The level-spacing distribution of consecutive energy eigenvalues is calculated numerically at the metal insulator transition for 3d systems with different cuboid shapes. It is found that the scale independent critical P c (s) changes as a function of the aspect ratio of the samples while the critical disorder W c /V = 16.4 remains the same. We use our data to test whether an expression for the small-s behaviour of the level statistics proposed by Kravtsov and Mirlin for the metallic regime is applicable also at the critical point. For this reason, a shape dependent dimensionless critical conductance g c has been extracted from the small-s behaviour of the critical level statistics. Our result for a cubic sample, g c = 0.112±0.005, is in good agreement with a value obtained previously from calculations using the Kubo-formula.

The disorder driven quantum Hall to insulator transition is investigated for a two-dimensional lattice model. The Hall conductivity and the localization length are calculated numerically near the transition. For uncorrelated and weakly correlated disorder potentials the current carrying states are annihilated by the negative Chern states originating from the band center. In the presence of correlated disorder potentials with correlation length larger than approximately half the lattice constant the floating up of the critical states in energy without merging is observed. This behavior is similar to the levitation scenario proposed for the continuum model.

We investigate a disordered two-dimensional lattice model for noninteracting electrons with long-range power-law transfer terms and apply the method of level statistics for the calculation of the critical properties. The eigenvalues used are obtained numerically by direct diagonalization. We find a metal-insulator transition for a system with orthogonal symmetry. The exponent governing the divergence of the correlation length at the transition is extracted from a finite size scaling analysis and found to be ϭ2.6Ϯ0.15. The critical eigenstates are also analyzed and the distribution of the generalized multifractal dimensions is extrapolated.

We investigate the influence of the boundary conditions on the scale invariant critical level statistics at the metal insulator transition of disordered three-dimensional orthogonal and two-dimensional unitary and symplectic tight-binding models. The distribution of the spacings between consecutive eigenvalues is calculated numerically and shown to be different for periodic and Dirichlet boundary conditions whereas the critical disorder remains unchanged. The peculiar correlations of the corresponding critical eigenstates leading to anomalous diffusion seem not to be affected by the change of the boundary conditions.

The crossover from the quantum Hall regime to the Hall-insulator is investigated by varying the strength of the diagonal disorder in a 2d tight-binding model. The Hall and longitudinal conductivities and the behavior of the critical states are calculated numerically. We find that with increasing disorder the current carrying states close to the band center disappear first. Simultaneously, the quantized Hall conductivity drops monotonically to zero also from higher quantized values.