From bulk descriptions to emergent interfaces: Connecting the Ginzburg-Landau and elastic-line models

Controlling interfaces is highly relevant from a technological point of view. However, their rich and complex behavior makes them very difficult to describe theoretically, and hence to predict. In this work, we establish a procedure to connect two levels of descriptions of interfaces: for a bulk description, we consider a two-dimensional Ginzburg-Landau model evolving with a Langevin equation, and boundary conditions imposing the formation of a rectilinear domain wall. At this level of description no assumptions need to be done over the interface, but analytical calculations are very difficult to handle, especially for disordered systems. On a different level of description, we consider a one-dimensional elastic line model evolving according to the Edwards-Wilkinson equation, which only allows one to study continuous and univalued interfaces, but which was up to now one of the most successful tools to treat interfaces analytically. To establish the connection between the bulk description and the interface description, we propose a simple method which has the advantage to be readily applicable to disordered systems. We probe the connection by numerical simulations at both levels for clean and disordered systems, and our simulations, in addition to making contact with experiments, allow us to test and provide insight to develop new analytical approaches to treat interfaces.

Figure 1

Snapshot of part of a system after solving numerically the Langevin equation (see text) for a 2D Ginzburg-Landau model [Eq. (3), with $\eta =\alpha =\delta =\gamma =1$, $T=0.05$, $t={10}^5$] to obtain the evolution of the order parameter $\phi (x,y)$. The obtained interface for this system is also shown in black. One of the fitted soliton profiles ${\phi }^{*}\left(x\right)$ (for fixed $y$) is highlighted in dashed blue line. Inset: The hyperbolic profile ${\phi }^{*}\left(x\right)$ from Eq. (7), its derivative (which characterizes the “density” of the interface), and three typical states in the local double-well potential.

Figure 2

Time dependence of the roughness $B(r,t)$, computed for interfaces in a 2D Ginzburg-Landau system (bottom figures) and for an equivalent 1D Edwards-Wilkinson system (top figures), obtained for ten realizations (left figures) and for the average over ten realizations of simulations which evolved during a time $t={10}^j$, $j=1,...,6$ (indicated by different colors), starting from a completely flat configuration. The analytical prediction of the evolution of $B(r,t)$ [Eq. (19)] for an equivalent one-dimensional interface is shown on dashed colored lines for different evolution times. The asymptotic value $\frac{T}{c}r$, expected for a completely stationarized interface, is shown in black dotted lines. On the right, the final extracted interfaces for one of the realizations after each evolution time $t$ are shown for both models. A portion of length $25.6\times 25.6$ of a Ginzburg-Landau simulated system is also shown after the evolution times of $t=10$ and $t={10}^6$, along with the detected interface.

Figure 3

Temperature dependence of the roughness $B(r,T)$ for a 2D Ginzburg-Landau system (bottom figures) and for an equivalent 1D Edwards-Wilkinson system (top figures), obtained for ten realizations (left figures) and for the average over ten realizations of simulations which evolved during a time $t={10}^3$ at temperatures $T=0.05,0.07,0.1,0.15,0.2,0.3$ (indicated by different colors), starting from a completely flat configuration. The analytical predictions of the evolution in time of $B(r,T)$ [Eq. (19)] for an equivalent one-dimensional interface is shown on dashed colored lines for different temperatures. The final interfaces obtained for one realization are also shown for both models at different temperatures. A portion of the Ginzburg-Landau system is also shown at $T=0.05$ and $T=0.3$, along with the detected interface, shown in black.

Figure 4

Computed correlations of 256 independent realizations of pinning forces $F_p(u,y)$ (in gray). The average of these correlations is plotted in pink, showing an excellent agreement with the expected correlations given by $\mathrm{\Gamma }\left(u\right)$ [Eq. (26)], shown in the dashed black line. On the inset, four different realizations of pinning forces are shown.

Figure 5

Comparison of observables for a 2D Ginzburg-Landau system (continuous lines) and for an equivalent 1D Edwards-Wilkinson system (dot-dashed lines), obtained after averaging over ten realizations of simulations which evolved during different times $t$, indicated by different colors, at temperature $T=0.05$ and with disorder intensity $\epsilon =0.1$ starting from a completely flat configuration. On the top figure, $B(r,t)$ for the larger simulation times show deviations from the thermal regime (dotted black line). For these larger times, $B(r,t)$ is characterized by the roughness exponent ${\zeta }_{RB}=2/3$. On the bottom figure, we show the structure factor $S(q,t)$, defined in Eq. (18).