In the presence of a conservation law, these arguments fail. Conservative dynamics is appropriate for many systems. Many CDWs have completely gapped Fermi surfaces at low temperatures, and thus have essentially no free carriers to act as a bath. They are, however, also complicated in this regime by long-range Coulomb interactions, which can no longer be screened without this bath. In vortex lattices, one has conservation of magnetic flux (and hence the vortex density) at essentially all temperatures. In addition, the inter-vortex interactions are cutoff by screening on the scale of the magnetic penetration depth.
An immediate consequence of conservative dynamics is that the
Coppersmith-Millis argument no longer goes through. The problem is
that their argument is based on isolated rare regions of the
sample which are anomalously weakly pinned and therefore undergo phase
slip. However, phase slip implies a non-zero current, since
. Clearly current cannot exist in isolated
regions of the sample for long times without an unbounded building of
charge.
The flaw is with the use of the RFXY model, which is inconsistent with conservation of charge. Unfortunately, it is not so clear how to build the required physics into such a model. Clearly, the completely elastic limit is conservative. It is also, however, too highly constrained. Some phase slip processes are certainly allowed even in a number-conserving system. For instance, phase slips along the direction transverse to the chain axis correspond to a large shear of the system by one lattice spacing. Likewise, equal and opposite phase slips along the x axis correspond to the formation of vacancy-interstitial pairs, which are also allowed. Currently, the only consistent models of conservative dynamics for any of these systems are formulated at the level of individual equations of motion for each particle in the system. This approach is adequate for simulation purposes, and is being pursued by several researchers. It is not particularly amenable, however, to analytic approaches.
Without such a formulation, I will instead simply review some of the phenomenology of these systems. Since the Coppersmith-Millis scenario is ruled out, we may wonder whether a non-zero depinning threshold might indeed exist in this case. Indeed, this seems likely, since conservation of particle number would seem to imply that motion must always occur in connected channels which span the entire system along the x direction. These paths can then carry the current. Because the length of these channels scale with the size of the system, the probability that an entire path is sufficiently weakly pinned would be expected to vanish in the thermodynamic limit. It would be interesting to try to formulate a precise version of this argument.
Instead, I will assume that it is the case that for small enough applied
forces, the system remains pinned with probability one. As the force
is increased, it will certainly depin, but this may now occur in at
least two different ways. First, there may simply be an elastic
depinning transition, in which topological defects are in some sense
irrelevant to the critical properties. The system then would behave
very similarly to depinning in elastic models of interfaces and CDWs.
That is, one would have larger and larger, roughly isotropic
avalanches occuring as F is increased, eventually diverging at the
critical force 
. Above threshold, the whole system flows, and
this velocity increases as F is further increased. The critical
behavior above threshold is characterized by large regions of the
sample which remain stuck for long times before getting once again
caught up in the flow.
An alternative possibility occurs in simulations. At some value of the applied force, connected channels can depin, while other regions of the sample remain stuck. As the force is increased beyond threshold, these channels widen, others form, and the ``network'' of flowing regions becomes more dense. Here the critical behavior above threshold is presumably characterized by a coarsening of this network and a increasing sparsity of flowing channels. I will call this scenario ``plastic depinning''.
Both mechanisms may be possible. Physically, the elastic depinning is more likely when the disorder in the sample is weak and relatively smoothly distributed in space. Plastic depinning should be favored when there is a broad distribution of pinning strengths in very sharply defined positions in space. Then one imagines that strong pins can hold some regions fixed while the rest of the system flows about them.
Although there is no theory developed, we may speculate that elastic depinning may even be in the universality class of purely elastic models. Plastic depinning, on the other hand, is entirely novel. Some theories of onset of flow have been developed in fluid models, but it is presently unclear to what extent these will apply here, where elasticity of the fixed regions is clearly important.
Rather simple considerations suggest that the two types of depinning might in fact be distinguishable from their velocity-force curves v(F). In the elastic case, the entire region moves as a whole. Furthermore, it is known that in this case (see section ??)
with 
. This implies that the v(F) curve is convex,
i.e. has infinite derivative at threshold.
In plastic depinning, the situation is rather different. Near
threshold, we expect a very coarse network of channels, with large
length scales 
and 
parallel and
perpendicular to the applied force. If
then the total fraction of sites which are moving thus goes to zero at
as
For reasonable values of 
(usually these are larger than 1/2,
this fraction is very small. We also expect that the velocity within
a channel goes to zero at threshold as some other power,
so the mean velocity behaves as
with 
.
For physical d and reasonable 
, we expect then 
. This implies that the v(F) curve is concave in this
case, and dv/dF is zero at threshold.
Qualitative behavior of this type has indeed been observed in experiments and simulations on vortex lattices. A second distinguishing feature of plastic depinning is metastability and hysteresis in the ``moving state''. Middleton's theorem for elastic models essentially rules this out without plasticity. In the channel picture, metastability is quite natural, since there are stationary regions which may be pinned in various metastable states. Some recent experiments by Eva Andrei show evidence of this (as well as tantalizing similarities to other spin-glass phenomena!).