Effects of symmetry on localization on two-dimensional square lattices are studied numerically. The inverse localization length is determined by the system-size dependence of the Thouless number in magnetic fields or in the presence of strong spin-orbit interactions. A finite-size-scaling method is also applied to the case of spin-orbit interactions. Extended states, present in each Landau level in strong magnetic fields or in the case of small randomness, are found to merge together with increasing randomness and disappear beyond a certain critical randomness. In weak magnetic fields, the field tends to reduce the localization near the band center, while the localization is enhanced in the band-tail region. Spin-orbit interactions cause effects similar to, but much stronger than, that due to a weak magnetic field. The critical randomness and exponent for a metal-insulator transition are determined in the presence of strong spin-orbit interactions.

• Received 6 April 1989

DOI:https://doi.org/10.1103/PhysRevB.40.5325