Strong coupling phases of conserved growth models are crumpled

TL;DR

The paper analyzes strong-coupling phases in a class of stochastically driven conserved growth models with nonlocal chemical potentials, showing that sufficiently strong nonlocality drives crumpled, non-KPZ-like strong-coupling phases in dimensions below and at the critical dimension $d_c=4+y$. Using one-loop dynamic renormalization group and mode-coupling theory, the authors derive flow equations, identify fixed points, and predict logarithmic corrections at $d=d_c$ and crumpling in unstable regimes. They corroborate the theory with direct numerical simulations in $d=2$ for CKPZ$^+$ and MBE$^+$, finding algebraic roughening in stable regions and divergence consistent with crumpling in unstable windows, thereby distinguishing these strong-coupling phases from KPZ behavior. The results advance understanding of how nonlocality and conservation laws impact surface roughening and reveal a crumpled strong-coupling phase as a generic feature of conserved growth models with nonlocal chemical potentials.

Abstract

We show that stochastically driven nonequilibrium conserved growth models admit generic strong coupling phases for sufficiently strong nonlocal chemical potentials underlying the dynamics. The models exhibit generic roughening transitions between perturbatively accessible weak coupling phases satisfying an exact relation between the scaling exponents in all dimensions $d$, and strong coupling phases. In dimensions below the critical dimension $d_c$, the latter phases are unstable and argued to be crumpled, and thus distinct from the well-known strong coupling rough phase of the Kardar-Parisi-Zhang equation in dimensions $d\geq 2$. At $d_c$, conventional spatio-temporal scaling in the weak coupling phase is logarithmically modulated and are exactly obtained.

Figures (9)

• Figure 1: ($\bar{\nu}=0$) (a) Stable (violet) and unstable (white) regions in the $\gamma$-$y$ plane as obtained from RG and MCT calculations at $d =d_c= 4 + y$. (b) RG flow diagram at $d < d_c$. The blue planes represent stable fixed planes toward which the RG flow lines (black dashed arrows) converge. The red region denotes the stable phase accessible via perturbation theory, while the central white region is speculated to be unstable. (c) RG flow diagram for $d > d_c$. The central white region contains a blue plane representing an unstable fixed plane, where a roughening transition occurs between a smooth phase (green region) and a perturbatively inaccessible strong coupling rough phase, as indicated by the RG flow lines. Outside this region, the roughening transition disappears, and only the smooth phase remains.
• Figure 2: ($\bar{\nu}=0$). RG flow diagram in $d = 2$ for the (a) CKPZ+ model. The violet region is the perturbatively accessible logarithmic rough phase, while the central white region is unstable. In the accessible region, the RG flow lines (indicated by arrows) flow toward the stable fixed line $g = 0$. (b) MBE+ model. The blue line within the stable (red) region denotes the stable fixed line, toward which the RG flow lines converge.
• Figure 3: For MBE+ model in $d=2$, log-log plots (with $\bar{\nu}=0, \nu=1$, $D=1$, $\nu_r=0.2$, $\lambda=2$, $\lambda_1 = 1$, and time step $(\delta t) = 10^{-3}$) of (a) ${\mathcal{W}}$ versus $t$. The black dashed line indicates a linear fit in the growth regime, yielding $\beta = 0.207 \pm 7.26 \times 10^{-5}$. (b) ${\mathcal{W}}_{\text{sat}}$ versus $L$. The red dashed line represent linear fit to the numerical data displayed on a log-log scale, with slope $\chi = 0.707 \pm 0.006$.
• Figure 4: Log-log plots of ${\mathcal{W}}$ versus $t$ for various values of $\gamma$ in the unstable region of Fig. \ref{['rg_flow_2d']}, for a fixed realization ($R1$) and fixed $L$. Inset: log-log plots of ${\mathcal{W}}$ versus $t$ for various values of $\gamma$ in the stable region of Fig. \ref{['rg_flow_2d']}, where ${\mathcal{W}}$ (averaged over many realizations) saturates, for the (a) CKPZ+ model (b) MBE+ model. Log-log plots of ${\mathcal{W}}$ versus $t$ for various system sizes but fixed realization and fixed $\gamma$ in the unstable region. Inset: same but for a fixed $L$, fixed $\gamma$, and over different realizations, for the (c) CKPZ+ model and (d) MBE+ model. Parameters $\bar{\nu}=0$, $\nu=1$, $D=1$, $\nu_r=0.2$, $\lambda=2$, with various values of $\lambda_1$ (i.e., $\gamma$), and time step $\delta t = 10^{-3}\,(10^{-4})$ in the stable (unstable) region.
• Figure 5: Diagrammatic representations of two point functions.