Summary, Omissions and Unanswered Questions

Abstract

We have seen in the previous chapters that the IQHE is, in its essence, a single-particle localization phenomenon associated with the finite mobility gap between extended states in different Landau levels. The gap is a single-particle effect in that it is associated with the kinetic energy. The only many-particle effect which is essential to include is the Pauli exclusion principle which allows the mobility gap to produce dissipationless current flow at zero temperature [Problem 1]. It is this lack of dissipation which is the key to the effect [Note 1]. Theoretical work on this problem has largely centered on proving the fundamental quantization theorem:

$${\sigma _{xx}} = 0 \Rightarrow {\sigma _{xy}} = i{e^2}/h$$

(10.1.1)

where σ is the macroscopic conductance. The original approach taken by Laughlin (1981) has provided a new language with which to address this problem and has led ultimately to the very beautiful picture that the macroscopic conductance is quantized because it is a topological invariant (Thouless Chap. IV).