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Type: Book chapter
Title: Towards the fractional quantum Hall effect: a noncummutative geometry perspective
Author: Marcolli, M.
Varghese, M.
Citation: Noncommutative geometry and number theory, 2006 / Consani, C., Marcolli, M. (ed./s), pp.235-262
Publisher: Springer
Publisher Place: Germany
Issue Date: 2006
ISBN: 3834801704
Editor: Consani, C.
Marcolli, M.
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.
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