Quantum Hall Effect and Dyson-Swinger Equation

TL;DR

This paper explores the Quantum Hall Effect using electron Green functions in planar electrodynamics, deriving quantized conductance and investigating fractional cases through Dyson-Swinger equations and scale approximation.

Contribution

It introduces a Green function approach to describe both integer and fractional Quantum Hall Effects, employing Dyson-Swinger equations and scale approximation for interacting systems.

Findings

Quantized Hall conductivity derived from free electron propagator.

Connection between fractional Quantum Hall Effect and interacting Green functions.

Application of Dyson-Swinger equations in a gauge-invariant framework.

Abstract

In this paper we make attempt to obtain a description of the Quantum Hall Effect (both integer and fractional) by means of electron's Green functions of three-dimensional (planar) electrodynamics. We show that expression for the free electron propagator yields an integer number for the second Chern-Simons term, that corresponds to the quantized Hall conductivity in the approximation of non-interacting particles for integer filling factors, when there exists a gap for all excitations in the system. Then we try to check correspondence between fractional case and "interacting" Green functions, so it requires taking into consideration "full-fledged" propagators, including interactions. We are going to obtain them from Dyson-Swinger equations. We attempt to reach out from the perturbation theory regime using a specific method, called scale approximation. Our solutions are found in general gauge.