Conformal invariance of isoheight lines in a two-dimensional Kardar-Parisi-Zhang surface

The statistics of isoheight lines in the $\left(2+1\right)$-dimensional Kardar-Parisi-Zhang (KPZ) model is shown to be conformally invariant and equivalent to those of self-avoiding random walks. This leads to a rich variety of exact analytical results for the KPZ dynamics. We present direct evidence that the isoheight lines can be described by the family of conformally invariant curves called Schramm-Loewner evolution (or ${\text{SLE}}_{\kappa }$) with diffusivity $\kappa =8/3$. It is shown that the absence of the nonlinear term in the KPZ equation will change the diffusivity $\kappa$ from 8/3 to 4, indicating that the isoheight lines of the Edwards-Wilkinson surface are also conformally invariant and belong to the universality class of domain walls in the $O\left(2\right)$ spin model.

Figure 1

(Color online) Main frame: Interface width $w_L\left(t\right)$ vs time $t$ of the KPZ equation in $\left(2+1\right)$ dimensions and for $ϵ=10$, for different square lattice sizes $L$. The slope of the straight line yields the growth exponent $\beta =0.23\pm 0.01$. Upper left inset: Saturation width $w_L\left(\infty \right)$ for systems of different sizes $L$. The slope of the solid line fit yields the roughness exponent $\alpha =0.37\pm 0.01$. Lower right inset: Rescaled $w_L$ vs rescaled $t$.

Figure 2

(Color online) The clusters with positive heights are shown for the 2D KPZ interface with different colors. Negative-height regions are colored with black.

Figure 3

(Color online) Cluster and loop statistics for the isoheight lines of the 2D KPZ and EW surfaces are shown by the lower (different colors) and upper (green) graphs, respectively. The results for the EW have been shifted by a constant of 1 in order to distinguish them from the 2D KPZ results. (a) The average area $M$ vs the radius of gyration $R$. (b) The length of a loop $l$ vs the radius of gyration $R$. (c) Number of clusters of area between $M$ and $\lambda M$. (d) Number of loops of length between $l$ and $\lambda l$. (e) Number of loops of radius of gyration between $R$ and $\lambda R$. (f) Number of loops of area between $A$ and $\lambda A$. (g) The average area of loops $A$ vs the length $l$. In all figures $\lambda \simeq 1.05$. Solid lines show the corresponding analytical results for the SAWs (bottom) and $O\left(2\right)$ model (top). The error bars are almost the same size as the symbols in the scaling regions and have not been drawn. The change in the exponents relative to the SAWs in (c), (d), (e), and (f) is due to the roughness exponent [17].

Figure 4

(Color online) Main frame: The probability that a contour line of the 2D KPZ surface in the upper half-plane does not hit the slits of height $0.05\le h\le 0.5$ placed at $\xi$ from origin on the real axis vs $\left|{\Psi }_S^{\prime }\left(0\right)\right|=\left|\xi \right|{\left({\xi }^2+h^2\right)}^{-1/2}$. The solid line is the corresponding analytical prediction for ${\text{SLE}}_{8/3}$ curves. The error bars are almost the same size as the symbols in the scaling region. Inset: The probability that such a contour line passes to the left of a point $z=\rho e^{i\theta }$ for $\rho =0.05$, 0.1, 0.15, 0.2, 0.25, and 0.3. The solid line shows the prediction of SLE for $\kappa =8/3$.

Figure 5

(Color online) Statistics of the driving function $\zeta \left(t\right)$. Main frame: The linear behavior of $⟨\zeta {\left(t\right)}^2⟩$ with slope $\kappa =8/3\pm 1/10$. Lower right inset: The probability density of $\zeta \left(t\right)$ rescaled by its variance $\kappa$ at times $t=0.01,0.015,0.02$ which is Gaussian. Upper left inset: The correlation function of the increments of the driving function $\delta \zeta \left(t\right)$. A Kolmogorov-Smirnov (KS) goodness of fit for a normal distribution of the noise $\zeta \left(t\right)/\sqrt{\kappa t}$ for $\kappa =8/3$ and $0\le t\le 0.05$ yields $K-S=0.017$.

Figure 6

(Color online) Positive height connected domains of the 2D KPZ interface without the nonlinear term, corresponding to the EW model. Negative height regions are black.