Restricted Solid-on-solid Research Articles

Extensive numerical simulations on competitive growth between the Edwards–Wilkinson and Kardar–Parisi–Zhang universality classes

Extensive numerical simulations and scaling analysis are performed to investigate competitive growth between the linear and nonlinear stochastic dynamic growth systems, which belong to the Edwards–Wilkinson (EW) and Kardar–Parisi–Zhang (KPZ) universality classes, respectively. The linear growth systems include the EW equation and the model of random deposition with surface relaxation (RDSR), the nonlinear growth systems involve the KPZ equation and typical discrete models including ballistic deposition (BD), etching, and restricted solid on solid (RSOS). The scaling exponents are obtained in both the (1 + 1)- and (2 + 1)-dimensional competitive growth with the nonlinear growth probability p and the linear proportion 1 – p. Our results show that, when p changes from 0 to 1, there exist non-trivial crossover effects from EW to KPZ universality classes based on different competitive growth rules. Furthermore, the growth rate and the porosity are also estimated within various linear and nonlinear growths of cooperation and competition.

Factorization of density matrices in the critical RSOS models

We study reduced density matrices of the integrable critical restricted solid-on-solid (RSOS) model in a particular topological sector containing the ground state. Similar as in the spin- Heisenberg model it has been observed that correlation functions of this model on short segments can be ‘factorized’: they are completely determined by a single nearest-neighbour two-point function ω and a set of structure functions. While ω captures the dependence on the system size and the state of the system the structure functions can be expressed in terms of the possible operators on the segment, in the present case representations of the Temperley–Lieb algebra , and are independent of the model parameters. We present explicit results for the function ω in the infinite system ground state of the model and compute multi-point local height probabilities for up to four adjacent sites for the RSOS model and the related three-point correlation functions of non-Abelian anyons.

Superconcentration in surface growth

Height functions of growing random surfaces are often conjectured to be superconcentrated, meaning that their variances grow sublinearly in time. This article introduces a new concept—called subroughness—meaning that there exist two distinct points such that the expected squared difference between the heights at these points grows sublinearly in time. The main result of the paper is that superconcentration is equivalent to subroughness in a class of growing random surfaces. The result is applied to establish superconcentration in a variant of the restricted solid‐on‐solid (RSOS) model and in a variant of the ballistic deposition model, and give new proofs of superconcentration in directed last‐passage percolation and directed polymers.

Characterization of topological phase transitions from a non-Abelian topological state and its Galois conjugate through condensation and confinement order parameters

Topological phases exhibit unconventional order that cannot be detected by any local order parameter. In the framework of Projected Entangled Pair States(PEPS), topological order is characterized by an entanglement symmetry of the local tensor which describes the model. This symmetry can take the form of a tensor product of group representations, or in the more general case a correlated symmetry action in the form of a Matrix Product Operator(MPO), which encompasses all string-net models. Among other things, these entanglement symmetries allow for the description of ground states and anyon excitations. Recently, the idea has been put forward to use those symmetries and the anyonic objects they describe as order parameters for probing topological phase transitions, and the applicability of this idea has been demonstrated for Abelian groups. In this paper, we extend this construction to the domain of non-Abelian models with MPO symmetries, and use it to study the breakdown of topological order in the double Fibonacci (DFib) string-net and its Galois conjugate, the non-hermitian double Yang-Lee (DYL) string-net. We start by showing how to construct topological order parameters for condensation and deconfinement of anyons using the MPO symmetries. Subsequently, we set up interpolations from the DFib and the DYL model to the trivial phase, and show that these can be mapped to certain restricted solid on solid(RSOS) models, which are equivalent to the $((5\pm\sqrt{5})/2)$-state Potts model, respectively. The known exact solutions of the statistical models allow us to locate the critical points, and to predict the critical exponents for the order parameters. We complement this by numerical study of the phase transitions, which fully confirms our theoretical predictions; remarkably, we find that both models exhibit a duality between the order parameters for condensation and deconfinement.

A new way of measuring the correlation length in surface growth models

We propose a new way of measuring the correlation length in surface growth processes. We define a quantity in d = 1 + 1, where w(t) and L are the surface width of a given sample at time t and the system size, respectively, and ⟨…⟩ denotes the average over samples. The quantity R(L, t) is proportional to the correlation length and follows the relation R(L, t) ∼ t 1/z before the saturation for various growth models, where z is the dynamic exponent. The applicability of the method was examined for the restricted solid-on-solid (RSOS) model, equilibrium solid-on-solid model, and restricted curvature model. We also estimated z by using the relation R(t) ∼ t 1/z for the RSOS model in higher dimensions.

