Coulomb disorder in periodic systems: Effect of unscreened charged impurities

We study the effect of unscreened charged impurities on periodic systems. We show that the long-wavelength component of the disorder becomes long ranged and dominates static correlation functions. On the other hand, because of the statistical tilt symmetry, dynamical properties such as pinning remain unaffected. As a concrete example, we focus on the effect of Coulombian disorder generated by charged impurities on three-dimensional charge density waves with nonlocal elasticity. We calculate the x-ray intensity and find that it is identical to the one produced by thermal fluctuations in a disorder-free smectic-$A$ phase. We discuss the consequences of these results for experiments.

Figure 1

A sketch of the x-ray intensity in a CDW with Coulomb elasticity and charged impurities for $K,Q$ parallel to the chain direction. We have taken $\kappa =0.5$ in the expressions of $I_a$ and $I_d$.

Figure 2

The graph of the function $F_{\delta }(qa∕\pi )$, of compact support for a value of $\delta =0.1$ (solid line). Dotted lines represent $F_{\delta }(qa∕\pi \pm 2)$. We can graphically check the validity of Eq. (A3).

Figure 3

The ground state of an array of CDWs coupled by a repulsive interaction.