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

We investigate the universal behavior of the Kardar-Parisi-Zhang (KPZ) equation with temporally correlated noise. The presence of time correlations in the microscopic noise breaks the statistical tilt symmetry, or Galilean invariance, of the original KPZ equation with $\delta$-correlated noise (denoted SR-KPZ). Thus, it is not clear whether the KPZ universality class is preserved in this case. Conflicting results exist in the literature, some advocating that it is destroyed even in the limit of infinitesimal temporal correlations, while others find that it persists up to a critical range of such correlations. Using nonperturbative and functional renormalization group techniques, we study the influence of two types of temporal correlators of the noise: a short-range one with a typical timescale $\tau$, and a power-law one with a varying exponent $\theta$. We show that for the short-range noise with any finite $\tau$, the symmetries (the Galilean symmetry, and the time-reversal one in $1+1$ dimension) are dynamically restored at large scales, such that the long-distance and long-time properties are governed by the SR-KPZ fixed point. In the presence of a power-law noise, we find that the SR-KPZ fixed point is still stable for $\theta$ below a critical value ${\theta }_{\mathrm{th}}$, in accordance with previous renormalization group results, while a long-range fixed point controls the critical scaling for $\theta >{\theta }_{\mathrm{th}}$, and we evaluate the $\theta$-dependent critical exponents at this long-range fixed point, in both $1+1$ and $2+1$ dimensions. While the results in $1+1$ dimension can be compared with previous studies, no other prediction was available in $2+1$ dimension. We finally report in $1+1$ dimension the emergence of anomalous scaling in the long-range phase.

Figure 1

(a) Evolution of the function $f_{\kappa }^D(\varpi ,0)$ with the RG scale for different values of $\tau =0,0.01,0.05,0.1$. For each $\tau$, the function is represented at successive RG times $s=-log(\kappa /\mathrm{\Lambda })$: $s=0$ where they are Gaussians of different width, $s=3$ where the functions for the different $\tau$ are already almost superimposed (erasure of the initial conditions), and $s=40$ where the fixed-point shape is reached, characterized by a power-law decay very close to the KPZ one $\sim {\varpi }^{1/3}$ (indicated as a guideline). (b) Evolution of the functions $f_{\kappa }^D(\varpi ,0)$ and $f_{\kappa }^{\nu }(\varpi ,0)$ with the RG scale for $\tau =0.01$, represented for the RG times $s=0,3,40$. Although they start at $s=0$ from different shapes $f_{\kappa =\mathrm{\Lambda }}^D(\varpi ,0)\ne f_{\kappa =\mathrm{\Lambda }}^{\nu }(\varpi ,0)$, the time-reversal symmetry is restored at $s\lesssim 3$ where they already coincide, up to the fixed point $f_{*}^D(\varpi ,0)=f_{*}^{\nu }(\varpi ,0)$.

Figure 2

Evolution with the RG time $s=-log(\kappa /\mathrm{\Lambda })$ of (a) ${\eta }_{\kappa }^{\lambda }$ and (b) $|{\eta }_{\kappa }^D-{\eta }_{\kappa }^{\nu }|$, for different values of $\tau =0,0.01,0.05,0.1$. One observes that: (a) the time-reversal symmetry is dynamically restored along the flow since ${\eta }_{*}^D={\eta }_{*}^{\nu }$ at the fixed point, which implies that $\chi =1/2$, (b) the Galilean invariance is almost restored: ${\eta }_{*}^{\lambda }$ takes a very small value for all $\tau$. As explained in the text, this residual nonzero violation of Galilean invariance is induced by the ${\mathrm{NLO}}_{\omega }$ ansatz, which implies that $z=2-\chi -{\eta }_{*}^{\lambda }$ slightly deviates (by less than 0.5%) from the SR-KPZ value $z=3/2$.

Figure 3

Evolution with the RG time $s$ in $d=1$ of (a) the LR coupling $w_{\kappa }^{\theta }$ and (b) the violation of Galilean invariance ${\eta }_{\kappa }^{\lambda }$, for different values of $\theta =0,0.1,0.16,0.166,0.17,0.22,0.24$ (from bottom to top) in $d=1$. For $\theta <{\theta }_{\mathrm{th}}$, the flow reaches the SR-KPZ fixed point with $w_{*}^{\theta }=0,{\eta }_{*}^{\lambda }=0$, while for $\theta >{\theta }_{\mathrm{th}}$, a LR fixed point with $w_{*}^{\theta }\ne 0,{\eta }_{*}^{\lambda }\ne 0$ is reached. The critical value is ${\theta }_{\mathrm{th}}=0.166$ confirming the theoretical prediction ${\theta }_{\mathrm{th}}=1/6$. Interestingly, it is clearly identified as the value leading to an algebraic decay of $w_{\kappa }^{\theta }$ and ${\eta }_{\kappa }^{\lambda }$ in the RG time $s$ (bold orange line).

