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Phase slips

The simplest illustration is, as usual, the CDW. Topological defects may arise here because tex2html_wrap_inline3438 is a phase, and is hence defined only modulo tex2html_wrap_inline2792 . The elementary defects are therefore ``phase slips'', in which tex2html_wrap_inline3438 jumps discontinuously by tex2html_wrap_inline3550 . All physical variables remain continuous despite such jumps. From Eq. 148, we see that a gradient in tex2html_wrap_inline3438 corresponds, however, to a shift in the density,

equation1185

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 tex2html_wrap_inline3438 by tex2html_wrap_inline2792 to accomodate the extra particle. Since the only physical energetics comes from local alignment of the electrons, this extra tex2html_wrap_inline2792 ``phase slip'' only costs some finite energy locally, and the difference of tex2html_wrap_inline2792 of adjoining chains far away from the interstitial is perfectly acceptable. A similar picture shows that phase slips of tex2html_wrap_inline3550 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 tex2html_wrap_inline3050 . This is because, again, tex2html_wrap_inline3438 appears only in the exponential, and far from the location of a point defect tex2html_wrap_inline3576 , 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 tex2html_wrap_inline3438 jumps by tex2html_wrap_inline2792 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 tex2html_wrap_inline3588 containing the origin, the net variation

equation1189

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.


next up previous contents
Next: Dislocations Up: Topological defects Previous: Topological defects

Leon Balents
Thu May 30 08:21:44 PDT 1996