arXiv1997
We study periodic lattices, such as vortex lattices in type II superconductors in a random pinning potential. For the static case we review the prediction that the phase diagram of such systems consists of a topologically ordered Bragg glass phase, with quasi long range translational order, at low fields. This Bragg glass phase undergoes a transition at higher fields into another glassy phase, with dislocations, or a liquid. This proposition is compatible with a large number of experimental results on BSCCO or Thalium compounds. Further experimental consequences of our results and relevance to other systems will be discussed. When such vortex systems are driven by an external force, we show that, due to periodicity in the direction transverse to motion, the effects of static disorder persist even at large velocity. In $d=3$, at weak disorder, or large velocity the lattice forms a topologically ordered glass state, the ``moving Bragg glass'', an anisotropic version of the static Bragg glass. The lattice flows through well-defined, elastically coupled, static channels. We determine the roughness of the manifold of channels and the positional correlation functions. The channel structure also provides a natural starting point to study the influence of topological defects such as dislocations. In $d=2$ or at strong disorder the channels can decouple along the direction of motion leading to a ``smectic'' like flow. We also show that such a structure exhibits an effective transverse critical pinning force due to barriers to transverse motion, and discuss the experimental consequences of this effect.