Faceted-rough surface with disassembling of macrosteps in nucleation-limited crystal growth

To clarify whether a surface can be rough with faceted macrosteps that maintain their shape on the surface, crystal surface roughness is studied by a Monte Carlo method for a nucleation-limited crystal-growth process. As a surface model, the restricted solid-on-solid (RSOS) model with point-contact-type step–step attraction (p-RSOS model) is adopted. At equilibrium and at sufficiently low temperatures, the vicinal surface of the p-RSOS model consists of faceted macrosteps with (111) side surfaces and smooth terraces with (001) surfaces (the step-faceting zone). We found that a surface with faceted macrosteps has an approximately self-affine-rough structure on a ‘faceted-rough surface’; the surface width is strongly divergent at the step-disassembling point, which is a characteristic driving force for crystal growth. A ‘faceted-rough surface’ is realized in the region between the step-disassembling point and a crossover point where the single nucleation growth changes to poly-nucleation growth.

Quantum Groups for Restricted SOS Models

We introduce the notion of restricted dynamical quantum groups through their category of representations, which are monoidal categories with a forgetful functor to the category of π-graded vector spaces for a groupoid π.

Lattice models, deformed Virasoro algebra and reduction equation

We study the fused currents of the deformed Virasoro algebra. By constructing a homotopy operator we show that for special values of the parameter of the algebra fused currents pairwise coincide on the cohomologies of the Felder resolution. Within the algebraic approach to lattice models these currents are known to describe neutral excitations of the solid-on-solid (SOS) models in the transfer-matrix picture. It allows us to prove the closeness of the system of excitations for a special nonunitary series of restricted SOS models. Though the results of the algebraic approach to lattice models were consistent with the results of other methods, the lack of such proof had been an essential gap in its construction.

Random Walk Diffusion Model with a Restricted Solid-on-Solid Condition in Higher Dimensions

We study a random walk diffusion model under the restricted solid-on-solid (RSOS) condition in d = 3 + 1 and d = 4 + 1 dimensions. A dropped particle is allowed to take a random walk until satisfying the RSOS condition. The surface width W increases as tβ in early time with β = 0.081 ± 0.006 and becomes saturated at Lα with α = 0.29 ± 0.03 in d = 3 + 1. In d = 4 + 1, W (t) grows as 2b ln t at the beginning and becomes saturated at 2a ln L. The dynamic exponent z ≈ 3.79 is obtained from the relation $$z = {a \over b}$$ . This value is slightly less than, but consistent with, the value obtained from the related Langevin equation. Our results imply that the upper critical dimension of the model is d =4+1.

Relationship between Macrostep Height and Surface Velocity for a Reaction-Limited Crystal Growth Process

This work examined the effect of macrostep height on the growth velocity of a vicinal surface during reaction- (interface-) limited crystal growth under non-equilibrium steady state conditions. The Monte Carlo method was employed, based on a restricted solid-on-solid (RSOS) model with point-contact-type step-step attraction (termed the p-RSOS model). Although this is a simple lattice model, the model surface shows a variety of distinctive configurations depending on the temperature and the driving force for crystal growth. The results demonstrate that the surface velocity decreases as the height of the faceted macrostep increases. In addition, the significant variation in surface velocity recently reported by Onuma {\it et al.} in a study based on 4H-SiC was reproduced. This work also shows that the terrace slope, elementary step velocity and elementary step kinetic coefficient are all affected by the faceted macrostep height.

Numerical integration of KPZ equation with restrictions

In this paper, we introduce a novel integration method of Kardar–Parisi–Zhang (KPZ) equation. It is known that if during the discrete integration of the KPZ equation the nearest-neighbor height-difference exceeds a critical value, instabilities appear and the integration diverges. One way to avoid these instabilities is to replace the KPZ nonlinear-term by a function of the same term that depends on a single adjustable parameter which is able to control pillars or grooves growing on the interface. Here, we propose a different integration method which consists of directly limiting the value taken by the KPZ nonlinearity, thereby imposing a restriction rule that is applied in each integration time-step, as if it were the growth rule of a restricted discrete model, e.g. restricted-solid-on-solid (RSOS). Taking the discrete KPZ equation with restrictions to its dimensionless version, the integration depends on three parameters: the coupling constant g, the inverse of the time-step k, and the restriction constant ε which is chosen to eliminate divergences while keeping all the properties of the continuous KPZ equation. We study in detail the conditions in the parameters’ space that avoid divergences in the 1-dimensional integration and reproduce the scaling properties of the continuous KPZ with a particular parameter set. We apply the tested methodology to the d-dimensional case () with the purpose of obtaining the growth exponent β, by establishing the conditions of the coupling constant g under which we recover known values reached by other authors, particularly for the RSOS model. This method allows us to infer that d = 4 is not the critical dimension of the KPZ universality class, where the strong-coupling phase disappears.

