Quantum Hall Droplets on Disc and Effective Wess-Zumino-Witten Action for Edge States

TL;DR

This paper analyzes quantum Hall droplets on a disc using geometric quantization, identifies edge states with a Wess-Zumino-Witten action, and explores their properties and wavefunctions in a noncommutative geometric framework.

Contribution

It introduces an algebraic approach to quantum Hall effects on the disc, deriving edge state dynamics via an effective Wess-Zumino-Witten action and connecting wavefunctions to noncommutative geometry.

Findings

Edge excitations are described by a Wess-Zumino-Witten action.

LLL wavefunctions are shown to be intelligent states.

The state density and excitation potential are characterized.

Abstract

We algebraically analysis the quantum Hall effect of a system of particles living on the disc ${\bf B}^1$ in the presence of an uniform magnetic field $B$. For this, we identify the non-compact disc with the coset space $SU(1,1)/U(1)$. This allows us to use the geometric quantization in order to get the wavefunctions as the Wigner ${\cal D}$-functions satisfying a suitable constraint. We show that the corresponding Hamiltonian coincides with the Maass Laplacian. Restricting to the lowest Landau level, we introduce the noncommutative geometry through the star product. Also we discuss the state density behavior as well as the excitation potential of the quantum Hall droplet. We show that the edge excitations are described by an effective Wess-Zumino-Witten action for a strong magnetic field and discuss their nature. We finally show that LLL wavefunctions are intelligent states.