Dimensional crossover in surface growth on rectangular substrates

Abstract

In a recent work [Phys. Rev. E 109, L042102 (2024)], interesting dimensional crossovers [from two- to one-dimensional (2D to 1D) scaling] were found in the growth of Kardar-Parisi-Zhang (KPZ) interfaces on rectangular substrates, with lateral sizes $L_y > L_x$. Here, we extend this study to other universality classes for interface growth -- specifically, the Edwards-Wilkinson (EW), the Mullins-Herring (MH), and the Villain-Lai Das Sarma (VLDS) classes. From extensive simulations, we demonstrate that, in all systems with sufficiently large aspect ratio $\mathcal{R}=L_y/L_x$, the roughness $W$ scales with time $t$ in the growth regime as $W \sim t^{\beta_{\text{2D}}}$ for $t \ll t_c$ and $W \sim t^{\beta_{\text{1D}}}$ for $t \gg t_c$, where $t_c \sim L_x^{z_{2\text{D}}}$ in most cases. For the VLDS class, this crossover is also observed in the height distribution (HD), which approaches its characteristic probability density function for the 2D case at short times ($t \ll t_c$) and then crosses over to the asymptotic 1D HD. Dimensional crossovers are also found in the steady state regime, both in the roughness scaling as well as in the VLDS HD, which interpolate between the 2D and 1D ones as $\mathcal{R}$ increases. The particular case $L_x = L_y^{\delta}$, with $0 < \delta < 1$, is also discussed in detail and reveals interesting features of the investigated systems. For instance, there exist a `special' exponent $\delta^* = z_{1\text{D}}/z_{2\text{D}}$ such that the temporal crossover cannot be observed for $\delta > \delta^*$. Moreover, this leads the saturation roughness to display a nonuniversal scaling: $W_s \sim L_y^{\Lambda}$, with $\Lambda = (1-\delta) \alpha_{1\text{D}} + \delta \alpha_{2\text{D}}$.