One-Dimensional Theory of the Quantum Hall System

TL;DR

This paper develops a one-dimensional theoretical framework for the quantum Hall system on a torus, revealing phase transitions and effective structures as the system's circumference varies, bridging gapped states and metallic phases.

Contribution

It introduces a one-dimensional theory of the quantum Hall system on a torus, elucidating phase transitions and effective Landau level structures as a function of system size.

Findings

At small circumference, the ground state is a crystal (Tao-Thouless state).

At half-filling, a transition to a Fermi sea of neutral dipoles occurs around 5 magnetic lengths.

The effective Landau level structure arises from magnetic symmetries within the lowest Landau level.

Abstract

We consider the lowest Landau level on a torus as a function of its circumference $L_1$. When $L_1\to 0$, the ground state at general rational filling fraction is a crystal with a gap--a Tao-Thouless state. For filling fractions $\nu=p/(2pm+1)$, these states are the limits of Laughlin's or Jain's wave functions describing the gapped quantum Hall states when $L_1\to \infty$. For the half-filled Landau level, there is a transition to a Fermi sea of non-interacting neutral dipoles, or rather to a Luttinger liquid modification thereof, at $L_1\sim5$ magnetic lengths. This state is a version of the Rezayi-Read state, and develops continuously into the state that is believed to describe the observed metallic phase as $L_1\to \infty$. Furthermore, the effective Landau level structure that emerges within the lowest Landau level follows from the magnetic symmetries.