Strong Coupling in Conserved Surface Roughening: A New Universality Class?

The Kardar-Parisi-Zhang (KPZ) equation defines the main universality class for nonlinear growth and roughening of surfaces. But under certain conditions, a conserved KPZ equation (CKPZ) is thought to set the universality class instead. This has non-mean-field behavior only in spatial dimension $d<2$. We point out here that CKPZ is incomplete: It omits a symmetry-allowed nonlinear gradient term of the same order as the one retained. Adding this term, we find a parameter regime where the one-loop renormalization group flow diverges. This suggests a phase transition to a new growth phase, possibly ruled by a strong-coupling fixed point and thus described by a new universality class, for any $d>1$. In this phase, numerical integration of the model in $d=2$ gives clear evidence of non-mean-field behavior.

Figure 1

(a) Contour plot of the surface $\varphi (x,y)=2\cos x+y$ and the current ${\mathbf{J}}_{\zeta }=-\zeta ({\nabla }^2\varphi )\nabla \varphi$ (vectors). ${\mathbf{J}}_{\zeta }$ resembles a shear flow and thus has nonzero curl. In (b), the blue line is the locus of points $\mathcal{C}$ on the surface equidistant from the origin in the natural metric of the surface. The red dotted line is the intersection between the vertical plane $y=0$ and the surface: Because it does not cut $\mathcal{C}$ in half, it induces ${\mathbf{J}}_{\zeta }\ne 0$ if a particle in (0,0) jumps with equal probability to any site on $\mathcal{C}$.

Figure 2

The diagrams present at one loop. Those in (b)–(d) do not generate any relevant coupling for $\epsilon$ small and close to the Gaussian fixed point.

Figure 3

One-loop RG flow of the (a) CKPZ model as a function of space dimension $d$ and of the $\mathrm{CKPZ}+$ model for (b) $1<d<2$, (c) $d=2$, and (d) $d>2$. (CKPZ and $\mathrm{CKPZ}+$ are the same model in $d=1$). The red lines are the fixed points of the RG flow given by solutions of (12) and the dashed lines their asymptotes; the origin is the Gaussian fixed point. In (b)–(d), the RG flow is radial and its direction is given by the arrows in the plots.

Figure 4

Growth of the width of the interface $W(L,t)$ with time for different system sizes $L$ of the $\mathrm{CKPZ}+$ equation in $d=2$. The parameters $2\lambda =\zeta =1$ were chosen to lie in the region where the RG flow diverges. Each line is an average over several noise realizations (from 1600 for $L=15$ to 80 for $L=45$). The value of $k_6=0.2$. The difference between the present case and the one where the RG flow converges towards the Gaussian fixed point is apparent. Here, at late times (decreasing with $k_6$ increasing [38]), the width grows faster than logarithmically. The growth law seems independent from the system size.