Predrag's notes, cribbed from here and there Nov 11 2012 The Integer Quantum Hall effect, first observed by K. von Klitzing, is used to determine the fine structure constant with precision that is comparable to the precision one gets from atomic physics. It is also used as a practical and fundamental way to define the Ohm. It is instructive to look at the experimental data. The graph that looks like a staircase function has remarkably flat plateaus. The ordinates of the plateaus correspond to integer integer multiple of the quantum unit of conductance, and can be measured very precisely. An intriguing aspect of this phenomenon is that a precision measurement of fundamental constants is carried on a system that is only poorly characterized: Little is actually known about the microscopic details of the system, which is artificially fabricated, and whose precise composition and shape are not known with a precision that is anywhere comparable with the precision that comes out of the experiment. Laughlin first suggested that IQHE should have a geometric explanation. More precisely, the fact that the quantization of the Hall conductance appears as a very robust phenomenon, insensitive to changes in the sample and its geometry, or to the presence of impurities, suggests the fact that the effect should have the same qualities of the index theorem, which assigns an integer invariant to an elliptic differential operator, in a way that is topological and independent of perturbations. The prototype of such index theorems is the Gauss-Bonnet theorem, which extracts from an infinitesimally variable quantity, the curvature of a closed surface, a robust topological invariant, its Euler characteristic. The idea of modelling the integer quantum Hall effect on an index theorem started fairly early after the discovery of the effect. Laughlin's formulation can already be seen as a form of Gauss-Bonnet, while this was formalized more precisely in such terms shortly afterwards by Thouless et al. (1982) and by Avron, Seiler, and Simon (1983). One of the early successes of Connes' noncommutative geometry was a rigorous mathematical model of the integer quantum Hall effect, developed by Bellissard, van Elst, and Schulz-Baldes. Unlike the previous models, this accounts for all the aspects of the phenomenon: integer quantization, localization, insensitivity to the presence of disorder, and vanishing of direct conductivity at plateaux levels. Again the integer quantization is reduced to an index theorem, albeit of a more sophisticated nature, involving the Connes.Chern character, the K-theory of C*-algebras and cyclic cohomology). Bellissard's theory of the IQHE is rigorous. There is no rigorous explanation of the Fractional Quantum Hall Effect. Bellissard says that one needs at least two 1-hour seminars to motivate the theory for audience of condensed matter theorists. For graduate students, it takes at least two weeks of lectures. ---------------- http://www.nobelprize.org/nobel_prizes/physics/laureates/1985/press.html http://www.nobelprize.org/nobel_prizes/physics/laureates/1998/press.html J. Bellissard A. van Elst and H. Shultz-Baldes, "The Non commutative Geometry of the Quantum Hall effect", J. Math. Phys. 35, 5373, (1994). http://arxiv.org/abs/cond-mat/9411052 (at least 400 citations)