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.

Recent experiments in the integer quantum Hall regime seem to find direct transitions from quantum Hall states with Hall conductance σ=ne2/h with integer n>1, to an insulating state in weak magnetic fields. We study the issue of quantmn Hall transitions with change in Hall conductance ne2/h with n>1 using a variation of the tight-binclJng lattice model for noninteracting electrons. Although such transitions do exit in our model for special tuning of parameters, they generically split, with the Hall conductance changing by e2/h at each transition. This suggests that a generic multiple step quantum Hall transition is incompatible with a noninter-acting electron picture.

We study the interplay of electron-electron interaction, confining potential and effects of finite temperature at the edge of a quantum Hall liquid. Our exact diagonalization calculation indicates that edge reconstruction occurs in the fractional quantum Hall regime for a variety of confining potential, including ones that correspond to a "sharp" edge. Our finite temperature Hartree-Fock calculation for integer quantum Hall edges indicates that reconstruction is suppressed above a certain temperature. We discuss the implication of our results on recent edge tunneling and microwave absorption experiments.

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.