Semiclassical theory of the quantum Hall effect

The localization-delocalization transition occurring in the quantum Hall effect is studied for noninteracting, spinless electrons subject to a disordered long-range potential. Within the Chalker-Coddington network we compute the localization length at various energies and for a wide distribution of disorder strengths W. A scaling analysis with respect to the system size and the disorder reveals a crossover from quantum-mechanical to classical behavior that can be studied by means of a length scale xi(irr)(W) which is "irrelevant" in the usual field-theoretical sense. We show that pronounced classical structures arise at parameter values W an order of magnitude below the classical limit. They are stabilized by interference effects and give rise to xi(irr)proportional to W-4/3, whereas for the localization length we still find xi(epsilon)proportional to epsilon(nu) with nu approximate to 7/3 in the entire investigated scaling regime. By relating our observations to recent results on the dynamical conductivity, we propose that the irrelevant scale actually leads to a "long-time tail" of sigma(xx)(omega) in long-range potentials.