A Mean Field Theory for the Quantum Hall Liquid. II --- The Vortex Solution

TL;DR

This paper develops a mean field theory for the fractional quantum Hall state, introducing vortex solutions with fractional charge and angular momentum, and analyzes their stability and energy gaps under varying magnetic fields.

Contribution

It presents a novel bi-local mean field approach to vortex solutions in the fractional quantum Hall effect, including numerical self-consistent solutions and stability analysis.

Findings

Vortex solutions have fractional charge and angular momentum.

Finite energy gaps are observed at certain magnetic fields.

The uniform mean field becomes unstable below a critical magnetic field.

Abstract

In the Fractional Quantum Hall state, we introduce a bi-local mean field and get vortex mean field solutions. Rotational invariance is imposed and the solution is constructed by means of numerical self-consistent method. It is shown that vortex has a fractional charge, a fractional angular momentum and a magnetic field dependent energy. In $\nu=1/3$ state, we get finite energy gap at $B=10,15,20[T]$. We find that the gap vanishes at $B=5.5[T]$ and becomes negative below it. The uniform mean field becomes unstable toward vortex pair production below $B=5.5[T]$.