Band of critical states in anderson localization in a strong magnetic field with random Spin-orbit scattering

The Anderson localization problem for a noninteracting two-dimensional electron gas subject to a strong magnetic field, disordered potential, and spin-orbit coupling is studied numerically on a square lattice. The nature of the corresponding localization-delocalization transition and the properties of the pertinent extended states depend on whether the spin-orbit coupling is uniform or fully random. For uniform spin-orbit coupling (such as Rashba coupling due to a uniform electric field), there is a band of metallic extended states in the center of a Landau band as in a "standard" Anderson metal-insulator transition. However, for fully random spin-orbit coupling, the familiar pattern of Landau bands disappears. Instead, there is a central band of critical states with definite fractal structure separated at two critical energies from two side bands of localized states. Moreover, finite size scaling analysis suggests that for this novel transition, on the localized side of a critical energy Ec, the localization length diverges as ξ(E) exp(α/E-Ec|), a behavior which, together with the emergence of a band of critical states, is reminiscent of a Berezinskii-Kosterlitz-Thouless transition.