Matrix Model Description of Laughlin Hall States

TL;DR

This paper models Laughlin quantum Hall states using a matrix Chern-Simons approach, revealing a symmetry-based description of their stability and properties.

Contribution

It introduces a matrix model framework for Laughlin states, connecting non-commutative Chern-Simons theory with W-infinity symmetry to describe quantum Hall ground states.

Findings

Eigenvalues represent electron coordinates in lowest Landau level.

A statistical interaction stabilizes the ground state.

W-infinity symmetry characterizes incompressible quantum Hall states.

Abstract

We analyze Susskind's proposal of applying the non-commutative Chern-Simons theory to the quantum Hall effect. We study the corresponding regularized matrix Chern-Simons theory introduced by Polychronakos. We use holomorphic quantization and perform a change of matrix variables that solves the Gauss law constraint. The remaining physical degrees of freedom are the complex eigenvalues that can be interpreted as the coordinates of electrons in the lowest Landau level with Laughlin's wave function. At the same time, a statistical interaction is generated among the electrons that is necessary to stabilize the ground state. The stability conditions can be expressed as the highest-weight conditions for the representations of the W-infinity algebra in the matrix theory. This symmetry provides a coordinate-independent characterization of the incompressible quantum Hall states.