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

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