THESIS
2016

xvi, 125 pages : illustrations ; 30 cm

**Abstract**
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...[

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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.

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