| Summary: | We derive a novel topological expression for the Hall conductivity. To that degree 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. Properties related to symmetry and topology are revealed in the fractional quantization of the Hall conductivity. It is assumed that the ground state of the system is degenerate. We represent the QHE conductivity as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfrac><msup><mi>e</mi><mn>2</mn></msup><mi>h</mi></mfrac><mo>×</mo><mfrac><mi mathvariant="script">N</mi><mi>K</mi></mfrac></mrow></semantics></math></inline-formula>, where <i>K</i> is the degeneracy of the ground state, while <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">N</mi></semantics></math></inline-formula> is the topological invariant composed of the Wigner-transformed multi-leg Green functions, which takes discrete values.
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