Localization, percolation, and the quantum Hall effect

Noninteracting electrons in a smooth two-dimensional random potential are localized in the large magnetic field limit. In contrast to Anderson localization, eigenstates with large localization lengths occur with a probability proportional to a universal power of their size, with the power given in terms of percolation critical exponents. Adding a parallel electric field $\overrightarrow{\mathcal{E}}$ causes extended states to appear in numbers proportional to a power of $\mathcal{E}$. This implies a nonlinear broadening of steps in the quantized Hall conductivity. The results for a parallel electric field are obtained by considering a graded percolation problem, in which the probability that a site is occupied varies with position.