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Towards the fractional quantum Hall effect: a noncommutative geometry perspective

  • Matilde Marcolli
  • Varghese Mathai
Part of the Aspects of Mathematics book series (ASMA)

Abstract

In this paper we give a survey of some models of the integer and fractional quantum Hall effect based on noncommutative geometry. We begin by recalling some classical geometry of electrons in solids and the passage to noncommutative geometry produced by the presence of a magnetic field. We recall how one can obtain this way a single electron model of the integer quantum Hall effect. While in the case of the integer quantum Hall effect the underlying geometry is Euclidean, we then discuss a model of the fractional quantum Hall effect, which is based on hyperbolic geometry simulating the multi-electron interactions. We derive the fractional values of the Hall conductance as integer multiples of orbifold Euler characteristics. We compare the results with experimental data.

Keywords

Noncommutative Geometry Fuchsian Group Spectral Projection Hall Conductance Fractional Quantum 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Friedr. Vieweg & Sohn Verlag ∣ GWV Fachverlage GmbH, Wiesbaden 2006

Authors and Affiliations

  • Matilde Marcolli
    • 1
  • Varghese Mathai
    • 2
  1. 1.Max-Planck Institut für MathematikBonnGermany
  2. 2.Department of Pure MathematicsUniversity of AdelaideAustralia

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