Random walk diffusion model with restricted solid on solid condition on fractal substrates

A random walk diffusion model with a restricted solid-on-solid (RSOS) condition is studied on both Sierpinski gasket and checkerboard fractal substrates. A deposited particle is allowed to hop randomly until finding a site satisfying the RSOS condition. The interface width W of the model grows as at the beginning and becomes saturated at for , where L is the system size. We obtain , , and the dynamic exponent for a Sierpinski gasket substrate. A scaling relation is well satisfied, where zrw is the random walk exponent of the fractal substrate. Also, they are in good agreement with the predicted values of a suggested Langevin equation.

Dynamic scaling behaviors of the restricted-solid-on-solid model on honeycomb and square-octagon lattice substrates

The dynamic scaling behaviors of the restricted-solid-on-solid (RSOS) model on two new types of substrate, which are honeycomb and square-octagon lattice substrates, are studied by means of Kinetic Monte Carlo simulations. The growth exponent β and the roughness exponent α defined, respectively, by the surface width via W ~ t β and the saturated width via W sat ~ L α , L being the system size, were obtained by a power-counting analysis. Our simulation results show that the Family-Vicsek scaling is still satisfied. However, the structures of the substrates indeed affect the dynamic behavior of the growth model. The values of the roughness exponents fall between regular and fractal lattices. Deeper analysis show that the coordination number of the substrates play an crucial role.

Harmonic Pinnacles in the Discrete Gaussian Model

The 2D Discrete Gaussian model gives each height function \({\eta : {\mathbb{Z}^2\to\mathbb{Z}}}\) a probability proportional to \({\exp(-\beta \mathcal{H}(\eta))}\), where \({\beta}\) is the inverse-temperature and \({\mathcal{H}(\eta) = \sum_{x\sim y}(\eta_x-\eta_y)^2}\) sums over nearest-neighbor bonds. We consider the model at large fixed \({\beta}\), where it is flat unlike its continuous analog (the Discrete Gaussian Free Field). We first establish that the maximum height in an \({L\times L}\) box with 0 boundary conditions concentrates on two integers M, M + 1 with \({M\sim \sqrt{(1/2\pi\beta)\log L\log\log L}}\). The key is a large deviation estimate for the height at the origin in \({\mathbb{Z}^{2}}\), dominated by “harmonic pinnacles”, integer approximations of a harmonic variational problem. Second, in this model conditioned on \({\eta\geq 0}\) (a floor), the average height rises, and in fact the height of almost all sites concentrates on levels H, H + 1 where \({H\sim M/\sqrt{2}}\). This in particular pins down the asymptotics, and corrects the order, in results of Bricmont et al. (J. Stat. Phys. 42(5–6):743–798, 1986), where it was argued that the maximum and the height of the surface above a floor are both of order \({\sqrt{\log L}}\). Finally, our methods extend to other classical surface models (e.g., restricted SOS), featuring connections to p-harmonic analysis and alternating sign matrices.

Random walk diffusion model for a restricted solid-on-solid condition

We study a random-walk diffusion model for a restricted solid-on-solid (RSOS) condition, where a deposited particle is allowed to walk randomly until visiting a site satisfying the RSOS condition. The interface width W grows as t β in early time, with β = 0.317 ± 0.005 in d = 1 + 1 dimension (β = 0.178±0.009 in d = 2+1 dimensions) and saturates at L α with α = 0.913±0.009 in d = 1+1 dimension (α = 0.638±0.008 in d = 2+1 dimensions). α and β satisfy a scaling relation α/β +α = 4 and are in good agreement with the values predicted by the conserved Langevin equation.

