One-Dimensional Theory of the Quantum Hall System

We consider the lowest Landau level on a torus as a function of its circumference $L_1$. When $L_1\rightarrow 0$, 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 $L_1\rightarrow \infty$. 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 $L_1\rightarrow \infty$. Furthermore, the effective Landau level structure that emerges within the lowest Landau level follows from the magnetic symmetries.