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Charge density waves

The simplest case is the CDW. The charge density of a CDW is

  equation1092

where we have begun a Fourier expansion. This may be rewritten in a suggestive way as

equation1097

where

equation1102

The complex variable tex2html_wrap_inline3050 will make a suitable CDW order parameter. It's transformation properties are determined by the requirement that the density be a scalar. Under spatial translations tex2html_wrap_inline3476 ,

  equation1106

so that a non-zero expectation value for tex2html_wrap_inline3050 indeed indicates broken translational symmetry. It vanishes in the disordered (liquid) phase, where the density is uniform. In the ordered phase, the phase of tex2html_wrap_inline3050 simply becomes the local phase of the CDW - i.e. it gives the displacement of the density wave.

One may push this idea further by writing a continuum Landau theory for the CDW melting transition. Generally, we expect the free energy to be an analytic function of tex2html_wrap_inline3050 at finite temperature. Near tex2html_wrap_inline2674 , where tex2html_wrap_inline3486 , we should therefore be able to make a Taylor and gradient expansion. The lowest non-trivial terms are then

equation1110

For r<0, the system orders and most of the fluctuations occur in the phase tex2html_wrap_inline3438 . The transformation properties under translations guarantee the XY model form above. Neglecting amplitude fluctuations then recovers the CDW elastic Hamiltonian, Eq. 147. For r>0, by contrast, we have a disordered (liquid) phase, where tex2html_wrap_inline3494 , and order parameter correlations decay exponentially,

equation1116

where the correlation length tex2html_wrap_inline3496 .

One may wonder why I have truncated the Fourier series in Eq. 148 at the first harmonic. A general periodic distortion of the density is not simply sinusoidal, and therefore contains higher harmonics. One might well define order parameters tex2html_wrap_inline3498 by

equation1124

so that tex2html_wrap_inline3500 . In fact, these additional ``order parameters'' are indeed non-zero for the CDW. However, they are in a sense less fundamental.

To see this, consider the generalization of Eq. 151,

equation1130

We might well include the tex2html_wrap_inline3498 in the Hamiltonian above. The transformation properties, however, allow for terms of the form

equation1133

These couplings act like uniform fields on the tex2html_wrap_inline3498 , so that, once tex2html_wrap_inline3050 orders, all the higher m ``order parameters'' are slaved to it and order in a non-critical way. In words, the tex2html_wrap_inline3498 are not independent. This is as expected, since there should only be a single displacement field or phase to describe the low-temperature phase.


next up previous contents
Next: Vortex lattices and Wigner Up: Spatial order parameters and Previous: Spatial order parameters and

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