Kardar-Parisi-Zhang equation with short-range correlated noise: Emergent symmetries and nonuniversal observables

We investigate the stationary-state fluctuations of a growing one-dimensional interface described by the Kardar-Parisi-Zhang (KPZ) dynamics with a noise featuring smooth spatial correlations of characteristic range $\xi$. We employ nonperturbative functional renormalization group methods to resolve the properties of the system at all scales. We show that the physics of the standard (uncorrelated) KPZ equation emerges on large scales independently of $\xi$. Moreover, the renormalization group flow is followed from the initial condition to the fixed point, that is, from the microscopic dynamics to the large-distance properties. This provides access to the small-scale features (and their dependence on the details of the noise correlations) as well as to the universal large-scale physics. In particular, we compute the kinetic energy spectrum of the stationary state as well as its nonuniversal amplitude. The latter is experimentally accessible by measurements at large scales and retains a signature of the microscopic noise correlations. Our results are compared to previous analytical and numerical results from independent approaches. They are in agreement with direct numerical simulations for the kinetic energy spectrum as well as with the prediction, obtained with the replica trick by Gaussian variational method, of a crossover in $\xi$ of the nonuniversal amplitude of this spectrum.

Figure 1

(a), (c), (e), and (g): RG flow of the effective noise correlation ${\widehat{f}}_k^D(0,\widehat{p})$ and (b), (d), (f), and (h): RG flow of the effective dissipation ${\widehat{f}}_k^{\nu }(0,\widehat{p})$. Different values of the rescaled correlation length are shown: $\widehat{\xi }=0$ (black solid line), $\widehat{\xi }=0.05$ (blue dashed line), $\widehat{\xi }=0.1$ (red dotted line), and $\widehat{\xi }=1$ (green dot-dashed line). The two functions are plotted as functions of the rescaled momentum $\widehat{p}$, for $\widehat{\omega }=0$ and four different values of the RG “time” $s=ln(k/\mathrm{\Lambda })$, as the cutoff scale is lowered (a) and (b) $s=0$, (c) and (d) $s=-1$, (e) and (f) $s=-6$, and (g) and (h) $s=-13$. The axes of all the plots are scaled in the same way. One sees that the time-reversal symmetry (${\widehat{f}}_k^D={\widehat{f}}_k^{\nu }$), which is broken at $s=0$, has emerged at $s\le -6$ and that the same fixed point is reached for all values of $\widehat{\xi }$ when $s\le -13$.

Figure 2

Collapse of two-point correlation functions on large scales for two different values of the microscopic noise correlation length: $\xi =1.6$ (gray) and $\xi =12.8$ (black). (a) The scaling function $F_{\xi }^{\mathrm{\Lambda }}\left(x\right)$ [see Eq. (46)] is represented as a function of $x$ for all momenta and frequencies. This plot shows that different values of $p$ lead to different functions of $x$. There is no data collapse. (b) The universal scaling function $\widehat{F}\left(\widehat{x}\right)={\nu }_{\xi }^2/D_{\xi }{F}_{\xi }^{0.001}\left({\nu }_{\xi }\widehat{x}\right)$ is represented as a function of $\widehat{x}$ for $p<K$ and $\omega <K^{3/2}$ with $K=0.001$. The normalization factors ${\nu }_{\xi }$ and $D_{\xi }$ are obtained from the fitting procedure detailed in Appendix pp4 and are consistent with the solution of the RG flow equations (34). It is clear from (b) that when $K=0.001$, enough UV modes are excluded and $F_{1.6}^{0.001}$ and $F_{12.8}^{0.001}$ are equivalent (up to nonuniversal normalization factors).

Figure 3

Kinetic energy spectrum $\overline{R}$. (a) $\overline{R}$ for different values of $\xi$. (b) $\overline{R}$ (solid black line) as well as its two dominating contributions [see Eq. (51)] ${\overline{R}}_1$ (dotted blue line) and ${\overline{R}}_2$ (dashed red line) and the microscopic noise correlation function $R_{\xi }\left(p\right)$ (dash-dotted green line) for a representative value of $\xi =1.6$. The UV physics is well captured by the bare noise correlator $R_{\xi }$, while the universal KPZ physics (given by ${\overline{R}}_1$) emerges in the IR domain.

Figure 4

Comparison of the NPFRG calculation with direct numerical simulations. The kinetic energy spectrum $\overline{R}$ is computed for $\xi =0.2622$ and ${\xi }_{\tau }=0.0579$ (blue solid line), $\xi =1.7285$ and ${\xi }_{\tau }=1.3428$ (red dashed line), and $\xi =4.9725$ and ${\xi }_{\tau }=7.8134$ (green dotted line). The lines show the NPFRG result and the circles are the results of [66].

Figure 5

Nonuniversal amplitude $\tilde{D}$, in log-log scale, from NPFRG and from numerical simulations: (a) $\tilde{D}$ computed with spatial noise correlations (2), with the NPFRG, and for different values of $\xi$ (blue solid line), as well as a power-law fit of the tail of the data (for $\xi \ge 18$) (black dashed line), and (b) $\tilde{D}$ computed with spatiotemporal correlations (55), with the NPFRG for different values of $\xi$ and ${\xi }_{\tau }\cong 0.539{\xi }^{5/3}$ (blue solid line) with its power-law fit (for $\xi \ge 43$) (black dashed line) as well as two different estimations of $\tilde{D}$ from the numerical computation of [66] (black circles and red squares). The vertical axes of the two plots are scaled in the same way. The horizontal axes are not. The decay of $\tilde{D}\left(\xi \right)$ with $\xi$ is a large-scale signature of the microscopic correlation length [see Eq. (58)]. It can be observed experimentally by varying the coupling $g={\lambda }^{2}D/{\nu }^3$.

Figure 6

Principle of minimum sensitivity. The circles represent the NPFRG results for $\tilde{D}\left(0\right)$ as a function of $\alpha$. The line is a third-order polynomial fit.

Figure 7

Fitting parameters as functions of $K$: (a) ${\alpha }_{\xi }^K$ and (b) ${\beta }_{\xi }^K$. The different levels of gray represent different values of $\xi$, with the same legend as in Fig. 3. If enough of the small-scale physics $K⪅\text{min}(1/\xi ,\mathrm{\Lambda })$ is excluded, Eq. (D2) provides an adequate fitting form for all the scaling functions.