Composite Fermion Approach to the Quantum Hall Hierarchy: When it Works and Why

TL;DR

This paper analyzes the composite fermion approach to the quantum Hall hierarchy, explaining its qualitative success and identifying the specific subset of states it effectively describes, based on angular momentum considerations.

Contribution

It provides a theoretical justification for the composite fermion picture by linking it to the selection of low angular momentum multiplets with small pair coefficients.

Findings

The MFCF picture selects a subset of angular momentum multiplets.

Selected states have small or vanishing pair coefficients for small relative angular momentum.

This subset corresponds to the lowest energy sector of the spectrum.

Abstract

The mean field composite Fermion (MFCF) picture has been qualitatively successful when applied to electrons (or holes) in the lowest Landau level. Because the energy scales associated with Coulomb interactions and with Chern-Simons gauge field interactions are different, there is no rigorous justification of the qualitative success of the MFCF picture. Here we show that what the MFCF picture does is to select from all the allowed angular momentum (L) multiplets of N electrons on a sphere, a subset with smaller values of L. For this subset, the coefficients of fractional parentage for pair states with small relative angular momentum $R$ (and therefore large repulsion) either vanish or they are small. This set of states forms the lowest energy sector of the spectrum.