Quantum Hall Effect on the Hyperbolic Plane in the presence of disorder

Details Quantum Hall Effect on the Hyperbolic Plane in the presence of disorder Quantum Hall Effect on the Hyperbolic Plane in the presence of disorder

1998-04-27

arxiv.org/abs/math/9804128v1

Letters in Math. Phys. 47 no. 3 (1999) 215-236

Mathematics

Differential Geometry

LaTeX2e, 17 pages

Scientific paper

We study both the continuous model and the discrete model of the integer quantum Hall effect on the hyperbolic plane in the presence of disorder, extending the results of an earlier paper [CHMM]. Here we model impurities, that is we consider the effect of a random or almost periodic potential as opposed to just periodic potentials. The Hall conductance is identified as a geometric invariant associated to an algebra of observables, which has plateaus at gaps in extended states of the Hamiltonian. We use the Fredholm modules defined in [CHMM] to prove the integrality of the Hall conductance in this case. We also prove that there are always only a finite number of gaps in extended states of any random discrete Hamiltonian. [CHMM] A. Carey, K. Hannabuss, V. Mathai and P. McCann, Quantum Hall Effect on the Hyperbolic Plane, Communications in Mathematical Physics, 190 vol. 3, (1998) 629-673.

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