Quantum Spaces

Quantum Hall Effect and Non-commutative Geometry (p. 1-2)

Vincent Pasquier

1. Introduction

Our aim is to introduce the ideas of non-commutative geometry through the example of the Quantum Hall Effect (QHE). We present a few concrete situations where the concepts of non-commutative geometry find physical applications. The Quantum Hall Effect [1-4] is a remarkable example of a purely experimental discovery which "could" have been predicted because the tools required are not extremely sophisticated and were known at the time of the discovery. What was missing was a good understanding of topological rigidity produced by quantum mechanics whose consequences can be tested at a macroscopic level: The quantized integers of the conductivity are completely analogous to the topological numbers one encounters in the study of fiber-bundles.

One can give a schematic description of the Quantum Hall Effect as follows. It deals with electrons constrained to move in a two-dimensional semiconductor sample in a presence of an applied magnetic field perpendicular to the sample. Due to the magnetic field, the Hilbert space of an electron is stratified into Landau levels separated by an energy gap (called the cyclotron frequency and proportional to the applied field). Each Landau level has a macroscopic degeneracy given by the area of the sample divided by a quantum of area (inversely proportional to the field) equal to 2?l2 where the length l is the so-called magnetic length. It is useful to think of the magnetic length as a Plank constant l2 ¡« . The limit of a strong magnetic field is very analogous to a classical limit.

The electrons behave much like incompressible objects occupying a quantum of area. Thus, when their number times 2?l2 is exactly equal a multiple of the area, it costs the energy gap to add one more electron. This discontinuity in the energy needed to add one more electron is at the origin of the incompressibility of the electron fluid. The number of electrons occupying each unit cell is called the filling factor, and the transverse conductivity is quantized each time the filling factor is exactly an integer.

We shall stick to this simple explanation, although this cannot be the end of the story. Indeed, if it was correct, the filling factor being linear in the magnetic field, the quantization of the conductance should be observed only at specific values of the magnetic field. In fact, it is observed on regions of finite width called plateaux, and it is necessary to invoke the impurities and localized states to account for these plateaus. Roughly speaking, some of the states are localized and do not participate to the conductance. These states are populated when the magnetic field is in a plateau. We refer the reader to a previous Poincar´e seminar for an introduction to these effects [4]. What is important for us here is that (although counter-intuitive) it is possible to realize experimentally situations where the filling factor is exactly an integer (or a fraction as we see next).

It came as a surprise (rewarded by the Nobel prize1) when the Quantum Hall Effect was observed at non-integer filling factors which turn out to always be simple fractions. To explain these fractions, it was necessary to introduce some very specific wave functions and to take into account the interactions between the electrons. The proposed wave functions are in some sense variational, although they carry no adjustable free parameters.