Faceting diagram for sticky steps

Faceting diagrams for the step-faceting zone, the step droplet zone, and the Gruber-Mullins-Pokrovsky-Talapov (GMPT) zone for a crystal surface are obtained by using the density matrix renormalization group method to calculate the surface tension. The model based on these calculations is the restricted solid-on-solid (RSOS) model with a point-contact-type step-step attraction (p-RSOS model) on a square lattice. The point-contact-type step-step attraction represents the energy gain obtained by forming a bonding state with orbital overlap at the meeting point of the neighboring steps. In the step-faceting zone, disconnectedness in the surface tension leads to the formation of a faceted macrostep on a vicinal surface at equilibrium. The disconnectedness in the surface tension also causes the first-order shape transition for the equilibrium shape of a crystal droplet. The lower zone boundary line (ZBL), which separates the step-faceting zone and the step droplet zone, is obtained by the condition γ1=limn→∞γn/n, where γn is the step tension of the n-th merged step. The upper ZBL, which separates the GMPT zone and the step droplet zone, is obtained by the condition Aq,eff = 0 and Bq,eff = 0, where Aq,eff and Bq,eff represent the coefficients for the |q→|2 term and the |q→|3 term, respectively, in the |q→|-expanded form of the surface free energy feff(q→). Here, q→ is the surface gradient relative to the (111) surface. The reason why the vicinal surface inclined in the 〈101〉 direction does not exhibit step-faceting is explained in terms of the one-dimensional spinless quasi-impenetrable attractive bosons at absolute zero.

RSOS quantum chains associated with off-critical minimal models and parafermions

We consider the φ1,3 off-critical perturbation of the general non-unitary minimal models where 2 ⩽ m ⩽ m′ and m, m′ are coprime and t measures the departure from criticality corresponding to the φ1,3 integrable perturbation. We view these models as the continuum scaling limit in the ferromagnetic Regime III of the Forrester–Baxter Restricted Solid-On-Solid (RSOS) models on the square lattice. We also consider the RSOS models in the antiferromagnetic Regime II related in the continuum scaling limit to parfermions with n = m′ − 2. Using an elliptic Yang–Baxter algebra of planar tiles encoding the allowed face configurations, we obtain the Hamiltonians of the associated quantum chains defined as the logarithmic derivative of the transfer matrices with periodic boundary conditions. The transfer matrices and Hamiltonians act on a vector space of paths on the Am′−1 Dynkin diagram whose dimension is counted by generalized Fibonacci numbers.

Inversion identities for inhomogeneous face models

We derive exact inversion identities satisfied by the transfer matrix of inhomogeneous interaction-round-a-face (IRF) models with arbitrary boundary conditions using the underlying integrable structure and crossing properties of the local Boltzmann weights. For the critical restricted solid-on-solid (RSOS) models these identities together with some information on the analytical properties of the transfer matrix determine the spectrum completely and allow to derive the Bethe equations for both periodic and general open boundary conditions.

Parallel discrete-event simulation schemes with heterogeneous processing elements.

To understand the effects of nonidentical processing elements (PEs) on parallel discrete-event simulation (PDES) schemes, two stochastic growth models, the restricted solid-on-solid (RSOS) model and the Family model, are investigated by simulations. The RSOS model is the model for the PDES scheme governed by the Kardar-Parisi-Zhang equation (KPZ scheme). The Family model is the model for the scheme governed by the Edwards-Wilkinson equation (EW scheme). Two kinds of distributions for nonidentical PEs are considered. In the first kind computing capacities of PEs are not much different, whereas in the second kind the capacities are extremely widespread. The KPZ scheme on the complex networks shows the synchronizability and scalability regardless of the kinds of PEs. The EW scheme never shows the synchronizability for the random configuration of PEs of the first kind. However, by regularizing the arrangement of PEs of the first kind, the EW scheme is made to show the synchronizability. In contrast, EW scheme never shows the synchronizability for any configuration of PEs of the second kind.

Pinning of steps near equilibrium without impurities, adsorbates, or dislocations

Discontinuous surface tension of a vicinal surface is calculated numerically based on a statistical mechanical method. The microscopic model is a restricted solid-on-solid (RSOS) model with a point-contact type step–step attraction (p-RSOS model). The discontinuity in the surface tension leads to step faceting at low temperatures near equilibrium, where macrosteps with smooth side surfaces are formed, resulting in two-surface coexistence. Using the Monte Carlo method with the Metropolis algorithm, we show the formation of the merged steps (macrosteps) and the slow velocity of the merged steps. We demonstrate that steps on a vicinal surface are pinned without impurities, adsorbates, or dislocations. We also propose simple ideas to avoid non-uniform growing of a crystal caused by the discontinuous surface tension of the crystal.