Scaling Theory of the Integer Quantum Hall Effect

TL;DR

This paper reviews the scaling theory of the integer Quantum Hall effect, emphasizing numerical studies that align well with experimental results on disorder-induced localization-delocalization transitions.

Contribution

It provides a comprehensive overview of numerical methods and finite-size scaling theory for understanding the quantum Hall transitions, connecting theory with experimental observations.

Findings

Numerical studies successfully describe experimental quantum Hall transitions.

Finite-size scaling theory relates to Anderson localization.

Multifractal measures characterize local observables at the transition.

Abstract

The scaling theory of the transitions between plateaus of the Hall conductivity in the integer Quantum Hall effect is reviewed. In the model of two-dimensional noninteracting electrons in strong magnetic fields the transitions are disorder-induced localization-delocalization transitions. While experimental and analytical approaches are surveyed, the main emphasis is on numerical studies, which successfully describe the experiments. The theoretical models for disordered systems are described in detail. An overview of the finite-size scaling theory and its relation to Anderson localization is given. The field-theoretical approach to the localization problem is outlined. Numerical methods for the calculation of scaling quantities, in particular the localization length, are detailed. The properties of local observables at the localization-delocalization transition are discussed in terms of multifractal measures. Finally, the results of extensive numerical investigations are compared with experimental findings.