Quantum Hall Effect on the Hyperbolic Plane

Abstract

Abstract. In this paper, we study both the continuous model and the discrete model of the Quantum Hall Effect (QHE) on the hyperbolic plane. The Hall conductivity is identified as a geometric invariant associated to an imprimitivity algebra of observables. We define a twisted analogue of the Kasparov map, which enables us to use the pairing between K-theory and cyclic cohomology theory, to identify this geometric invariant with a topological index, thereby proving the integrality of the Hall conductivity in this case.

Towards the fractional quantum Hall effect: a noncommutative geometry perspective

“…In this survey we will only discuss a proposed model [23] [24], which is based on extending the validity of the Bellissard approach to the setting of hyperbolic geometry as in [5], where passing to a negatively curved geometry is used as a device to simulate the many-electrons Coulomb interaction while remaining within a single electron model. What is expected of any proposed mathematical model?…”

“…This will be useful later, in our model of the fractional quantum Hall effect, but we introduce it here for convenience. For the general setup for finitely generated discrete groups recalled here below, we follow [5]. Suppose given a finitely generated discrete group Γ and a multiplier σ :…”

“…We will not describe in detail the derivation of the quantization of the Hall conductance in this model of the integer quantum Hall effect. In fact, we will concentrate mostly on a model for the fractional quantum Hall effect and we will show how to recover the integer quantization within that model, using the results of [5].…”

“…The result in the case of torsion-free Fuchsian groups was established in [5]. In more recent work, Mathai generalized this result to discrete subgroups of rank 1 groups and to all amenable groups, and more generally whenever the BaumConnes conjecture with coefficients holds for the discrete group, [25].…”

“…Consequently, the Hall conductance is seen to be stable under small deformations of the Hamiltonian. Thus, this model can be generalized to systems with disorder as in [6], and then the hypothesis that the Fermi level is in a spectral gap of the Hamiltonian can be relaxed to the assumption that it is in a gap of extended states. This is a necessary step in order to establish the presence of plateaux.…”