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Inclusion of disorder


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 tex2html_wrap_inline3630 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, tex2html_wrap_inline3632 and tex2html_wrap_inline3634 are Fourier components of the original random potential. For slowly varying tex2html_wrap_inline3438 , they may all be taken to be roughly independent random variables. The full Hamiltonian can be written


where tex2html_wrap_inline3638 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, tex2html_wrap_inline3640 , we need only consider the behavior for tex2html_wrap_inline3642 . As we have seen for random manifolds, in this case perturbation theory in tex2html_wrap_inline3644 breaks down, and it is a relevant operator in the sense of the RG.

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