Under this assumption, the phase (for the CDW) and displacement (for
the lattice) fields are single-valued and give a faithful description
of the configuration of the system. The only additional ingredient
needed from the previous sections elastic description is the coupling
to disorder. This is of the random-field type, and may be obtained
physically by including a random potential 
via
I will work out here the form for the case of a CDW. Using the Fourier expansion for the density,
one finds, up to a constant, that
Here, 
and 
are Fourier components of the original
random potential. For slowly varying 
, they may all be taken to
be roughly independent random variables. The full Hamiltonian can
be written
where 
and
is a random periodic potential. Eq. 172 is the continuum Hamiltonian of the random-field XY model, neglecting vortices. A similar Hamiltonian obtains for the case of the lattice.
Since, as we have already remarked, 
, we need only
consider the behavior for 
. As we have seen for random
manifolds, in this case perturbation theory in 
breaks down, and
it is a relevant operator in the sense of the RG.
