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Topological quantization of Fractional Quantum Hall conductivity

We consider the quantum Hall effect (QHE) in a system of interacting electrons. Our formalism is valid for systems in the presence of an external magnetic field, as well as for systems with a nontrivial band topology. That is, the expressions for the conductivity derived are valid for both the ordinary QHE and for the intrinsic anomalous QHE. The expression for the conductivity applies to external fields that may vary in an arbitrary way, and takes into account disorder. It is assumed that the ground state of the system is degenerate. We represent the QHE conductivity as $\frac{e^2}{h} \times \frac{\cal N}{K}$, where $K$ is the degeneracy of the ground state, while $\cal N$ is the topological invariant composed of the Wigner - transformed multi - leg Green functions. $\cal N$ takes discrete values, which gives rise to quantization of the fractional QHE conductivity.