# Topological invariant in terms of the Green functions for the Quantum Hall Effect in the presence of varying magnetic field

## TL;DR

This paper extends the topological invariant approach to the Quantum Hall Effect, demonstrating that the electric current in systems with varying magnetic fields can be expressed through a phase space topological invariant involving Green functions.

## Contribution

It introduces a phase space topological invariant based on Wigner transformed Green functions for the Quantum Hall Effect under arbitrary magnetic field variations.

## Key findings

- Electric current proportional to phase space topological invariant.
- Green functions depend on both coordinates and momenta.
- Extension of topological invariant formalism to non-uniform magnetic fields.

## Abstract

Recently the Wigner - Weyl formalism has been applied to the lattice models of solid state physics and to the lattice regularized quantum field theory. This allows to demonstrate that the electric current of intrinsic Anomalous Quantum Hall effect is expressed through the momentum space topological invariant composed of the Green functions both for the two - and the three - dimensional systems. Here we extend this consideration to the case of the Quantum Hall Effect existing in the presence of arbitrarily varying external magnetic field. The corresponding electric current appears to be proportional to the topological invariant in phase space composed of the Wigner transformed Green function that depends both on coordinates and momenta.