Two-Dimensional Electrons in a Strong Magnetic Field with Disorder: Divergence of the Localization Length

TL;DR

This paper models electrons in a magnetic field with disorder using a 2D Dirac theory, revealing critical delocalization behavior and estimating localization length through advanced theoretical methods.

Contribution

It introduces a Dirac-based effective description for electrons in a magnetic field with disorder, analyzing localization and phase transitions with novel theoretical approaches.

Findings

Localization length diverges at critical points

Two Hall transition points are suggested by percolation theory

Delocalization linked to spontaneous symmetry breaking

Abstract

Electrons on a square lattice with half a flux quantum per plaquette are considered. An effective description for the current loops is given by a two-dimensional Dirac theory with random mass. It is shown that the conductivity and the localization length can be calculated from a product of Dirac Green's functions with the {\it same} frequency. This implies that the delocalization of electrons in a magnetic field is due to a critical point in a phase with a spontaneously broken discrete symmetry. The estimation of the localization length is performed for a generalized model with $N$ fermion levels using a $1/N$--expansion and the Schwarz inequality. An argument for the existence of two Hall transition points is given in terms of percolation theory.