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.

This paper, based on a presentation at the Spintronics 2001 conference, provides a review of our studies on II-VI and III-V Mn-doped Diluted Magnetic Semiconductors. We use simple models appropriate for the low carrier density (insulating) regime, although we believe that some of the unusual features of the magnetization curves should qualitatively be present at larger dopings (metallic regime) as well. Positional disorder of the magnetic impurities inside the host semiconductor is shown to have observable consequences for the shape of the magnetization curve. Below the critical temperature the magnetization is spatially inhomogeneous, leading to very unusual temperature dependence of the average magnetization as well as specific heat. Disorder is also found to enhance the ferromagnetic transition temperature. Unusual spin and charge transport is implied.