Quantum Hall Effect and Noncommutative Geometry

Details Quantum Hall Effect and Noncommutative Geometry Quantum Hall Effect and Noncommutative Geometry

2000-08-16

arxiv.org/abs/math/0008115v2

Journal of Geometry and Symmetry in Physics, vol 6 (2006) 16-37

Mathematics

Operator Algebras

18 pages, paper rewritten

Scientific paper

We study magnetic Schrodinger operators with random or almost periodic electric potentials on the hyperbolic plane, motivated by the quantum Hall effect 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 [CHMM] in order to prove the integrality of the Hall conductance in this case.

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