Towards the fractional quantum Hall effect: a noncommutative geometry perspective

Details Towards the fractional quantum Hall effect: a noncommutative geometry perspective Towards the fractional quantum Hall effect: a noncommutative geometry perspective

2005-02-15

arxiv.org/abs/cond-mat/0502356v2

"Noncommutative Geometry and Number Theory". Editors C.Consani, M. Marcolli, Aspects of Mathematics, Vieweg Verlag, Wiesbaden,

Physics

Condensed Matter

Mesoscale and Nanoscale Physics

27 pages, LaTeX, 9 eps figures, v2: minor changes

Scientific paper

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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