Kardar-Parisi-Zhang equation with temporally correlated noise: A nonperturbative renormalization group approach

Emergence in kinetic roughening with long-range temporal correlations

The role of long-range temporal correlations in kinetic roughening processes is significant, potentially influencing surface morphologies and dynamic scaling properties. This type of correlation, acting as a nonlocal interaction during growth processes, disrupts the self-affine fractal structure by causing the emergence of nontrivial global properties from individual local interactions. In this work, two typical discrete growth models, including random deposition and ballistic deposition, are investigated numerically in the presence of long-range temporal correlations. Two quantitative measurement methods, the Hellinger distance and the novelty detection, are used independently to determine whether emergence occurs. Our simulation results indicate that, above a critical threshold of temporal correlation exponent, the systems exhibit emergence behavior, consistent with the existence of nontrivial scaling properties and surface morphologies when long-range temporal correlations are presented.

Fragility of Kardar-Parisi-Zhang universality class in the presence of temporally correlated noise

We study numerically a family of surface growth models that are known to be in the universality class of the Kardar-Parisi-Zhang equation when driven by uncorrelated noise. We find that, in the presence of noise with power-law temporal correlations with exponent θ, these models exhibit critical exponents that differ both quantitatively and qualitatively from model to model. The existence of a threshold value for θ below which the uncorrelated fixed point is dominant occurs for some models but not for others. In some models the dynamic exponent z(θ) is a smooth decreasing function, while it has a maximum in other cases. Despite all models sharing the same symmetries, critical exponents turn out to be strongly model dependent. Our results clearly show the fragility of the universality class concept in the presence of long-range temporally correlated noise.

Finite-size and finite-time scaling for kinetic rough interfaces

We consider discrete models of kinetic rough interfaces that exhibit space-time scale invariance in height-height correlation. We use the generic scaling theory of Ramasco et al. [Phys. Rev. Lett. 84, 2199 (2000)0031-900710.1103/PhysRevLett.84.2199] to confirm that the dynamical structure factor of the height profile can uniquely characterize the underlying dynamics. We apply both finite-size and finite-time scaling methods that systematically allow an estimation of the critical exponents and the scaling functions, eventually establishing the universality class accurately. The finite-size scaling analysis offers an alternative way to characterize the anomalous rough interfaces. As an illustration, we investigate a class of self-organized interface models in random media with extremal dynamics. The isotropic version shows a faceted pattern and belongs to the same universality class (as shown numerically) as the Sneppen model (version A). We also examine an anisotropic version of the Sneppen model and suggest that the model belongs to the universality class of the tensionless Kardar-Parisi-Zhang (tKPZ) equation in one dimension.

Unpredicted Scaling of the One-Dimensional Kardar-Parisi-Zhang Equation

The celebrated Kardar-Parisi-Zhang (KPZ) equation describes the kinetic roughening of stochastically growing interfaces. In one dimension, the KPZ equation is exactly solvable and its statistical properties are known to an exquisite degree. Yet recent numerical simulations in the tensionless (or inviscid) limit of the KPZ equation [C. Cartes et al., The Galerkin-truncated Burgers equation: Crossover from inviscid-thermalized to Kardar-Parisi-Zhang scaling, Phil. Trans. R. Soc. A 380, 20210090 (2022).PTRMAD1364-503X10.1098/rsta.2021.0090; E. Rodríguez-Fernández et al., Anomalous ballistic scaling in the tensionless or inviscid Kardar-Parisi-Zhang equation, Phys. Rev. E 106, 024802 (2022).PRESCM2470-004510.1103/PhysRevE.106.024802] unveiled a new scaling, with a critical dynamical exponent z=1 different from the KPZ one z=3/2. In this Letter, we show that this scaling is controlled by a fixed point which had been missed so far and which corresponds to an infinite nonlinear coupling. Using the functional renormalization group (FRG), we demonstrate the existence of this fixed point and show that it yields z=1. We calculate the correlation function and associated scaling function at this fixed point, providing both a numerical solution of the FRG equations within a reliable approximation, and an exact asymptotic form obtained in the limit of large wave numbers. Both scaling functions accurately match the one from the numerical simulations.

Correlated noise and critical dimensions

In equilibrium, the Mermin-Wagner theorem prohibits the continuous symmetry breaking for all dimensions d≤2. In this work, we discuss that this limitation can be circumvented in nonequilibrium systems driven by the spatiotemporally long-range anticorrelated noise. We first compute the lower and upper critical dimensions of the O(n) model driven by the spatiotemporally correlated noise by means of the dimensional analysis. Next we consider the spherical model, which corresponds to the large-n limit of the O(n) model and allows us to compute the critical dimensions and critical exponents, analytically. Both results suggest that the critical dimensions increase when the noise is positively correlated in space and time and decrease when anticorrelated. We also report that the spherical model with the correlated noise shows the hyperuniformity and giant number fluctuation even well above the critical point.

Stirred Kardar-Parisi-Zhang Equation with Quenched Random Noise: Emergence of Induced Nonlinearity

We study the stochastic Kardar-Parisi-Zhang equation for kinetic roughening where the time-independent (columnar or spatially quenched) Gaussian random noise f(t,x) is specified by the pair correlation function ⟨f(t,x)f(t′,x′)⟩∝δ(d)(x−x′), d being the dimension of space. The field-theoretic renormalization group analysis shows that the effect of turbulent motion of the environment (modelled by the coupling with the velocity field described by the Kazantsev-Kraichnan statistical ensemble for an incompressible fluid) gives rise to a new nonlinear term, quadratic in the velocity field. It turns out that this “induced” nonlinearity strongly affects the scaling behaviour in several universality classes (types of long-time, large-scale asymptotic regimes) even when the turbulent advection appears irrelevant in itself. Practical calculation of the critical exponents (that determine the universality classes) is performed to the first order of the double expansion in ε=4−d and the velocity exponent ξ (one-loop approximation). As is the case with most “descendants” of the Kardar-Parisi-Zhang model, some relevant fixed points of the renormalization group equations lie in “forbidden zones”, i.e., in those corresponding to negative kinetic coefficients or complex couplings. This persistent phenomenon in stochastic non-equilibrium models requires careful and inventive physical interpretation.

Extensive numerical simulations of surface growth with temporally correlated noise

Surface growth processes can be significantly affected by long-range temporal correlations. In this work, we perform extensive numerical simulations of a (1+1)- and (2+1)-dimensional ballistic deposition (BD) model driven by temporally correlated noise, which is regarded as the temporal correlated Kardar-Parisi-Zhang universality class. Our results are compared with the existing theoretical predictions and numerical simulations. When the temporal correlation exponent is above a certain threshold, BD surfaces develop gradually faceted patterns. We find that the temporal correlated BD system displays nontrivial dynamic properties, and the characteristic roughness exponents satisfy α≃α_{loc}<α_{s} in (1+1) dimensions, which is beyond the existing dynamic scaling classifications.