Nonperturbative renormalization group for the stationary Kardar-Parisi-Zhang equation: Scaling functions and amplitude ratios in 1+1, 2+1, and 3+1 dimensions

We investigate the strong-coupling regime of the stationary Kardar-Parisi-Zhang equation for interfaces growing on a substrate of dimension $d=1$, 2, and 3 using a nonperturbative renormalization group (NPRG) approach. We compute critical exponents, correlation and response functions, extract the related scaling functions, and calculate universal amplitude ratios. We work with a simplified implementation of the second-order (in the response field) approximation proposed in a previous work [Phys. Rev. E 84, 061150 (2011) and Phys. Rev. E 86, 019904(E) (2012)], which greatly simplifies the frequency sector of the NPRG flow equations, while keeping a nontrivial frequency dependence for the two-point functions. The one-dimensional scaling function obtained within this approach compares very accurately with the scaling function obtained from the full second-order NPRG equations and with the exact scaling function. Furthermore, the approach is easily applicable to higher dimensions and we provide scaling functions and amplitude ratios in $d=2$ and $d=3$. We argue that our ansatz is reliable up to $d\simeq 3.5$.

Figure 1

Variations of the roughness exponent $\chi$ with the cutoff parameter $\alpha$ from the NPRG approach within the LO and NLO approximations in integer dimensions $d$. In $d=1$ the curves are computed with the NLO ansatz, either imposing the time-reversal Ward identity Eq. (26) (FDT)—in which case NLO coincides with LO—or relaxing it (no FDT).

Figure 2

Comparison of the one-dimensional scaling function $\tilde{f}\left(k\right)$; exact result [5], NPRG at NLO (respecting FDT, this work), and NPRG at SO [42]. The inset shows the stretched exponential behavior of the tail with the superimposed oscillations, developing on the same scale $k^{3/2}$. Note the vertical scale: this behavior sets in with amplitudes below typically ${10}^{-6}$.

Figure 3

Comparison of the one-dimensional scaling function $f\left(y\right)$; exact result [5], NPRG at NLO (respecting FDT, this work), and NPRG at SO [42].

Figure 4

Normalized scaling function ${\mathring{F}}_N$ (relative to the correlation function) from NPRG at NLO in dimensions 1, 2, and 3.

Figure 5

Real and imaginary parts of the normalized scaling function ${\mathring{H}}_N$ (relative to the response function) from NPRG at NLO in dimensions 1, 2, and 3.

Figure 6

Ratio of the scaling function ${\mathring{F}}_N$ and of the real part of the scaling function ${\mathring{H}}_N$ in dimensions 1, 2, and 3, which illustrates the departure from the generalized FDT.

Figure 7

Comparison of one-dimensional scaling functions ${\mathring{F}}_N$ obtained from different NPRG approximations: SO, NLO with FDT, and NLO without FDT. The scaling function from the SO ansatz [42] also respects FDT. (Inset) Violation of generalized FDT for the NLO ansatz without the explicit constraint ${\widehat{f}}_{\kappa }^{\lambda }=1$.