Furthermore, any dynamics problem is also sensitive to conservation laws, which play no role in equilibrium. We must therefore study the dynamics separately in the presence or absence of a conserved density. Consider first the case in which no conservation law applies. This is appropriate for certain CDW phases. A well studied example is the higher-temperature CDW in NBSe . This material has three chains in its unit cell. In a temperature range of 59K to 144K, only one of the three chains forms a CDW, and the others provide itinerant metallic carriers which coexist with it. Charge can thus be exchanged back and forth between the CDW and the Fermi sea.
Provided thermal effects can be neglected (this is far from obvious in real materials, but seems to be a good approximation in some instances), an appropriate model would appear to be simply a driven random-field XY (RFXY) model, i.e.
where F is a uniform force, and H is the RFXY Hamiltonian,
Here the angular brackets denote a sum over nearest neighbors, K is a stiffness constant, and and are the magnitude and direction of the random field at site i. Because this lattice form involves only periodic functions, local phase slips are allowed. We will also need the expression for the current, which is
An argument due to Coppersmith and Millis tells us that once phase slips are allowed, this depinning transition is actually washed out even at zero temperature[16]. Let us suppose that the system is below threshold, . Then at long times, the system is static, and the net force on any region of the sample must be zero. This net force is the sum of the external force, the interaction forces, and the random forces acting on the region, i.e.
where represents a compact region of the sample. Because the interaction forces act pairwise, the elastic forces in the interior of the region cancel. We thus have
where denotes the set of boundary spins. The external elastic force on boundary spin i is
where the prime on the sum indicates that it is taken only over nearest neighbor sites not contained in . Clearly, is bounded,
One can now easily argue that this force balance cannot be satisfied everywhere in an infinite sample. Suppose we divide space up into regions (say cubes) of linear size L. There is some small, but finite probability that mean (spatially averaged) magnitude of the random fields in each region is less than F, no matter how small F is. If this is the case, the local random fields are not strong enough to balance the external force alone. The only hope is that the boundary interactions can complete the balance. But since this contribution only grows with the area of the regions, it cannot do so. Thus in some finite fraction of these regions, force balance is impossible, and some phase slip must occur. Because it occurs in a finite volume fraction of the system, the spatially averaged velocity must be non-zero for any F. In other words, there is no threshold field below which the CDW is completely pinned. The functional form of is, however, expected to be highly non-analytic, because it depends upons these extremely rare weakly pinned regions of the sample.