In this thesis, the Anderson localization problem for a noninteracting two-dimensional electron gas subject to a high magnetic field, disordered potential, and spin-orbit coupling is investigated theoretically. There are three issues: (1) the evolution of extended states in the integer quantum Hall systems, (2) the nature of quantum states in a disordered two-dimensional system in a magnetic field with a uniform spin-orbit interaction, (3) same issue as (2) but with a fully random spin-orbit interaction.

First, we investigate how the extended state in the lowest Landau band evolves with increasing the disorder strength W and/or decreasing the magnetic field B by using the Anderson model numerically. Focusing on the lowest Landau band, we establish an anti-levitation scenario: The...[ Read more ]

In this thesis, the Anderson localization problem for a noninteracting two-dimensional electron gas subject to a high magnetic field, disordered potential, and spin-orbit coupling is investigated theoretically. There are three issues: (1) the evolution of extended states in the integer quantum Hall systems, (2) the nature of quantum states in a disordered two-dimensional system in a magnetic field with a uniform spin-orbit interaction, (3) same issue as (2) but with a fully random spin-orbit interaction.

First, we investigate how the extended state in the lowest Landau band evolves with increasing the disorder strength W and/or decreasing the magnetic field B by using the Anderson model numerically. Focusing on the lowest Landau band, we establish an anti-levitation scenario: The energies of the extended states move below the Landau levels pertaining to a clean system as either increasingW or decreasing B. Moreover, for a high enough disorder, there is a disorder-dependent magnetic field B(W) and below which the system is the Anderson insulator. We also suggest a general phase boundary between the integer quantum Hall regime and Anderson insulator in the W − 1/B plane.

Then, we study the role of a uniform spin-orbit interaction in a disordered two-dimensional electron gas: How it turns an insulator to a conductor. We choose the spin-orbit interaction as a Rashba coupling. For B = 0, the Rashba model supports a band of extended states. We sketch the corresponding phase diagram in the disorder-energy plane with a boundary separated the metallic phase and insulating phase. At a strong magnetic field, we find a band of extended states in the center of the lowest Landau band with two mobility edges E_c1 and E_c2. At E_c1 or E_c2, the system undergoes a standard Anderson transition and states within the energy range [E_c1 , E_c2] are all extended. It is quite different from the quantum-Hall-type transition where the isolated critical level separates localized phases. With decreasing B, the extended band in the lowest Landau band becomes wider and wider and meets those from higher Landau bands at a disorder-dependent magnetic field. Eventually, there is only one extended band at an extremely low magnetic field where the Landau bands are strongly mixed. Schematic phase diagrams in the field-energy plane with a fixed disorder and the disorder-energy plane with a fixed magnetic field are drawn to demonstrate the evolution of extended states with changing B or W.

Finally, we explore how a fully random spin-orbit interaction affects the metal-insulator transition in a disordered two-dimensional system. We refer the corresponding model as the SU(2) model. In the absence of magnetic field, the SU(2) model can undergo a standard Anderson transition. Remarkably, at a strong magnetic field, we find a band of critical 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) ∝ exp[α/√ - E_c , which is a typical feature of the Berezinskii-Kosterlitz-Thouless transition. At a weak magnetic field, it is found that the metallic phase (known to exists at B = 0) also persists at albeit weak magnetic fields and eventually crosses over into the critical phase, which has already been confirmed at high magnetic fields. A schematic phase diagram drawn in the E − B plane elucidates the occurrence of localized, metallic and critical phases. Also, it is shown that nearest level spacing distribution is determined solely by the symmetry and follows the Wigner surmise irrespective of whether states are metallic or critical.