Dimensional crossover in Kardar-Parisi-Zhang growth

Abstract

Two-dimensional (2D) KPZ growth is usually investigated on substrates of lateral sizes $L_x=L_y$, so that $L_x$ and the correlation length ($\xi$) are the only relevant lengths determining the scaling behavior. However, in cylindrical geometry, as well as in flat rectangular substrates $L_x\ne L_y$ and, thus, the surfaces can become correlated in a single direction, when $\xi \sim L_x\ll L_y$. From extensive simulations of several KPZ models, we demonstrate that this yields a dimensional crossover in their dynamics, with the roughness scaling 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 $t_c\sim L_x^{1/z_{2\text{D}}}$. The height distributions (HDs) also cross over from the 2D flat [cylindrical] HD to the asymptotic Tracy-Widom GOE [GUE] distribution. Moreover, 2D-to-1D crossovers are found also in the asymptotic growth velocity and in the steady state regime of flat systems, where a family of universal HDs exists, interpolating between the 2D and 1D ones as $L_y/L_x$ increases. Importantly, the crossover scalings are fully determined and indicate a possible way to solve 2D KPZ models.

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Cited by in corpus (1)

• Dimensional crossover in surface growth on rectangular substrates