The Physical Significance of Singularities in the Chern--Simons Fermi Liquid Description of a Partially Filled Landau Level

TL;DR

This paper investigates the response of a half-filled Landau level using Fermi liquid theory for composite fermions, deriving key physical quantities and revealing a continuous excitation spectrum under certain conditions.

Contribution

It provides an exact expression for the composite fermion effective mass and conjectures the form of the Landau interaction function, advancing understanding of quantum Hall systems.

Findings

Derived an exact expression for the composite fermion effective mass.

Conjectured the form of the Landau interaction function.

Showed the excitation spectrum is continuous for certain parameters.

Abstract

We analyze the linear response of a half filled Landau level to long wavelength and low frequency driving forces, using Fermi liquid theory for composite fermions. This response is determined by the composite fermions quasi--particle effective mass, $m^*$, and quasi--particle Landau interaction function $f(\theta-\theta')$. Analyzing infra--red divergences of perturbation theory, we get an exact expression for $m^*$, and conjecture the form of the $f(\theta-\theta')$. We then conclude that in the limit of infinite cyclotron frequency, and small ${\bf q},\omega$, the composite fermion excitation spectrum is continuous for $0<\omega<\gamma \frac{e^2}{\epsilon h}q$, with $\gamma$ an unknown number. For fractional quantum Hall states near a half filled Landau level, we derive an exact expression for the energy gap.