Figure 4

(a) Roughness exponent $\chi$ and (b) dynamical exponent $z$ as a function of $\theta$ in $d=1$ and for different approximation schemes. The exponents take the SR-KPZ values $\chi =1/2$ and $z=3/2$ up to a critical value of $\theta$ very close to the theoretical prediction ${\theta }_{\mathrm{th}}=1/6$. Beyond this value, the exponents vary continuously with $\theta$. The LPA results improves the analytical RG result at one-loop given in Eq. (5), where we recall the authors set ${\eta }_{\kappa }^{\lambda }=0$, but still differ from the NLO results. The NLO and ${\mathrm{NLO}}_{\omega }$ results are very close, despite the slight shift in $z$ at ${\mathrm{NLO}}_{\omega }$ due the the residual breaking a Galilean invariance within this sheme. At NLO, we find an almost linear behavior of $\chi \left(\theta \right)\simeq 1.43\theta +0.26$. It turns out to be reasonably close to numerical estimation from the approximate RG Eqs. (7) for $\chi$, but not for $z$. All the estimates from NPRG lie within the bounds given in Eq. (9) (shaded region).

Figure 5

(a) The LR coupling $w_{\kappa }^{\theta }$ and (b) ${\eta }_{\kappa }^{\lambda }$, for different values of $\theta =0,0.1,0.2,0.3,0.34,0.35,0.4,0.45$ (from bottom to top) and ${\theta }_{\mathrm{th}}=0.346$ (bold orange line) as a function of the RG time $s$, in $d=2$. Two distinct behaviors, corresponding to the SR and the LR fixed points are observed, separated by the critical value ${\theta }_{\mathrm{th}}$ for which $w_{\kappa }^{\theta }$ and ${\eta }_{\kappa }^{\lambda }$ vanish algebraically with $s$.

Figure 6

(a) The roughness $\chi \left(\theta \right)$ and (b) dynamical $z\left(\theta \right)$ critical exponents as a function of the LR exponent $\theta$ in $d=2$, and for different approximation schemes. The reference values for the pure KPZ case, obtained within NLO and from recent numerical simulations [71], are represented as the dashed lines SR-KPZ NLO and SR-KPZ Num., respectively. We find in $d=2$ at NLO two regimes, as in $d=1$. The critical exponents coincide with the SR-KPZ ones below a critical value ${\theta }_{\mathrm{th}}\simeq 0.346$, while beyond this value, we obtain $\theta$-dependent critical exponents. Within the simplified NLO approximation with $f_{\kappa }^{\lambda }=1$, the qualitative picture is the same, although the curves are shifted because the values for the exponents at the SR-KPZ fixed point differ a bit (less than 10 %) within this scheme. The one-loop analytical result Eq. (5) from DRG is also represented for comparison, although in this case, the value for ${\theta }_{\mathrm{th}}$ cannot be obtained from the perturbative analysis. It can be estimated here as the intersection between this prediction and the SR-KPZ values for the exponents.

Figure 7

(a) Nonlinear part $I_{*}^D({\varpi }_0,p)$ of the flow of $f_{\kappa }^D$ at the fixed point as a function of the momentum $p$. For $\theta =0.4$, the contribution of $I_{*}^D$ in the $p\gg 1$ regime is almost three orders of magnitude larger than in the SR-KPZ case $\theta <{\theta }_{\text{th}}$, which indicates that the decoupling is much less effective. (b) Scaling function ${\zeta }^D$ associated with $f_{\kappa }^D$ as a function of the scaling variable $y=\varpi /p^z$. The enhanced contribution of the nonlinear part of the flow at large $\theta$ translates into a nonzero slope in the $y\to 0$ limit. The curves for $\theta =0$ and $\theta =0.15$, which both belong to the SR-KPZ phase, are superimposed.

Figure 8

(a) Critical roughness exponent: ${\chi }^{\left(\kappa \right)}$ determined from the scaling dimension of the renormalization functions $f_{\kappa }^X$ and ${\chi }^{\left(p\right)}$ determined from their large momentum behavior. We observe that while ${\chi }^{\left(\kappa \right)}$ displays a quasi-linear variation with $\theta$ in the LR regime, ${\chi }^{\left(p\right)}$ seems to saturate at large $\theta$. (b) Comparison of the anomalous scaling computed in this work and found in the numerical simulations of Ref. [41]. (The corresponding data points are courtesy of the authors.)

Figure 9

Sketch of the numerical procedure: the blue triangles and green circles represent the grid points where the flows of the functions $f_{\kappa }^X(\varpi ,p)$ are computed from given initial conditions at $s=0$. The orange squares represent the grid points used in the Gauss-Legendre algorithm to compute numerically the integrals over the internal frequency and momentum (the grid for the integration over the angle is not represented in this sketch). The values of the functions in the whole orange-shaded domain are calculated using a spline interpolation. The flows of the functions on the boundaries (green circles) are approximated by only the dimensional (linear) flow.

Figure 10

Roughness exponent $\chi$ as a function of the cutoff parameter $\alpha$ in $d=2$ for different values of $\theta$.

Figure 11

Roughness exponent $\chi$ for $\theta =0.45$ as a function of the normalization point ${\varpi }_0$ in $d=2$.