Fermions in the Lowest Landau Level: Bosonization, $W_{\infty}$ Algebra, Droplets, Chiral Boson

TL;DR

This paper develops field theoretical descriptions of (2+1)D nonrelativistic fermions in magnetic fields, revealing an infinite $W_{ abla}$ algebra and a universal chiral boson Lagrangian, connecting to string theory models.

Contribution

It introduces a fermionic and bosonic second quantized framework and links the $W_{ abla}$ algebra to the droplet approximation and chiral boson theory.

Findings

$W_{ abla}$ algebra governs transformations preserving the lowest Landau level.

Droplet approximation leads to a universal chiral boson Lagrangian.

Bosonic droplet approximation corresponds to the strong magnetic field limit.

Abstract

We present field theoretical descriptions of massless (2+1) dimensional nonrelativistic fermions in an external magnetic field, in terms of a fermionic and bosonic second quantized language. An infinite dimensional algebra, $W_{\infty}$, appears as the algebra of unitary transformations which preserve the lowest Landau level condition and the particle number. In the droplet approximation it reduces to the algebra of area-preserving diffeomorphisms, which is responsible for the existence of a universal chiral boson Lagrangian independent of the electrostatic potential. We argue that the bosonic droplet approximation is the strong magnetic field limit of the fermionic theory. The relation to the $c=1$ string model is discussed.