For the more general problem of a two-dimensionally periodic system, we have somewhat more general defects known as dislocations. In this case, such defects are allowed because the displacement fields are defined only modulo a lattice vector. The analogous topological constraint is
where is a real-space lattice vector.
Vortices and dislocations necessarily have long-range phase and displacement fields, and therefore have a much stronger effect upon long-range order of the system. In the absence of disorder, it can be argued that they are not present at large scales in the CDW and vortex lattice phases. However, these arguments must be reexamined after we ``solve'' the problem of the purely elastic medium interacting with randomness.
One may view the existence of the disordered or liquid phase (as discussed in the previous subsection) as due to the proliferation of such defects. Without these defects to relax the topological constraints of a perfectly connected lattice, it would be impossible for the CDW or vortex lattice to melt. Indeed, one may now worry that a particular system may never order at any finite temperature in the presence of disorder. If this is the case, we say that the system is below its lower critical dimension, . In the pure systems, , so we can still adopt the elastic description at long scales. In the random case, however, the lower critical dimension is not generally known, but surely exists. We expect that the pure system result provides a lower bound,