Effects of a local defect on one-dimensional nonlinear surface growth

The slow-bond problem is a long-standing question about the minimal strength ${\epsilon }_{\mathrm{c}}$ of a local defect with global effects on the Kardar-Parisi-Zhang (KPZ) universality class. A consensus on the issue has been delayed due to the discrepancy between various analytical predictions claiming ${\epsilon }_{\mathrm{c}}=0$ and numerical observations claiming ${\epsilon }_{\mathrm{c}}>0$. We revisit the problem via finite-size scaling analyses of the slow-bond effects, which are tested for different boundary conditions through extensive Monte Carlo simulations. Our results provide evidence that the previously reported nonzero ${\epsilon }_{\mathrm{c}}$ is an artifact of a crossover phenomenon which logarithmically converges to zero as the system size goes to infinity.

Figure 1

1D TASEP (lower) and the corresponding BCSOS growth model (upper) with a slow bond of strength $\epsilon \in [0,1]$ (marked by a red segment) and open boundaries. The numbers above the arrows indicate hopping rates, and the site indices are shown at the bottom.

Figure 2

Average height profiles of the BCSOS growth model (upper) and density profiles of the TASEP (lower) under the assumption of ${\epsilon }_{\mathrm{c}}>0$. In the strong SB regime ($\epsilon >{\epsilon }_{\mathrm{c}}$), the TASEP (BCSOS growth model) develops a queue (nonzero slope) spanning the system. In the weak SB regime ($0<\epsilon <{\epsilon }_{\mathrm{c}}$), the queue (nonzero slope) disappears on a large scale, leaving algebraically decaying corrections around the slow bond. In the absence of the slow bond ($\epsilon =0$), the queue (nonzero slope) vanishes.

Figure 3

The steady-state current $J$ and the reduced current $\delta J$ as the SB strength $\epsilon$ and the system size $L$ are varied from $L=2^6$ to $2^{16}$, with darker colors used for larger $L$. (a) For the OBC ($\circ$), $J$ monotonically converges to $1/4$ (black solid line) as $\epsilon$ goes to 0, regardless of $L$. (b) For large $\epsilon$, we observe $\delta J\sim exp(-b/\epsilon )$ (black solid line), with $b\simeq 2.24$ estimated by curve fitting. (c) For small $\epsilon$, we observe $\delta J\simeq \epsilon /L$ (black solid line). (d)–(f) Similar plots are shown for the PBC ($\times$).

Figure 4

Data collapse by Eq. (13), where the crossover scale is given by ${\epsilon }_{\times }\left(L\right)=b/lnL$. Using the estimates of $b$ shown in Fig. 3, the data for the OBC ($\circ$) and the PBC ($\times$) are compared with the asymptotic scaling function ${\mathrm{\Psi }}_J\left(x\right)$ (dashed black lines) given by Eq. (14). The system size is varied from $L=2^6$ to $2^{16}$, with darker colors used for larger $L$ (see the legends of Fig. 3). The inset is a log-log replot of the main figure.

Figure 5

Relations between the excess particle density ${\mathrm{\Delta }}_{L/4}$ and the reduced current $\delta J$. The black diagonal line corresponds to ${\mathrm{\Delta }}_{L/4}=\delta J$. The inset shows the data collapse by Eq. (16), with the black solid (dashed) line indicating $y=x\left(y=x^2\right)$. The system size ranges from $L=2^6$ to $2^{16}$, with darker colors used for larger $L$ (see the legends of Fig. 3).

Figure 6

Two conflicting FSS theories for the OBC ($\circ$) and the PBC ($\times$). (a) Data collapse for Eq. (19), which corresponds to ${\epsilon }_{\mathrm{c}}=0.15$. (b) Data collapse for Eq. (20), which implies ${\epsilon }_{\mathrm{c}}=0$. The system size is varied from $L=2^6$ to $2^{16}$, with darker colors used for larger $L$ (see the legends of Fig. 3).