Disorder and spin-orbit coupling in the integer quantum Hall effect

The physics of two-dimensional electron gas (2DEG) in the presence of a perpendicular magnetic field, disordered potential, and spin-orbit coupling (SOC) is very rich. It touches upon numerous fundamental concepts such as Anderson localization, the integer quantum Hall effect, and random matrix ensembles (Gaussian, unitary, and symplectic). At strong magnetic field the system is extensively studied. It is characterized by isolated Landau levels wherein the energy is linear with the magnetic field and the corresponding wave functions are extended, while between two Landau levels, the corresponding wave functions are localized. In most cases, for strong magnetic field, pertinent calculations are based on the projection of a single Landau level. The first topic to be discussed below is the Anderson localization at weak magnetic field and strong, albeit uniform SOC. In fact, the physics at weak magnetic field seems to be even richer than that at strong magnetic field. Indeed, projection on a single Landau level is not justified, since the energy distance between adjacent levels compares with the strength of disorder and the SOC energy. The second topic to be discussed below is the Anderson localization in a strong magnetic field and with random SOC.