Critical phenomena of the disorder driven localization-delocalization transition
The critical behavior of non-interacting electrons in disordered systems is investigated. The scaling functions of the localization lengths in two-dimensional systems with a magnetic field perpendicular to the plane and anisotropic hopping are rescaled to the isotropic one. The geometric mean of the critical values of the scaling functions equals the critical value of the isotropic scaling function. The critical exponent of the localization lengths in the two anisotropic directions is found to be equal and also equal to that of the isotropic system. The critical energy is independent of the direction of propagation. If disorder is strong enough, the critical energy does not coincide with the band center.;The probability distribution of the conductance at the critical energy of anisotropic systems is also rescaled to the corresponding distribution of the isotropic system. The ratio of side lengths of a two-dimensional anisotropic system should equal the square root of the ratio of the critical values of the scaling functions of the localization lengths.;The form of the critical distribution of the conductance is investigated for two-dimensional systems with magnetic fields or spin-orbit interaction, as well as three-dimensional systems, for both periodic as well as hard wall boundary conditions. The form for g < 1 can be explained reasonably well, whereas the understanding of the distribution for systems with g > 1 is still incomplete.;Finally, the fluctuations of the conductance for such systems are investigated from the ballistic regime to the localized regime, with an emphasis on the differences with respect to boundary conditions.;Parts of this dissertation have been published in Phys. Rev. B 63, 085102 (2001), Phys. Rev. B 64, 172202 (2001), Phys. Rev. B 64, 193103 (2001), and Phys. Rev. B (to be published).