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.