Journal of Geometry and Symmetry in Physics

Quantum Hall Effect and Noncommutative Geometry

Alan L. Carey, Keith C. Hannabuss, and Varghese Mathai

Full-text: Open access

Abstract

We study magnetic Schrödinger operators with random or almost periodic electric potentials on the hyperbolic plane, motivated by the quantum Hall effect (QHE) in which the hyperbolic geometry provides an effective Hamiltonian. In addition we add some refinements to earlier results. We derive an analogue of the Connes-Kubo formula for the Hall conductance via the quantum adiabatic theorem, identifying it as a geometric invariant associated to an algebra of observables that turns out to be a crossed product algebra. We modify the Fredholm modules defined in [4] in order to prove the integrality of the Hall conductance in this case.

Article information

Source
J. Geom. Symmetry Phys., Volume 6 (2006), 16-37.

Dates
First available in Project Euclid: 20 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.jgsp/1495245686

Digital Object Identifier
doi:10.7546/jgsp-6-2006-16-37

Mathematical Reviews number (MathSciNet)
MR2267772

Zentralblatt MATH identifier
1104.81088

Citation

Carey, Alan L.; Hannabuss, Keith C.; Mathai, Varghese. Quantum Hall Effect and Noncommutative Geometry. J. Geom. Symmetry Phys. 6 (2006), 16--37. doi:10.7546/jgsp-6-2006-16-37. https://projecteuclid.org/euclid.jgsp/1495245686


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