Beyond dynamic scaling: rare events break universality

Abstract

Surface growth driven by non-monomeric deposition has remained largely unexplored. We investigate a model based on the deposition of blobs with a power-law size distribution $P(s)\sim s^{-τ}$. We find that the critical exponents vary continuously with $τ$, recovering Kardar--Parisi--Zhang behavior only for $τ\ge 3$. For $τ<3$, roughness scaling exhibits strong corrections and scale invariance breaks down. We show that this behavior originates from the emergence of a second dynamical length scale $ζ$, corresponding to the linear size of the largest cluster, in addition to the usual correlation length $ξ$. The coexistence of these two relevant scales signals the breakdown of the usual Family--Vicsek scaling. These results point to a new phenomenology of surface growth beyond the standard scale-invariant paradigm.

Figures (10)

• Figure 1: Snapshots of the surface for $\tau=3.5$ (left) and $\tau=2.5$ (right). The system size is $L=1000$, and the number of deposited clusters is chosen such that the average height reaches $h \approx 1000$ in both cases. The red line indicates the final interface.
• Figure 2: Measured roughness and dynamic exponents $\alpha$ and $z$ as a function of $\tau$. For $\tau<3$, the critical exponents depend continuously on $\tau$, while for $\tau\geq3$ (highlighted in grey) the KPZ universality class is recovered. Dotted lines indicate the KPZ universal exponents. The determination of the exponents and the associated error bars are discussed in the SM.
• Figure 3: Dynamic scaling breakdown. (Top) Family-Vicsek collapse for $\tau =3$. Dotted line : predicted growth exponent $\beta_{\text{th}}=1/3$. (Bottom) Failure of the dynamic scaling ansatz. For $\tau =2$, the FV ansatz does not hold : the curves do not collapse on the whole time range on a mastercurve upon normalization, and there are strong, system-size dependent corrections to the roughness which effectively does not scale as $t^{\beta}$ (here $\beta=0.6$). Close to the saturation point, the curves collapse and can be described by a decreasing effective exponent $\beta_\text{e}(t)$. The dash-dotted line shows a local tangent to the curve with $\beta_\text{e} = 0.3$. The vertical dotted line represents the approximate (rescaled) time at which curves collapse again. The dashed line represents the scaling with the exponent $\beta_{th}$ predicted by the Family-Vicsek ansatz.
• Figure 4: Effective growth exponent $\beta_e$. The effective growth exponent $\beta_e=\mathrm{d} \log W(L,t)/\mathrm{d} \log t$ is shown as a function of the rescaled time $t/L^z$, for system sizes varying from $L=10^3$ to $10^5$. The exponent predicted by the Family-Vicsek ansatz $\alpha/z$ is displayed as a dashed line. (Left) In the case $\tau=3$, the effective slopes are independent of the system size and hence collapse on a master curve which is constant at small $t$, and null at large times. (Right) For $\tau=2$, (right) the corrections display a complex, system size-dependent shape : firstly, it decays from $\alpha/z$ slowly until reaching a point $\beta_c \approx 0.3$. Beyond this point, the curves collapse (upon appropriate time-rescaling by $L^z$) and there is a crossover to the saturation regime.
• Figure 5: Sketch of the crossover mechanism. For illustration we take $\tau=2$. The correlation length $\xi \sim t^{1/z}$ is shown as a purple dash--dotted line, using a prefactor inferred from Fig. \ref{['fig:collbreakdown']}. The additional scale $\zeta \sim (L t)^{1/[2(\tau - 1)]}$ (here for $L=10^5$) is shown as the blue dotted line. Changing $L$ shifts this curve vertically while preserving its slope. Axes are rescaled so as to factor out trivial system-size dependence and to display the saturation region where $\xi/L\sim1$.