Low-frequency behavior of the kinetic coefficients in localization regimes in strong magnetic fields

Using a method of integration over commuting and anticommuting variables, we study the motion of noninteracting electrons in a system with short-range disorder and in a strong magnetic field. An explicit calculation shows that the static conductivities ${\mathrm{\sigma }}_{\mathit{x}\mathit{x}}$ and ${\mathrm{\sigma }}_{\mathit{y}\mathit{x}}$ vanish if the Fermi energy is situated in the lower tail of the lowest Landau level. At the same time, the Hall resistivity ${\mathrm{\rho }}_{\mathit{x}\mathit{y}}$ remains finite, although the longitudinal resistivity diverges. These results are valid both for two-dimensional (2D) and for 3D systems. We compare our theoretical findings with experiments on magnetic-field-induced metal-insulator transitions in 3D systems. Furthermore, the result for the Hall conductivity of 2D systems is generalized to higher localization regions in order to study the deformation of the Hall plateaus in microwave experiments on GaAs-${\mathrm{Al}}_{\mathit{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As heterostructures.