One-Dimensional Theory of the Quantum Hall System

Emil J. Bergholtz; Anders Karlhede

doi:10.1088/1742-5468/2006/04/L04001

paper

One-Dimensional Theory of the Quantum Hall System

arXiv:cond-mat/0509434 · doi:10.1088/1742-5468/2006/04/L04001

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 $ν = p / (2 p m + 1)$ , 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 $L_{1} \sim 5$ 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.

4 pages, 1 figure