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. 
Lian Zheng and H. A. Fertig (Univ. Kentucky)