Quantum Hall system in Tao-Thouless limit

Abstract

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 $2\pi /L_1$, where $L_1$ is a circumference of the torus (in units of the magnetic length). In the Tao-Thouless limit, $L_1\to 0$, the interacting many-electron problem is exactly diagonalized at any rational filling factor $\nu =p/q\le 1$. For odd $q$, the ground state has the same qualitative properties as a bulk ($L_1\to \infty$) quantum Hall hierarchy state and the lowest energy quasiparticle exitations have the same fractional charges as in the bulk. These states are the $L_1\to 0$ limits of the Laughlin/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, {\it ie} that there is no phase transition as $L_1\to \infty$ for filling factors where such states can be observed. For even denominator fractions, a phase transition occurs as $L_1$ increases. For $\nu =1/2$ this leads to the system being mapped onto a Luttinger liquid of neutral particles at small but finite $L_1$, this then develops continuously into the composite fermion wave function that is believed to describe the bulk $\nu =1/2$ system. The analysis generalizes to non-abelian quantum Hall states.

25 pages, 9 figures

References in corpus (11)

• Particle-hole symmetry and the Pfaffian state
• Particle-Hole Symmetry and the $\nu =5/2$ Quantum Hall State
• Half-Filled Lowest Landau Level on a Thin Torus
• Abelian and Non-abelian Hall Liquids and Charge Density Wave: Quantum Number Fractionalization in One and Two Dimensions
• Paired composite fermion wavefunctions
• The Pfaffian quantum Hall state made simple--multiple vacua and domain walls on a thin torus
• Microscopic theory of the quantum Hall hierarchy
• Conformal Field Theory of Composite Fermions
• Quantum Hall wave functions on the torus
• Possible Reentrance of the Fractional Quantum Hall Effect in the Lowest Landau Level
• On the self-similarity in quantum Hall systems

Cited by in corpus (19)

• Fractional Chern Insulators in Topological Flat bands with Higher Chern Number
• Enhancing the stability of a fractional Chern insulator against competing phases
• Fibonacci Anyons From Abelian Bilayer Quantum Hall States
• From fractional Chern insulators to Abelian and non-Abelian fractional quantum Hall states: adiabatic continuity and orbital entanglement spectrum
• Fractional charge pumping of interacting bosons in one-dimensional superlattice
• Competing Abelian and non-Abelian topological orders in $\nu =1/3+1/3$ quantum Hall bilayers
• Hierarchy wave functions--from conformal correlators to Tao-Thouless states
• Composite Fermions on a Torus
• Zero modes, Bosonization and Topological Quantum Order: The Laughlin State in Second Quantization
• Solvable models for unitary and non-unitary topological phases
• Algebraic approach to the study of zero modes of Haldane pseudopotentials
• Symmetry breaking in Laughlin's state on a cylinder
• Effective spin chains for fractional quantum Hall states
• Spin-chain description of fractional quantum Hall states in the Jain series
• Edge properties of principal fractional quantum Hall states in the cylinder geometry
• Spin chain description of rotating bosons at $\nu =1$
• Quantum Hall Circle
• Exclusion statistics for quantum Hall states in Tao-Thouless limit
• Conformal field theory construction for nonabelian hierarchy wave functions