Roughening transition and universality of single step growth models in (2+1)-dimensions

We study (2+1)-dimensional single step model (SSM) for crystal growth including both deposition and evaporation processes parametrized by a single control parameter $p$. Using extensive numerical simulations with a relatively high statistics, we estimate various interface exponents such as roughness, growth and dynamic exponents as well as various geometric and distribution exponents of height clusters and their boundaries (or iso-height lines) as function of $p$. We find that, in contrary to the general belief, there exists a critical value $p_c\approx 0.25$ at which the model undergoes a roughening transition from a rough phase with $pp_c$, asymptotically in the Edwards-Wilkinson (EW) class. We validate our conclusion by estimating the effective roughness exponents and their extrapolation to the infinite-size limit.


I. INTRODUCTION
Growth processes and rough surfaces are interesting topics in physics from both theoretical and experimental points of view [1,2].
Various discrete models have been suggested to describe surface growth processes, for examples see [1,2]. These models produce a self-affine interface h(x) such that its probability distribution function remains invariant under scale transformation:  averaging. For a non-equilibrium growth, the width is expected to have the following scaling form [3] The scaling function f usually has the asymptotic form f (x → ∞) = constant and f (x → 0) ∼ x 2β . The saturation time t s has the scaling ansatz t s ∼ L z . The universality class can then be given by two independent roughness and growth exponents, α and β, respectively.
For some equilibrium-rough interfaces, the width behaves logarithmically [4], i.e., w 2 (t, L) ∼ ln t for t ≪ t sat , and w 2 (t, L) ∼ ln L for t ≫ t s , with z = 2.
Whether rough surfaces fall into discrete universality classes is unknown, however many self-affine surfaces are observed to belong to certain universality classes.
Two well known universality classes are described by continuous Langevin equations i.e., the Edwards-Wilkinson (EW) [4] and the Kardar-Parisi-Zhang (KPZ) [5] equations. The KPZ equation is given by where the relaxation term is caused by a surface tension ν, and the nonlinear term is due to the lateral growth. The noise η is uncorrelated Gaussian white noise in both space and time with zero average i.e., η(x, t) = 0 and . For λ = 0, the EW equation is retrieved whose exact solution α = (2 − d)/2 and z = 2, is known in (d + 1)-dimensions [4]. For KPZ equation, due to additional scaling relation α + z = 2, there remains only one independent exponent, say α. The exact solution only exists in 1d [5] which gives α = 1/3.
A new tool for study of domain walls in critical systems is the theory of Schramm-Loewner Evolution (SLE) [9]; for a review see [10]. The scaling behaviour of the twodimensional critical lattice models can be reflected in the statistics of non-crossing random curves which form the boundaries of clusters on the lattice.
In the 1920s, Loewner studied simple curves growing from the origin into the upper half-plane H [11]. Loewner's idea was to describe the evolution of these curves in terms of the evolution of the analytic function g t (z), which conformally maps the region H\K t to H, where K t is the hull (the union of the curve and the set of points which can not be reached from infinity without intersecting the curve). He showed that this function satisfies the following differential equation:  [12][13][14][15].
One column (i, j) is chosen randomly, if it is a local minimum then its height is increased by 2 with probability p + ; one can consider this process as deposition. If it is a local maximum then its height is decreased by 2 with probability p − , as a process of desorption or evaporation. This definition guarantees that in each step, the height difference between two neighboring sites is exactly one. Overhanging is not allowed in this model and the interface will not develop large slopes. We start with the initial condition h(i, j; t = 0) = [1 + (−1) i+j ]/2 where 1 ≤ i ≤ L x and 1 ≤ j ≤ L y . We impose the condition p + + p − = 1, so due to up/down symmetry p + ↔ (1 − p + ), we just need to consider p + ≤ 0.5. Here after we shall refer to p + as p.

II. INTERFACE EXPONENTS
We ran simulations on a square lattice of size 50 ≤ L ≤ 700, averaging w(t, L) over more than 200 independent runs. Size L = 4000 was used only for computing β. To calculate the exponents of iso-height clusters and loops, more than 10 4 height configurations were undertaken on square lattice with size L = 1000. Also in the part of SLE, strip geometry L x = 3L y and L y = L was considered with 100 ≤ L ≤ 1000. We define one time step as equivalent to L 2 tries for deposition or evaporation.
To check the efficacy of our simulations, we calculated the roughness exponent α The offset directions {e m } M m=1 are a fixed set of vectors summing to zero. In our case (square lattice), {e m } pointing along the {10} type directions. For a self-affine surface, the curvature satisfies the relation [19]: where ·· denotes spatial averaging. To       For an ensemble of proposed dipolar SLE curves with hull K t , in a strip geometry S ∆ = {z ∈ C, 0 < ℑz < π∆} of width π∆, the evolution of the conformal map g t (z) which maps S ∆ \K t to S ∆ is given by [30] with initial condition g 0 (z) = z. In order to calculate sequences ξ(t i ) for the iso-height curves, we used the algorithm introduced   in [31]. First we consider the sequence of the points of a spanning curve as z 0 i = (x 0 i , y 0 i ), i = 0 · · · N; the upper index represents time step, where t 0 = 0 and ξ 0 = ξ(t 0 ) = 0. At the ith step, we map the sequence to the transformed and shortened sequence z i i , z i i+1 , · · · z i N , by using the map appropriate for dipolar SLE, where ∆ i = πy i−1 i /2L and ξ i = x i−1 i . For SLE curves, the driving function satisfies ξ(t) = √ κB t [10,30], where B t is the Brownian motion.
Another test of conformal invariance, is to compute the left passage probability P (ρ, φ), defined as the probability that a curve in the upper half-plane, passes to the left of a given point at polar coordinates (ρ, φ). This is given by Schramm's formula [32]: where 2 F 1 is the hypergeometric function.
Although this equation is originally obtained for chordal SLE curves, it also holds for dipo- We considered a substrate of strip geome- previous results in table I. For instance, for p = 0.5, we obtain κ ≈ 2.24 (5), which is clearly different from κ EW = 4 [24].
Our guess is that the statistics of the span- We also simulated critical percolation model on a strip geometry with the same two boundary conditions. We found the diffusivity κ ≈ 6 [37] compatible with fractal dimension d f ≈ 7/4 [38], for both BCs. This result is expected since for the percolation model, the statistics of the domain walls are not affected by the BCs due to a locality property [10,28].
In SSM, only for p = 0, we obtain the compatibility between fractal dimension and diffusivity that is in SAW universality class. It is known that the outer perimeter of critical percolation clusters in 2d are described by SAW [39]. In our recent work [12], we have shown that the iso-height lines of various discrete models including BD, RSOS and EDEN models are in the SAW universality class.