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Quantum Hall system in Tao-Thouless limit
Stockholm University, Faculty of Science, Department of Physics.
Stockholm University, Faculty of Science, Department of Physics.
2008 (English)In: Physical Review B. Condensed Matter and Materials Physics, ISSN 1098-0121, E-ISSN 1550-235X, Vol. 77, no 15, 155308- p.Article in journal (Refereed) Published
Abstract [en]

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 2π/L1, where L1 is a circumference of the torus (in units of the magnetic length). In the Tao-Thouless limit L1→0, the interacting many-electron problem is exactly diagonalized at any rational filling factor ν=p/q≤1. 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 L1→0 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 ν=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 ν=1/2 system. The analysis generalizes to non-Abelian quantum Hall states.

Place, publisher, year, edition, pages
2008. Vol. 77, no 15, 155308- p.
National Category
Physical Sciences
URN: urn:nbn:se:su:diva-24941DOI: 10.1103/PhysRevB.77.155308ISI: 000255457400078OAI: diva2:198559
Available from: 2008-05-01 Created: 2008-05-01 Last updated: 2012-02-24Bibliographically approved
In thesis
1. One-dimensional theory of the quantum Hall system
Open this publication in new window or tab >>One-dimensional theory of the quantum Hall system
2008 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The quantum Hall (QH) system---cold electrons in two dimensions in a perpendicular magnetic field---is a striking example of a system where unexpected phenomena emerge at low energies. The low-energy physics of this system is effectively one-dimensional due to the magnetic field. We identify an exactly solvable limit of this interacting many-body problem, and provide strong evidence that its solutions are adiabatically connected to the observed QH states in a similar manner as the free electron gas is related to real interacting fermions in a metal according to Landau's Fermi liquid theory.

The solvable limit corresponds to the electron gas on a thin torus. Here the ground states are gapped periodic crystals and the fractionally charged excitations appear as domain walls between degenerate ground states. The fractal structure of the abelian Haldane-Halperin hierarchy is manifest for generic two-body interactions. By minimizing a local k+1-body interaction we obtain a representation of the non-abelian Read-Rezayi states, where the domain wall patterns encode the fusion rules of the underlying conformal field theory.

We provide extensive analytical and numerical evidence that the Laughlin/Jain states are continuously connected to the exact solutions. For more general hierarchical states we exploit the intriguing connection to conformal field theory and construct wave functions that coincide with the exact ones in the solvable limit. If correct, this construction implies the adiabatic continuation of the pertinent states. We provide some numerical support for this scenario at the recently observed fraction 4/11.

Non-QH phases are separated from the thin torus by a phase transition. At half-filling, this leads to a Luttinger liquid of neutral dipoles which provides an explicit microscopic example of how weakly interacting quasiparticles in a reduced (zero) magnetic field emerge at low energies. We argue that this is also smoothly connected to the bulk state.

Place, publisher, year, edition, pages
Stockholm: Fysikum, 2008. 70 p.
fractional quantum Hall effect, thin torus, spin chains, conformal field theory, strong correlations, non-abelian states
National Category
Condensed Matter Physics
Research subject
Theoretical Physics
urn:nbn:se:su:diva-7545 (URN)978-91-7155-627-1 (ISBN)
Public defence
2008-05-28, sal FB53, AlbaNova universitetscentrum, Roslagstullsbacken 21, Stockholm, 13:15 (English)
Available from: 2008-05-01 Created: 2008-05-01 Last updated: 2010-02-23Bibliographically approved

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