So far, we have been discussing systems which I called elastic *
manifolds*. These were objects with fixed topology embedded and
fluctuating in a larger disordered space. They arose fairly naturally
as topological defects in ordered phases. The ordered phases
themselves, however, would of course generically contain many of these
defects (a finite density in many situations). Indeed, the models we
have studied are in this sense not extensive. The simplest
classical analog is the behavior of a single classical particle.

We would like now to turn to cases in which the true thermodynamic limit of a system can be described, at least partially, using elasticity. Elasticity arises in bulk systems as a result of symmetry breaking. In particular, Goldstone's theorem tells us that if a system undergoes a spontaneous breaking of a continuous symmetry, low energy elastic modes will result. Furthermore, we would like the elastic degrees of freedom to couple strongly to disorder. Because they represent slow distortions of the order parameter in the low temperature phase, such strong coupling exists only if the disorder distinguishes different values of (and not just gradients in) the order parameter. This type of randomness is known as random-field disorder.

In a very rich set of systems, the broken symmetry in question is
translational invariance. Since translational symmetry is by
definition locally broken by disorder, these are indeed of
random-field type. Spontaneously translationally non-invariant
systems are all around us - every crystalline solid is an example.
For our purposes, however, the crystal must form inside some solid
matrix. The matrix is needed to keep the impurities fixed and provide
the quenched random potential. I will refer to such systems as pinned
periodic *media*.

Thu May 30 08:21:44 PDT 1996