Wigner crystal states for the two-dimensional electron gas in a double quantum well system

Using the Hartree-Fock approximation, we calculate the energy of different Wigner crystal states for the two-dimensional electron gas of a double quantum well system in a strong magnetic field. Our calculation takes interlayer hopping as well as an in-plane magnetic field into consideration. The ground The ground state at small layer separations is a one-component triangular lattice Wigner state. As the layer separation is increased, the ground state first undergoes a transition to two stacked square lattices, and then undergoes another transition at an even larger layer separation to a two-component triangular lattice. The range of the layer separation at which the two-component square lattice occurs as the ground state shrinks, and eventually disappears, as the interlayer hopping is increased. An in-plane magnetic field induces another phase transition from a commensurate to a incommensurate state, similar to that of $\nu=1$ quantum Hall state observed recently. We calculate the critical value of the in-plane field of the transition and find that the anisotropy of the Wigner state, {\it i.e.,}, the relative orientation of the crystal and the in-plane magnetic field, has a negligible effect on the critical value for low filling fractions. The effect of this anisotropy on the low-lying phonon energy is discussed. A novel exerimental geometry is proposed in which the parallel magnetic field is used to enhance the orientational correlations in the ground state when the crystal is subject toa random potential.Comment: RevTex 3.0, 22pages, 3figures available upon request. ukcm-xxx

Electronic States of Wires and Slabs of Topological Insulators: Quantum Hall Effects and Edge Transport

We develop a simple model of surface states for topological insulators, developing matching relations for states on surfaces of different orientations. The model allows one to write simple Dirac Hamiltonians for each surface, and to determine how perturbations that couple to electron spin impact them. We then study two specific realizations of such systems: quantum wires of rectangular cross-section and a rectangular slab in a magnetic field. In the former case we find a gap at zero energy due to the finite size of the system. This can be removed by application of exchange fields on the top and bottom surfaces, which lead to gapless chiral states appearing on the lateral surfaces. In the presence of a magnetic field, we examine how Landau level states on surfaces perpendicular to the field join onto confined states of the lateral surfaces. We show that an imbalance in the number of states propagating in each direction on the lateral surface is sufficient to stabilize a quantized Hall effect if there are processes that equilibrate the distribution of current among these channels.Comment: 14 pages, 9 figures include