The simplest case is the CDW. The charge density of a CDW is
where we have begun a Fourier expansion. This may be rewritten in a suggestive way as
where
The complex variable 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 ,
so that a non-zero expectation value for indeed indicates broken translational symmetry. It vanishes in the disordered (liquid) phase, where the density is uniform. In the ordered phase, the phase of 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 at finite temperature. Near , where , we should therefore be able to make a Taylor and gradient expansion. The lowest non-trivial terms are then
For r<0, the system orders and most of the fluctuations occur in the phase . 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 , and order parameter correlations decay exponentially,
where the correlation length .
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 by
so that . 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,
We might well include the in the Hamiltonian above. The transformation properties, however, allow for terms of the form
These couplings act like uniform fields on the , so that, once orders, all the higher m ``order parameters'' are slaved to it and order in a non-critical way. In words, the are not independent. This is as expected, since there should only be a single displacement field or phase to describe the low-temperature phase.