Coexistence of short- and large-scale phase variations in a charge-density wave weakly coupled to impurities

We reexamine the problem of phase pinning of a three-dimensional charge-density wave (CDW) weakly coupled to impurities. Within an analytical approach, we find that the phase is adjusted both at the short scale, around individual impurity sites, and at the large scale, over extended domains containing many impurities. Both phase adjustments are important for the physical behavior and contribute to the understanding of experimental data as well as numerical simulation results in CDW systems. The former give the main contribution to energy and are responsible for the white-line effect observed in x-ray patterns. The latter determine the functional dependence of the threshold field ${\mathit{E}}_{\mathrm{th}}$ for nonlinear conduction. Unlike what was widely accepted, we find that ${\mathit{E}}_{\mathrm{th}}$ is not related to the total energy gained by phase spatial variations. This picture is consistent with that emerging from the recent experimental investigations of doped ${\mathrm{NbSe}}_3$ and could contribute to clarifying the controversy on the pinning type in this material.