Dimensional crossover of localization and delocalization in a quantum Hall bar

The two- to one-dimensional crossover of the localization of electrons confined to a disordered quantum wire of finite width $L_y$ is studied in a model of electrons moving in the potential of uncorrelated impurities. The localization length is derived as a function of the perpendicular magnetic field B, the wire width $L_y,$ and the conductance parameter g. The analytical theory allows us to study the localization continuously from weak to strong magnetic fields. On the basis of these results, the scaling analysis of the quantum Hall effect in high Landau levels and the delocalization transition in a quantum Hall wire are reconsidered. We conclude that in quantum Hall bars, the quantum Hall transition is driven in all but the lowest two Landau bands by a noncritical dimensional crossover of the localization length.