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
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.