We present the results of numerical studies of spin quantum Hall transitions in disordered superconductors, in which the pairing order parameter breaks time-reversal symmetry. We focus mainly on p-wave superconductors in which one of the spin components is conserved. The transport properties of the system are studied by numerically diagonalizing pairing Hamiltonians on a lattice, and by calculating the Chern and Thouless numbers of the quasiparticle states. We find that in the presence of disorder, (spin-) current carrying states exist only at discrete critical energies in the thermodynamic limit, and the spin quantum Hall transition driven by an external Zeeman field has the same critical behavior as the usual integer quantum Hall transition of noninteracting electrons. These critical energies merge and disappear as disorder strength increases, in a manner similar to those in lattice models for integer quantum Hall transition.

We describe a Monte Carlo simulation study of the magnetic phase diagram of diluted magnetic semiconductors doped with shallow impurities in the low concentration regime. We show that because of a wide distribution of interaction strengths, the system exhibits strong quantum effects in the magnetically ordered phase. A discrete spin model, found to closely approximate the quantum system, shows long relaxation times, and the need for specialized cluster algorithms for updating spin configurations. Results for a representative system are presented.

We study the thermodynamics of a one-dimensional quantum spin-1/2 Heisenberg ferromagnetic system with random antiferromagnetic impurity bonds. In the dilute impurity limit, we generalize the modified spinwave theory for random spin chains, where local chemical potentials for spin waves in ferromagnetic spin segments are introduced to ensure zero magnetization at finite temperature. This approach successfully describes the crossover from the behavior of a pure one-dimensional ferromagnet at high temperatures to a distinct Curie behavior due to randomness at low temperatures. We discuss the effects of impurity bond strength and concentration on the crossover and low-temperature ehavior.

We study the behavior of two-dimensional electron gas in the fractional quantum Hall regime in the presence of finite layer thickness and correlated disordered potential. Generalizing the Chern number calculation to many-body systems, we determine the mobility gaps of fractional quantum Hall states based on the distribution of Chern numbers in a microscopic model. We find excellent agreement between experimentally measured activation gaps and our calculated mobility gaps, when combining the effects of both disordered potential and layer thickness. We clarify the difference between mobility gap and spectral gap of fractional quantum Hall states and explain the disorder-driven collapse of the gap and the subsequent transitions from the fractional quantum Hall states to the insulator.

The lattice dynamics of zinc-blende GaN and AlN were studied theoretically using a two-parameter Keating potential together with the long-range Coulomb interactions. Phonon frequencies transformed from those of the wurtzite counterparts were used to determine the two Keating parameters and the effective charge. We present for the first time the phonon dispersion curves for zinc-blende GaN and AlN. An interesting feature was found for the phonons propagating along the [110] direction, which does not exist in other III-V semiconductors.