Universal spatiotemporal scaling of distortions in a drifting lattice

We study the dynamical response to small distortions of a lattice about its uniform state, drifting through a dissipative medium due to an external force, and show, analytically and numerically, that the fluctuations, both transverse and longitudinal to the direction of the drift, exhibit spatiotemporal scaling belonging to the Kardar-Parisi-Zhang universality class. Further, we predict that a colloidal crystal drifting in a constant electric field is linearly stable against distortions and the distortions propagate as underdamped waves.

Figure 1

Schematic diagram to show displacement fields. $\delta$ and $\epsilon$ are the displacements of a particle in $x$ and $z$ direction from its original position.

Figure 2

(a) (top) Log-log plot of equal-time correlators $C^{xx}(l,t=0)$ and $C^{zz}(l,t=0)$ versus spatial separation $l$ and (b) (bottom) log-log plots of equal space-point time-dependent correlators $C^{xx}(l=0,t)$ and $C^{zz}(l=0,t)$ versus $t$. Slopes yield exponents ${\chi }_x,{\chi }_z$ and ${\chi }_x/z_x,{\chi }_z/z_z$, respectively (see text).

Figure 3

Log-log plots of (top) $C^{xx}$ and (bottom) $C^{zz}$ as functions of $t/L^z$, showing data collapse on to single curves after scaling.

Figure 4

Diagrammatic representation of the propagators and noise for ${\varphi }_{+}$ and ${\varphi }_{-}$. Perturbation expansion for the propagators $G_{+}(k,\omega )$, $G_{-}(k,\omega )$. Contracted noise $A_1$, $A_2$, $A_3$.

Figure 5

One-loop diagrams for propagator renormalization.

Figure 6

One-loop diagrams for the renormalization of $A_1$.

Figure 7

One-loop diagrams that contribute to the fluctuation-corrections to the vertex $a_1$.