The simplest illustration is, as usual, the CDW. Topological defects may arise here because is a phase, and is hence defined only modulo . The elementary defects are therefore ``phase slips'', in which jumps discontinuously by . All physical variables remain continuous despite such jumps. From Eq. 148, we see that a gradient in corresponds, however, to a shift in the density,
in units where the inter-chain spacing is 1.
Let us consider a situation in which an extra particle has been inserted on one chain at x=0 in an otherwise well-ordered lattice. As we follow the phase on an adjoing chain, it does not change as we go from the last particle with x<0 to the first with x>0. On the chain of insertion, however, we must change by to accomodate the extra particle. Since the only physical energetics comes from local alignment of the electrons, this extra ``phase slip'' only costs some finite energy locally, and the difference of of adjoining chains far away from the interstitial is perfectly acceptable. A similar picture shows that phase slips of correspond physically to interstitials and vacancies, respectively.
We note that the presence of a small concentration of such point defects does not disrupt the long-range order of . This is because, again, appears only in the exponential, and far from the location of a point defect , where N is an integer. Larger defects, however, can disrupt the order. In particular, consider a configuration in which a column of electrons are inserted for y>0 at x=0. Then jumps by along this line, but is continuous elsewhere. This configuration may be ``relaxed'' into an energetically more favorable one by small adjustments of the electron positions. It has, however, a topological character: on any clockwise closed loop containing the origin, the net variation
This type of topological defect in a phase field is known as a vortex. In two dimensions, it is a point defect, located at the terminus of the inserted line of electrons. In three dimensions, it is a line defect at the end of a plane of inserted particles.