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A Band of Critical States in Anderson Localization at Strong Magnetic Field with Random Spin-Orbit Scattering

Anderson localization problem for non-interacting two-dimensional electron gas subject to 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 the nature of the spin-orbit coupling (uniform or fully random). For uniform spin-orbit coupling (such as Rashba coupling), there is a band of 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 $E_c$, the localization length diverges as $ξ(E) \propto\exp(α/\sqrt{|E-E_c|})$, a behavior which, along with the band of critical states, is reminiscent of a Berezinskii-Kosterlitz-Thouless transition.