One-Dimensional Theory of the Quantum Hall System

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We consider the lowest Landau level on a torus as a function of its circumference L₁. When L₁→ 0, the ground state at general rational filling fraction is a crystal with a gap--a Tao-Thouless state. For filling fractions ν=p/(2pm+1), these states are the limits of Laughlin's or Jain's wave functions describing the gapped quantum Hall states when L₁→ ∞. 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 L₁ 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 L₁→ ∞. Furthermore, the effective Landau level structure that emerges within the lowest Landau level follows from the magnetic symmetries.