Infinite Symmetry in the Fractional Quantum Hall Effect

TL;DR

This paper extends the symmetry framework of the quantum Hall effect to fractional cases, linking edge excitations to a ${ m W}_{1+ ext{infinity}}$ algebra, and explores implications for wave functions and composite fermion theory.

Contribution

It explicitly constructs a ${ m W}_{1+ ext{infinity}}$ symmetry for the fractional quantum Hall effect and relates it to edge excitations and wave functions, generalizing previous integer case results.

Findings

${ m W}_{1+ ext{infinity}}$ symmetry underlies fractional quantum Hall edge excitations.

Wave functions of edge states are described as area-preserving deformations.

The approach applies to more general wave functions beyond Laughlin states.

Abstract

We have generalized recent results of Cappelli, Trugenberger and Zemba on the integer quantum Hall effect constructing explicitly a ${\cal W}_{1+\infty}$ for the fractional quantum Hall effect such that the negative modes annihilate the Laughlin wave functions. This generalization has a nice interpretation in Jain's composite fermion theory. Furthermore, for these models we have calculated the wave functions of the edge excitations viewing them as area preserving deformations of an incompressible quantum droplet, and have shown that the ${\cal W}_{1+\infty}$ is the underlying symmetry of the edge excitations in the fractional quantum Hall effect. Finally, we have applied this method to more general wave functions.