Kardar-Parisi-Zhang equation with spatially correlated noise: A unified picture from nonperturbative renormalization group

We investigate the scaling regimes of the Kardar-Parisi-Zhang (KPZ) equation in the presence of spatially correlated noise with power-law decay $D\left(p\right)\sim p^{-2\rho }$ in Fourier space, using a nonperturbative renormalization group approach. We determine the full phase diagram of the system as a function of $\rho$ and the dimension $d$. In addition to the weak-coupling part of the diagram, which agrees with the results from Europhys. Lett. 47, 14 (1999) and Eur. Phys. J. B 9, 491 (1999), we find the two fixed points describing the short-range- (SR) and long-range- (LR) dominated strong-coupling phases. In contrast with a suggestion in the references cited above, we show that, for all values of $\rho$, there exists a unique strong-coupling SR fixed point that can be continuously followed as a function of $d$. We show in particular that the existence and the behavior of the LR fixed point do not provide any hint for 4 being the upper critical dimension of the KPZ equation with SR noise.

Figure 1

RG trajectories in the (${\widehat{y}}^{\prime },\widehat{x}$) plane in $d=2$ for (a) $\rho =0.3$ and (b) $\rho =0.7$ obtained with the LPA${}^{\prime }$. Increasing $\rho$, the LR fixed point moves from the unphysical quadrant $\widehat{x}<0$ [not shown in panel (a)] to the physical one [panel (b)] through crossing the SR fixed point that thus simultaneously changes stability in the ${\widehat{y}}^{\prime }$ direction.

Figure 2

RG trajectories in the (${\widehat{y}}^{\prime },\widehat{x}$) plane in $d=3$ for increasing values of $\rho$: (a) $\rho =0.24$, (b) $\rho =0.3$, (c) $\rho =0.4$, (d) $\rho =0.52$, (e) $\rho =0.9$, obtained with the LPA${}^{\prime }$ for (a) to (d) and NLO for (e). Panels (a) and (b): The EWLR2 fixed point enters into the physical quadrant ${\widehat{y}}^{\prime }>0$ and EWLR1 changes its stability. Panel (c): The TLR fixed point enters into the physical quadrant $\widehat{x}>0$ and T changes stability. Panel (d): The TLR fixed point merges with EWLR2 that changes stability. Panel (e): The LR fixed point enters into the physical quadrant $\widehat{x}>0$ and the SR fixed point changes stability.

Figure 3

RG trajectories in the (${\widehat{y}}^{\prime },\widehat{x}$) plane in $d=3$ and for increasing values of $\rho$: (a) $\rho =0.24$, (b) $\rho =0.3$, (c) $\rho =0.5$, and (d) $\rho =0.52$, obtained with the perturbative flow equations (A1). The rapid move of TLR in Fig. 2 is replaced by a fixed line joining T and EWLR2 displayed in (c), the rest of the (weak-coupling part of the) flow diagrams being very similar to the nonperturbative ones. (Note the difference of the ${\widehat{y}}^{\prime }$ scale.)

Figure 4

RG trajectories in the (${\widehat{y}}^{\prime },\widehat{x}$) plane in $d=4$ for (a) $\rho =1.125$ and (b) $\rho =1.25$ obtained with NLO. The LR fixed point lies away from the Gaussian fixed point and is fully attractive for $\rho \gtrsim 1.14$.

Figure 5

Phase diagram of the KPZ equation with spatially LR-correlated noise in the ($\rho ,d$) plane. Bounds between the regions, which are indicated by black lines, are given by $d_{\mathrm{EWLR}}\left(\rho \right)=2(1+2\rho )$, $d_c\left(\rho \right)=2(1+\rho )$ and $d_{\mathrm{SR}}\left(\rho \right)$ defined by (28) where averaged values for ${\chi }_{\mathrm{SR}}$ are taken from numerical simulations (mean values from Refs. [15, 16, 23, 24, 25, 26, 27, 28]). LR is the unique fully attractive fixed point in the dark gray region. In the other regions there is either a phase transition between a smooth and a rough phase where the LR noise is irrelevant or, below $d_c\left(\rho \right)$, only a rough phase described by SR. The TLR fixed point exists in the gray region between the lines $d_c^T$ and $d_c$; see Sec. 3a4.