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Quantum Hall effect and noncommutative geometry. (English) Zbl 1104.81088

Summary: 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 A. L. Cary, K. C. Hannabuss, V. Mathai and P. McCann [Commun. Math. Phys. 190, 629–673 (1998; Zbl 0916.46057)] in order to prove the integrality of the Hall conductance in this case.

MSC:

81V70 Many-body theory; quantum Hall effect
81T75 Noncommutative geometry methods in quantum field theory
81R60 Noncommutative geometry in quantum theory
35R60 PDEs with randomness, stochastic partial differential equations

Citations:

Zbl 0916.46057
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