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.
