Motivated by the recent suggestion of anisotropic effective exchange interactions between Mn spins in Ga12xMnxAs (arising as a result of spin-orbit coupling), we study their effects in diluted Heisenberg spin systems. We perform Monte Carlo simulations on several phenomenological model spin Hamiltonians, and investigate the extent to which frustration induced by anisotropic exchanges can reduce the low-temperature magnetization in these models and the interplay of this effect with disorder in the exchange. In a model with low coordination number and purely ferromagnetic exchanges, we find that the low-temperature magnetization is gradually reduced as exchange anisotropy is turned on. As the connectivity of the model is increased, the effect of small-to-moderate anisotropy is suppressed, and the magnetization regains its maximum saturation value at low temperatures unless the distribution of exchanges is very wide. To obtain significant suppression of the low-temperature magnetization in a model with high connectivity, as is found for long-range interactions, we find it necessary to have both ferromagnetic and antiferromagnetic exchanges (e.g., as in the Ruderman-Kittel-Kasuya-Yosida interaction). This implies that disorder in the sign of the exchange interaction is much more effective in suppressing magnetization at low temperatures than exchange anisotropy.

The interplay of electron-electron interaction and confining potential can lead to the reconstruction of fractional quantum Hall edges. We have performed exact diagonalization studies on microscopic models of fractional quantum Hall liquids, in finite-size systems with disk geometry, and found numerical evidence of edge reconstruction under rather general conditions. In the present work we have taken into account effects such as layer thickness and Landau-level mixing, which are found to be of quantitative importance in edge physics. Due to edge reconstruction, additional nonchiral edge modes arise for both incompressible and compressible states. These additional modes couple to electromagnetic fields and thus can be detected in microwave conductivity measurements. They are also expected to affect the exponent of electron Green's function, which has been measured in tunneling experiments. We have studied in this work the electric dipole spectral function that is directly related to the microwave conductivity measurement. Our results are consistent with the enhanced microwave conductivity observed in experiments performed on samples with an array of antidots at low temperatures, and its suppression at higher temperatures. We also discuss the effects of the edge reconstruction on the single-electron spectral function at the edge.

We study the v=1/3 quantum Hall state in the presence of random disorder. We calculate the topologically invariant Chern number, which is the only quantity known at present to distinguish unambiguously between insulating and current carrying states in an interacting system. The mobility gap can be determined numerically this way and is found to agree with experimental value semiquantitatively. As the disorder strength increases towards a critical value, both the mobility gap and plateau width narrow continuously and ultimately collapse, leading to an insulating phase.

In a microscopic model of fractional quantum Hall liquids with electron-electron interactions and confinement, we calculate the edge Green's function via exact diagonalization. Our results for υ=1/3 and 2/3 suggest that, in the presence of Coulomb interaction, "extelalal" parameters such as the sharpness of the edge and the strength of the edge confining potential, which can lead to edge reconstruction, may cause deviations from universality in the edge-tunneling I-V exponent. In particular, we do not find any direct dependence of this exponent on the range of the interaction potential as suggested by recent calculations in contradiction to the topological nature of the edge.

We investigate tile effect of mass anisotropy on tile Wigner crvstallization transition in a two-dimensional (2D) electron gas. Tile static and dynamical properties of a 2D Wigner crysstal have been calculated for arbitrary 2D Bravais lattices in tile presence of anisotropic mass, as may be obtainable in Si metal-oxide-selniconductor field-effect transistors with (110) surface, as well as tile newly discovered organic senticonduc-tor systems. By studying the stability of all possible lattices, we find significant changes in tile crystal structure and melting density of the electron lattice with tile lowest ground state energy.