Kardar-Parisi-Zhang growth on square domains that enlarge nonlinearly in time

Fundamental properties of an interface evolving on a domain of size $L$, such as its height distribution (HD) and two-point covariances, are known to assume universal but different forms depending on whether $L$ is fixed (flat geometry) or expands linearly in time (radial growth). The interesting situation where $L$ varies nonlinearly, however, is far less explored and it has never been tackled for two-dimensional (2D) interfaces. Here, we study discrete Kardar-Parisi-Zhang (KPZ) growth models deposited on square lattice substrates, whose (average) lateral size enlarges as $L=L_0+\omega t^{\gamma }$. Our numerical simulations reveal that the competition between the substrate expansion and the increase of the correlation length parallel to the substrate, $\xi \simeq c{t}^{1/z}$, gives rise to a number of interesting results. For instance, when $\gamma <1/z$ the interface becomes fully correlated, but its squared roughness, $W_2$, keeps increasing as $W_2\sim t^{2\alpha \gamma }$, as previously observed for one-dimensional (1D) systems. A careful analysis of this scaling, accounting for an intrinsic width on it, allows us to estimate the roughness exponent of the 2D KPZ class as $\alpha =0.387\left(1\right)$, which is very accurate and robust, once it was obtained averaging the exponents for different models and growth conditions (i.e., for various ${\gamma }^{\prime }\mathrm{s}$ and ${\omega }^{\prime }\mathrm{s}$). In this correlated regime, the HDs and covariances are consistent with those expected for the steady-state regime of the 2D KPZ class for flat geometry. For $\gamma \approx 1/z$, we find a family of distributions and covariances continuously interpolating between those for the steady-state and the growth regime of radial KPZ interfaces, as the ratio $\omega /c$ augments. When $\gamma >1/z$ the system stays forever in the growth regime and the HDs always converge to the same asymptotic distribution, which is the one for the radial case. The spatial covariances, on the other hand, are $(\gamma ,\omega )$-dependent, showing a trend towards the covariance of a random deposition in enlarging substrates as the expansion rate increases. These results considerably generalize our understanding of the height fluctuations in 2D KPZ systems, revealing a scenario very similar to the one previously found in the 1D case.

Figure 1

(a) Illustration of a hole-like substrate for $\gamma >1$. (b) Vertical cross section of a deposit/surface (in red) growing on the substrate shown in (a).

Figure 2

Squared roughness, $W_2$, as a function of time, $t$, for the RSOS model with (a) $\omega =2, \gamma =0.2$, and initial sizes (from bottom to top) $L_0=0$, 4, 8, 16, 32, 64, and 128; and (b) $\omega =2, L_0=0$, and (from bottom to top) $\gamma =1/z$, 0.8, 1.0, and 1.1. The insertion in (a) shows the data in the main plot rescaled according to the Family-Vicsek scaling. In (b), the inset displays the temporal variation of the rescaled roughness $W_2/t^{2\beta }$. The exponents $\alpha =0.387$ and $\beta =0.24$ were used here.

Figure 3

(a) Squared roughness subtracted of the intrinsic width ($W_2-w_2^{*}$) as a function of time $t$, for the RSOS model with $\gamma =0.5$ and several values of $\omega$, as indicated by the legend. The dashed line has the indicated slope. (b) Effective roughness exponents ${\alpha }_{\mathrm{eff}}$ against ${\overline{t}}^{-\mathrm{\Delta }}$, for the SS (two upper curves) and RSOS (four lower curves) models, with several values of $\omega$ [following the same symbol and color schema as in (a)] for $\gamma =0.2$ (open) and 0.5 (closed symbols). The solid lines are linear fits used in the extrapolations. All results here are for an initial size $L_0=0$.

Figure 4

Temporal evolution of the skewness $S$ (left) and kurtosis $K$ (right panels) of the HDs for the RSOS and SS models. (a) and (b) show results for $\gamma <1/z$, while in (c) and (d) data for $\gamma >1/z$ are presented, as indicated by the legends. In all panels, the shaded regions represent the ranges of values found in the literature for $S$ and $K$ of the GR HDs in flat and curved geometries, and these moment ratios for the SSR HDs, for the 2D KPZ class. All data shown here are for $\omega =2$, except that for $\gamma =1.1$, for which $\omega =1$.

Figure 5

(a) Temporal evolution of the skewness $S$ for the RSOS model with $\gamma \approx 1/z$ and different values of $\omega$, as indicated by the legend. (b) Asymptotic values of $S$ (top) and $K$ (bottom) for both models as a function of $\omega /c$. The shaded regions represent the intervals estimated in the literature for $S$ and $K$ for the SSR HD and the GR HDs for flat and curved geometries.

Figure 6

(a) Rescaled spatial covariances $C_S/\left[A{\left(\omega t^{\gamma }\right)}^{2\alpha }\right]$ against ${[r/\left(\omega t^{\gamma }\right)]}^{2\alpha }$ for both models and several values of $\gamma \le 1/z$, as indicated by the legend. Curves measured at several times are shown, the longest time ones being denoted by symbols. The dashed line is the covariance for the SSR of fixed-size systems, for which $\left(\omega t^{\gamma }\right)\to L$ was used in rescaling. Rescaled spatial covariances $C_S/W_2$ versus $r/a$ for: (b) both models with $\omega =2$ and the indicated ${\gamma }^{\prime }\mathrm{s}$; (c) the RSOS model with $\gamma =0.8$ and several ${\omega }^{\prime }\mathrm{s}$; and (d) the SS model with $\gamma \approx 1/z$ and several values of $\omega$. Results for $\omega \in [0.25,4]$, increasing in the direction of the arrows, are shown in (c) and (d), where the insertions highlight the regions inside the boxes in the main panels. Data in (b)–(d) are for times yielding the deposition of $\sim 3\times {10}^9$ particles, which are compared with covariances for the SSR, for the GR in curved geometry ($\gamma =1$), and for a random deposition (RD) on expanding substrates (with $\gamma =1.1, \gamma =0.8$, and $\gamma \approx 1/z$, respectively), as indicated by the legend in (b).