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

Coulomb drag between disordered two-dimensional electron gas layers

We derive and evaluate expressions for the frictional Coulomb drag between disordered two-dimensional electron gas layers. Our derivation is based on the memory-function formalism and the expression for the drag reduces to previously known results in the ballistic limit. We find that Coulomb drag is appreciably enhanced by disorder at low temperatures when the mean-free-path within a layer is comparable to or shorter than the layer separation. In high mobility two-dimensional electron gas systems, where the drag has been studied experimentally, the effect of disorder on the drag is negligible at attainable temperatures. We predict that an enhancement due to disorder and a crossover in the temperature-dependence and layer-separation dependence will be observable at low temperatures in moderate and low mobility samples.Comment: 17 pages, revtex, iucm93-00

The Distance Coloring of Graphs

Let $G$ be a connected graph with maximum degree $\Delta \ge 3$. We investigate the upper bound for the chromatic number $\chi_\gamma(G)$ of the power graph $G^\gamma$. It was proved that $\chi_\gamma(G) \le\Delta\frac{(\Delta-1)^{\gamma}-1}{\Delta-2}+1=:M+1$ with equality if and only $G$ is a Moore graph. If $G$ is not a Moore graph, and $G$ holds one of the following conditions: (1) $G$ is non-regular, (2) the girth $g(G) \le 2\gamma-1$, (3) $g(G) \ge 2\gamma+2$, and the connectivity $\kappa(G) \ge 3$ if $\gamma \ge 3$, $\kappa(G) \ge 4$ but $g(G) >6$ if $\gamma =2$, (4) $\Delta$ is sufficiently large than a given number only depending on $\gamma$, then $\chi_\gamma(G) \le M-1$. By means of the spectral radius $\lambda_1(G)$ of the adjacency matrix of $G$, it was shown that $\chi_2(G) \le \lambda_1(G)^2+1$, with equality holds if and only if $G$ is a star or a Moore graph with diameter 2 and girth 5, and $\chi_\gamma(G) < \lambda_1(G)^\gamma+1$ if $\gamma \ge 3$