On Fractional Quantum Hall Effect (FQHE): A Chern-Simons and nonequilibrium quantum transport Weyl transform approach

TL;DR

This paper offers a macroscopic phase-space explanation of the fractional quantum Hall effect using nonequilibrium quantum transport and Chern-Simons theory, highlighting the topological invariants and correcting common assumptions about the hierarchy of fractions.

Contribution

It introduces a simplified, phase-space based approach to FQHE employing Weyl transform formalism and Chern-Simons gauge theory, clarifying the scaling hierarchy and topological nature.

Findings

Derived the k-factor scaling hierarchy in Chern-Simons gauge theory.

Identified the prime number condition for the fundamental scaling hierarchy.

Challenged the conventional hierarchy of denominators in FQHE fractions.

Abstract

We give a simple macroscopic phase-space explanation of fractional quantum Hall effect (FQHE), in a fashion reminiscent of the Landau-Ginsburg macroscopic symmetry breaking analyses. This is in contrast to the more complicated microscopic wavefunction approaches. Here, we employ a nonequilibrium quantum transport in the lattice Weyl transform formalism. This is coupled with the Maxwell Chern-Simons gauge theory for defining fractional filling of Landau levels. Flux attachment concept is inherent in fully occupied and as well as in partially occupied Landau levels. We derived the k-factor scaling hierarchy in Chern-Simons gauge theory, as the scaling hierarchy of the magnetic field or magnetic flux in FQHE. This is crucial in our simple explanation of FQHE as a topological invariant in phase space. For the fundamental scaling hierarchy, the integer k must be a prime number, and for fractions both the numerator and denominator of k must also be prime numbers. The assumption in the literature that a hierarchy of denominators of v = 1 k is given by the expression, (2n + 1), is wrong. Furthermore, even denominators for v cannot belong to fundamental scaling hierarchy and is often absent or less resolved in the experiments.