Roughening phase transition in surface growth

Kardar-Parisi-Zhang Physics and Phase Transition in a Classical Single Random Walker under Continuous Measurement

We introduce and study a new model consisting of a single classical random walker undergoing continuous monitoring at rate γ on a discrete lattice. Although such a continuous measurement cannot affect physical observables, it has a nontrivial effect on the probability distribution of the random walker. At small γ, we show analytically that the time evolution of the latter can be mapped to the stochastic heat equation. In this limit, the width of the log-probability thus follows a Family-Vicsek scaling law, N^{α}f(t/N^{α/β}), with roughness and growth exponents corresponding to the Kardar-Parisi-Zhang (KPZ) universality class, i.e., α_{KPZ}^{1D}=1/2 and β_{KPZ}^{1D}=1/3, respectively. When γ is increased outside this regime, we find numerically in 1D a crossover from the KPZ class to a new universality class characterized by exponents α_{M}^{1D}≈1 and β_{M}^{1D}≈1.4. In 3D, varying γ beyond a critical value γ_{M}^{c} leads to a phase transition from a smooth phase that we identify as the Edwards-Wilkinson class to a new universality class with α_{M}^{3D}≈1.

Efimov effect at the Kardar-Parisi-Zhang roughening transition

Surface growth governed by the Kardar-Parisi-Zhang (KPZ) equation in dimensions higher than two undergoes a roughening transition from smooth to rough phases with increasing the nonlinearity. It is also known that the KPZ equation can be mapped onto quantum mechanics of attractive bosons with a contact interaction, where the roughening transition corresponds to a binding transition of two bosons with increasing the attraction. Such critical bosons in three dimensions actually exhibit the Efimov effect, where a three-boson coupling turns out to be relevant under the renormalization group so as to break the scale invariance down to a discrete one. On the basis of these facts linking the two distinct subjects in physics, we predict that the KPZ roughening transition in three dimensions shows either the discrete scale invariance or no intrinsic scale invariance.

Growth model with restricted surface relaxation

We simulate a growth model with restricted surface relaxation process in d=1 and d=2, where d is the dimensionality of a flat substrate. In this model, each particle can relax on the surface to a local minimum, as the discrete surface relaxation model, but only within a distance s. If the local minimum is out from this distance, the particle evaporates through a refuse mechanism similar to the Kim-Kosterlitz nonlinear model. In d=1, the growth exponent beta, measured from the temporal behavior of roughness, indicates that in the coarse-grained limit, the linear term of the Kardar-Parisi-Zhang equation dominates in short times (low-roughness) and, in asymptotic times, the nonlinear term prevails. The crossover between linear and nonlinear behaviors occurs in a characteristic time t(c) which only depends on the magnitude of the parameter s, related to the nonlinear term. In d=2, we find indications of a similar crossover, that is, logarithmic temporal behavior of roughness in short times and power law behavior in asymptotic times.

Crossover effects in a discrete deposition model with Kardar-Parisi-Zhang scaling

We simulated a growth model in (1+1) dimensions in which particles are aggregated according to the rules of ballistic deposition with probability p or according to the rules of random deposition with surface relaxation (Family model) with probability 1-p. For any p>0, this system is in the Kardar-Parisi-Zhang (KPZ) universality class, but it presents a slow crossover from the Edwards-Wilkinson class (EW) for small p. From the scaling of the growth velocity, the parameter p is connected to the coefficient lambda of the nonlinear term of the KPZ equation, giving lambda approximately p(gamma), with gamma=2.1+/-0.2. Our numerical results confirm the interface width scaling in the growth regime as W approximately lambda(beta)t(beta) and the scaling of the saturation time as tau approximately lambda(-1)L(z), with the expected exponents beta=1/3 and z=3/2, and strong corrections to scaling for small lambda. This picture is consistent with a crossover time from EW to KPZ growth in the form t(c) approximately lambda(-4) approximately p(-8), in agreement with scaling theories and renormalization group analysis. Some consequences of the slow crossover in this problem are discussed and may help investigations of more complex models.

Scaling behavior of the surface in ballistic deposition

Using a dynamical scaling form for the surface fractal dimension as well as efficient algorithms for the simulation and analysis of the surface in three-dimensional ballistic deposition, we show that while the top of the surface is self-affine, the complete surface including overhangs has fractal dimension D(f) approximately 3. The existence of such a fractal surface is a consequence of the difficulty of closing off voids in three and higher dimensions. By studying a modified ballistic deposition model in which sticking is allowed with a given probability p, we show that the surface undergoes a phase transition from fractal to compact at a finite value of p. Our results also have implications for understanding the surface morphology in sedimentary rocks and low-temperature thin films.

Entropy fluctuations for directed polymers in 2+1 dimensions

We find numerically that the sample to sample fluctuation of the entropy DeltaS is a more sensitive tool in distinguishing low from high temperature behaviors than the common corresponding fluctuation in the free energy. In 1+1 dimensions we find a single phase for all temperatures, since (DeltaS)(2) is always extensive. In 2+1 dimensions we find a behavior that at first sight might appear to be a transition from a low temperature phase where (DeltaS)(2) is extensive to a high temperature phase where it is subextensive. This is observed in spite of the relatively large system we use. The observed behavior is explained not as a phase transition but as a strong crossover behavior. We use an analytical argument to obtain (DeltaS)(2) for high temperature, and find that while it is always extensive it is also extremely small, and that the leading extensive part decays very quickly to zero as the temperature is increased.