Influence of interactions on Integer Quantum Hall Effect

TL;DR

This paper demonstrates that the Hall conductivity in the Integer Quantum Hall Effect remains topologically invariant in the presence of interactions, extending the non-interacting Green function approach to interacting systems using Wigner-Weyl calculus.

Contribution

It extends the topological invariant formulation of IQHE to interacting systems by incorporating interactions into the Green function framework with Wigner-Weyl calculus.

Findings

Hall conductivity expressed via Green functions remains topologically invariant with interactions.

The proof is valid up to two loops and argued to hold to all orders in perturbation theory.

Wigner-Weyl calculus is used to incorporate interactions into the diagrammatic perturbation theory.

Abstract

Conductivity of Integer Quantum Hall Effect (IQHE) may be expressed as the topological invariant composed of the two - point Green function. Such a topological invariant is known both for the case of homogeneous systems with intrinsic Anomalous Quantum Hall Effect (AQHE) and for the case of IQHE in the inhomogeneous systems. In the latter case we may speak, for example, of the AQHE in the presence of elastic deformations and of the IQHE in presence of magnetic field. The topological invariant for the general case of inhomogeneous systems is expressed through the Wigner transformed Green functions and contains Moyal product. When it is reduced to the expression for the IQHE in the homogeneous systems the Moyal product is reduced to the ordinary one while the Wigner transformed Green function (defined in phase space) is reduced to the Green function in momentum space. Originally the mentioned above topological representation has been derived for the non - interacting systems. We demonstrate that in a wide range of different cases in the presence of interactions the Hall conductivity is given by the same expression, in which the noninteracting two - point Green function is substituted by the complete two - point Green function with the interactions taken into account. Several types of interactions are considered including the contact four - fermion interactions, Yukawa and Coulomb interactions. We present the complete proof of this statement up to the two loops, and argue that the similar result remains to all orders of perturbation theory. It is based on the incorporation of Wigner - Weyl calculus to the perturbation theory. We, therefore, formulate Feynmann rules of diagram technique in terms of the Wigner transformed propagators.