Single-particle states on a sphere with a magnetic field and disorder

We study the states of noninteracting electrons subject to a strong magnetic field and disorder in the two-dimensional spherical geometry. It is found that the critical behavior is manifested only very close to the (broadened Landau) band center, where one-parameter scaling seems to be valid. Evaluation of the critical exponent depends on our knowledge of the scaling behavior of the inverse participation ratio. The nearest level-spacing distribution is computed at the band center and near the band edge. In the first case, it deviates slightly from the Gaussian unitary ensemble prediction, most notably for large spacing, whereas in the band tail it displays an almost pure Poisson statistics. Multifractal analysis of wave functions in the center as well as in the tail of the broadened Landau band is carried out. It gives strong support to the conjecture that the former ones are critical (rather than simply extended).