Quantum Hall system in Tao-Thouless limit

We consider spin-polarized electrons in a single Landau level on a torus. The quantum Hall problem is mapped onto a one-dimensional lattice model with lattice constant 2/L₁, where L₁ is a circumference of the torus (in units of the magnetic length). In the Tao-Thouless limit L₁0, the interacting many-electron problem is exactly diagonalized at any rational filling factor =p/q1. For odd q, the ground state has the same qualitative properties as a bulk (L₁) quantum Hall hierarchy state and the lowest-energy quasiparticle excitations have the same fractional charges as in the bulk. These states are the L₁0 limits of the Laughlin and Jain wave functions for filling fractions where these exist. We argue that the exact solutions generically, for odd q, are continuously connected to the two-dimensional bulk quantum Hall hierarchy states---i. e. , that there is no phase transition as L₁ for filling factors where such states can be observed. For even-denominator fractions, a phase transition occurs as L₁ increases. For =1/2 this leads to the system being mapped onto a Luttinger liquid of neutral particles at small but finite L₁; this then develops continuously into the composite fermion wave function that is believed to describe the bulk =1/2 system. The analysis generalizes to non-Abelian quantum Hall states.