Pinned Wigner crystals

We study the effects of weak Gaussian disorder oil various physical and dyn amical properties of the pinned Wigner crystal. We use the replica based Ga ussian variational method to study the problem. The pinned crystal is the q uantum analog of the Bragg grass phase seen in vortex lattices. We obtain t he full frequency dependence of the dynamical magneto-conductivities. This allows us to extract various features of the pinning peak which can be comp ared directly with experimentally measured quantities. We also calculate th e magnetic field and density dependences of these quantities. We find that the physical properties of the crystal depend crucially on the relative mag nitudes of the disorder correlation length and the cyclotron length. This h as dramatic experimental consequences. Our results are of extreme relevance to the recent experiments on two dimensional Ga-As quantum Hall samples an d Si MOSFETS, where the insulating phase seen at low densities and/or high magnetic fields is expected to be a pinned Wigner crystal.