Nonperturbative renormalization group for the Kardar-Parisi-Zhang equation: General framework and first applications

We present an analytical method, rooted in the nonperturbative renormalization group, that allows one to calculate the critical exponents and the correlation and response functions of the Kardar-Parisi-Zhang (KPZ) growth equation in all its different regimes, including the strong-coupling one. We analyze the symmetries of the KPZ problem and derive an approximation scheme that satisfies the linearly realized ones. We implement this scheme at the minimal order in the response field, and show that it yields a complete, qualitatively correct phase diagram in all dimensions, with reasonable values for the critical exponents in physical dimensions. We also compute in one dimension the full (momentum and frequency dependent) correlation function, and the associated universal scaling function. We find a very satisfactory quantitative agreement with the exact result from Prähofer and Spohn [J. Stat. Phys. 115, 255 (2004)]. In particular, we obtain for the universal amplitude ratio $g_0\simeq 1.149\left(18\right)$, to be compared with the exact value $g_0=1.1504...$ (the Baik and Rain [J. Stat. Phys. 100, 523 (2000)] constant). We emphasize that all these results, which can be systematically improved, are obtained with sole input the bare action and its symmetries, without further assumptions on the existence of scaling or on the form of the scaling function.

Figure 1

(a) Flow diagram stemming from our (simplified) minimal order approximation in the $(d,\widehat{g}{v}_d)$ plane (with $v_d^{-1}=2^{d+1}{\pi }^{d/2}\Gamma \left[\frac{d}{2}\right]$ a normalization factor related to the integration volume). (Red circles) Renormalized value ${\widehat{g}}_{\mathit{SC}}^{*}$ at $F_{\mathit{SC}}$. (Dashed purple line) Gaussian fixed point $F_{\mathit{EW}}$. (Cyan squares) Bare value ${\widehat{g}}_c$, which separates the basins of attraction of $F_{\mathit{SC}}$ and $F_{\mathit{EW}}$. Gray arrows symbolize flow lines. (b) Variation with $d$ of $\chi =2-z$ for $F_{\mathit{SC}}$ (red circles, our results; orange squares, numerical values from [43, 44]; see also Table ).

Figure 2

Typical shape of the dimensionless fixed point function ${\widehat{f}}^{*}(\widehat{\varpi },\widehat{p})$ (recorded at $s=-25$ and for $\alpha =1$).

Figure 3

(Leftmost curve) Scaling function $\mathring{F}\left(\frac{\widehat{\varpi }}{{\widehat{p}}^{3/2}}\right)$ corresponding to the data collapse (62), with the two insets zooming different parts to show the very high quality of the collapse (note the vertical scales); the various colors (gray shades) differentiate the contributions of distinct values of $\widehat{\varpi }$. (Rightmost curve) Scaling function $\mathring{F}(\tau =\frac{\widehat{\varpi }}{{\widehat{p}}^{7/3}})$ multiplied by ${\tau }^{7/3}$ to illustrate the power-law behavior of the tail $\mathring{F}\sim {\tau }^{-7/3}$ (see text). Note that the small dispersion visible in the inset zooming the tail for $\tau \gtrsim 100$ is hence magnified by a factor of order ${100}^{7/3}$.

Figure 4

Illustration of the fitting procedure described in the text, using the same data for $\mathring{F}$ as in Fig. 3 but with unified colors [black curves, on the left $\mathring{F}\left(\tau \right)$; on the right, $\mathring{F}\left(\tau \right){\tau }^{7/3}$]. The four successive orders of the fit appear superimposed in the main graph; the two insets present large zooms (note the vertical scales) to differentiate them and highlight the convergence: the higher the order, the closer to the data (black-dotted curve) the fit lies. Note that for the right curve and inset $\mathring{F}\left(\tau \right){\tau }^{7/3}$ the differences are again magnified by a factor ${\tau }^{7/3}$.

Figure 5

Comparison of the scaling function $\tilde{F}\left(k\right)$ (red curve with dots) obtained in this work with the exact one $\tilde{f}\left(k\right)$ (black curve with squares) from [24]. 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 develops with amplitudes below typically ${10}^{-6}$ (see text for a detailed comment of the figure).

Figure 6

Comparison of the scaling function $F\left(y\right)$ (red curve with dots) obtained in this work with the exact one $f\left(y\right)$ (black curve with squares) from [24].