Chiral boson, $W_\infty$-coherent state and edge states in the quantum Hall effect

TL;DR

This paper develops a quantization framework for chiral bosons on the circle, constructs wave functions for quantum Hall edge states, and introduces $W_ abla$-coherent states to interpret the infinite-dimensional symmetries involved.

Contribution

It provides a consistent quantization method for chiral bosons with boundary parameters and introduces $W_ abla$-coherent states to describe edge states in quantum Hall systems.

Findings

Holomorphic wave functions explain neutral edge states at $ u=1$.

New wave functions for higher radial excitations of edge states.

Charged edge states described by wave functions with $|eta| eq 1$.

Abstract

We perform consistently the Gupta-Bleuler quantization combined with Dirac procedure for a chiral boson with the parameter ($\alpha$) on the circle, the boundary of the circular droplet. For $\alpha =1$, we obtain the holomorphic constraints. Using the representation of Bargmann-Fock space and the Schr\"odinger equation, we construct the holomorphic wave functions. In order to interpret these functions, we introduce the $W_\infty$-coherent state to account for the infinite-dimensional translation symmetry for the Fourier (edge) modes. The $\alpha=1$ wave functions explain the neutral edge states for $\nu =1$ quantum Hall fluid very well. In the case of $\alpha = -1$, we obtain the new wave functions which may describe the higher modes (radial excitations) of edge states. Finally, the charged edge states are described by the $|\alpha| \not=1$ wave functions.