Laughlin Wave Function and One-Dimensional Free Fermions

TL;DR

This paper demonstrates an exact correspondence between the Laughlin wave function in quantum Hall systems and a one-dimensional free fermion system, revealing connections to random matrix theory and Jain's parton picture.

Contribution

It introduces a novel exact mapping of the Laughlin wave function to a 1D free fermion system, incorporating Jain's parton picture and linking to random matrix theory.

Findings

Exact representation of Laughlin wave function as a 1D coherent state

Connection between quantum Hall effect and Gaussian unitary ensemble

Incorporation of Jain's parton picture in the correspondence

Abstract

Making use of the well-known phase space reduction in the lowest Landau level(LLL), we show that the Laughlin wave function for the $\nu = {1\over m}$ case can be obtained exactly as a coherent state representation of an one dimensional $(1D)$ wave function. The $1D$ system consists of $m$ copies of free fermions associated with each of the $N$ electrons, confined in a common harmonic well potential. Interestingly, the condition for this exact correspondence is found to incorporate Jain's parton picture. We argue that, this correspondence between the free fermions and quantum Hall effect is due to the mapping of the $1D$ system under consideration, to the Gaussian unitary ensemble in the random matrix theory.