Quantum Hall Circle

TL;DR

This paper models quantum Hall states on a cylinder as the circumference approaches infinity, revealing their adiabatic connection to bulk states and identifying a gapless Wigner crystal phase at low filling fractions.

Contribution

It introduces a one-dimensional quantum Hall circle model that captures the transition from 1D to bulk quantum Hall states and explores the emergence of a Wigner crystal at low fillings.

Findings

Quantum Hall states are exact ground states for certain short-range interactions.

States evolve adiabatically from the quantum Hall circle to bulk states as circumference increases.

A gapless Wigner crystal phase appears at low filling fractions.

Abstract

We consider spin-polarized electrons in a single Landau level on a cylinder as the circumference of the cylinder goes to infinity. This gives a model of interacting electrons on a circle where the momenta of the particles are restricted and there is no kinetic energy. Quantum Hall states are exact ground states for appropriate short range interactions, and there is a gap to excitations. These states develop adiabatically from this one-dimensional quantum Hall circle to the bulk quantum Hall states and further on into the Tao-Thouless states as the circumference goes to zero. For low filling fractions a gapless state is formed which we suggest is connected to the Wigner crystal expected in the bulk.