Mean Field Description of the Fractional Quantum Hall Effect Near $\nu=1/(2k+1)$

TL;DR

This paper analyzes the mean field solutions of the Chern--Simons Landau--Ginsberg model for the fractional quantum Hall effect near specific filling fractions, showing the stability of solutions and vortex configurations over certain parameter ranges.

Contribution

It demonstrates the robustness of mean field solutions and vortex configurations in the CSLG model near fractional filling fractions, including the conditions for vortex condensates.

Findings

Mean field solutions are stable over a finite range of chemical potential deviations.

Vortex configurations do not depend on chemical potential within this range.

Vortex or antivortex condensates occur when chemical potential exceeds critical values.

Abstract

The nature of Mean Field Solutions to the Equations of Motion of the Chern--Simons Landau--Ginsberg (CSLG) description of the Fractional Quantum Hall Effect (FQHE) is studied. Beginning with the conventional description of this model at some chemical potential $\mu_0$ and magnetic field $B$ corresponding to a ``special'' filling fraction $\nu=2\pi\rho/eB=1/n$ ($n=1,3,5\cdot \cdot\cdot$) we show that a deviation of $\mu$ in a finite range around $\mu_0$ does not change the Mean Field solution and thus the mean density of particles in the model. This result holds not only for the lowest energy Mean Field solution but for the vortex excitations as well. The vortex configurations do not depend on $\mu$ in a finite range about $\mu_0$ in this model. However when $\mu-\mu_0 < \mu_{cr}^-$ (or $\mu-\mu_0>\mu_{cr}^+$) the lowest energy Mean Field solution describes a condensate of vortices (or antivortices). We give numerical examples of vortex and antivortex configurations and discuss the range of $\mu$ and $\nu$ over which the system of vortices is dilute.