniform magnetic field. The proposed expression is the topological invariant in phase space composed of the Weyl symbols of the two-point Green's function. It is applicable to a wide range of non-uniform tight-binding models, including the interacting ones.