Restricted Solid-on-solid Model Research Articles

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.

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.

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.

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.

Coarse-grained approach for universality classification of discrete models

Discrete and continuous models belonging to a universality class share the same linearities and (or) nonlinearities. In this work, we propose a new approach to calculate coarse-grained coefficients of the continuous differential equation from discrete models. We apply small constant translations in a test space and show how to obtain these coefficients from the transformed average interface growth velocity. Using the examples of the ballistic deposition (BD) model and the restricted solid-on-solid (RSOS) model, both belonging to the Kardar–Parisi–Zhang (KPZ) universality class, we demonstrate how to apply our approach to calculate analytically the corresponding coefficients of the KPZ equation. Our analytical nonlinear coefficients are in agreement with numerical results obtained by Monte Carlo tilted simulations. In addition to the BD and the RSOS, we study a competitive RSOS model that shows crossover between the KPZ and Edwards–Wilkinson universality classes.

Scaling properties of surface growth on rough substrates generated by standard models

In surface growth simulations, usually the substrate is initially considered smooth with no roughness. But as known in reality, many substrates in nature are initially rough. In this article, we study the sensitivity of dynamic scaling of different models on rough substrates generated by standard surface growth models. We implement the three growth procedures which are as follows; for the first two procedures two separate substrates are grown with the Random deposition (RD) and Restricted solid on solid (RSOS) for different time steps t1 before growing them both with the Ballistic deposition model (BD). But in the third growth procedure, as an inverse of the second procedure, particles deposit on the substrates by the BD model and become rough by the RSOS model. We conclude that the surfaces need a characteristic growth time (tx) to lose information about the substrate. We show that the characteristic time depends on both the time needed for roughening the substrate and the special surface growth models. In addition, we show that the dependence of the two parameters ‘tx’ and ‘t1’ is a power law dependence only for adequate short times of t1 which is also calculated.

Exact solution of the discrete (1+1)-dimensional RSOS model in a slit with field and wall interactions

We present the solution of a linear restricted solid-on-solid (RSOS) model confined to a slit. We include a field-like energy, which equivalently weights the area under the interface, and also include independent interaction terms with both walls. This model can also be mapped to a lattice polymer model of Motzkin paths in a slit interacting with both walls including an osmotic pressure. This work generalizes the previous work on the RSOS model in the half-plane which has a solution that was shown recently to exhibit a novel mathematical structure involving basic hypergeometric functions 3ϕ2. Because of the mathematical relationship between the half-plane and slit this work hence effectively explores the underlying q-orthogonal polynomial structure to that solution. It also generalizes two other recent works: one on Dyck paths weighted with an osmotic pressure in a slit and another concerning Motzkin paths without an osmotic pressure term in a slit.

Exact solution of the discrete (1+1)-dimensional RSOS model with field and surface interactions

We present the solution of a linear restricted solid-on-solid (RSOS) model in a field. Aside from the origins of this model in the context of describing the phase boundary in a magnet, interest also comes from more recent work on the steady state of non-equilibrium models of molecular motors. While similar to a previously solved (non-restricted) SOS model in its physical behaviour, mathematically the solution is more complex. Involving basic hypergeometric functions 3ϕ2, it introduces a new form of solution to the lexicon of directed lattice path generating functions.

A restricted solid-on-solid model for growth on fractal substrates

The restricted solid-on-solid (RSOS) model for growth on twodifferent fractal substrates having the same fractal dimensiondf = 2 is studied. The Sierpinski gasket and the checkerboard fractal embedded in threedimensions are considered as the substrates. It is found that the interface widthW growsas tβ withβ≈0.26 for growth on a Sierpinskigasket and with β≈0.30 for growth on a checkerboard fractal. For the saturated regime,W followsW∼Lα,L being the size of thesystem, with α≈0.50 for aSierpinski gasket and α≈0.56 for a checkerboard fractal, implying that the growing surfaces of fractal substratesare rougher than that of a regular substrate. The growth exponent is notfully determined by the fractal dimension only, and the dynamic exponentz, obtained fromthe relation z = α/β, for both fractal lattices does not satisfy the scaling relationα+z = 2 due to the intrinsic fractal nature of the substrate. The RSOS model for growth on aregular lattice is generally believed to be described by the Kardar–Parisi–Zhang (KPZ)equation. However, the RSOS model for fractal substrates does not appear to follow theKPZ type universality. Generalization to the equilibrium RSOS model for growth on thefractal substrates is also investigated.

Initial-stage growth phenomena and distribution of local configurations of the restricted solid-on-solid model.

We take a detailed study on the restricted solid-on-solid (RSOS) model with finite nearest-neighbor height difference S. We numerically show that, for all finite values of S, the system belongs to the random-deposition (RD) class in the early time stage and then crossovers to the Kardar-Parisi-Zhang (KPZ) class. We find that the crossover time scales as Szeta with the crossover exponent zeta=2.06. Besides, we analytically study the RSOS model by grouping consecutive sites into local configurations to obtain the Markov chain describing the time evolution of the probability distribution of these local configurations. For demonstration, we use the RSOS model with S=2 as an explicit example and calculate the correlation functions and even scaling exponents based on the obtained probability distribution of local configurations. The results are very consistent with those obtained from direct simulation of the RSOS model.

RESTRICTED SOLID-ON-SOLID MODEL: FROM LOCAL MORPHOLOGY TO CORRELATION FUNCTIONS AND SCALING EXPONENTS

We study the restricted solid-on-solid (RSOS) model by grouping consecutive sites into local configurations and obtain the master equations of the probability distribution of these local configurations in closed forms. The obtained solutions to these equations fit very well with those from direct computer simulation of the RSOS model. To demonstrate the effectiveness of this new approach for studying interfacial phenomena, we then calculate the correlation functions and even scaling exponents based on this obtained probability distribution of local configurations. The results are also consistent very well with those obtained from the KPZ equation or direct simulation of the RSOS model.

Relaxation to steady states and dynamical exponents in deposition models

Considering some deposition models with limited mobility, we show that the typical decay of the interface width to its saturation value is exponential, which defines the crossover or saturation time τ. We present a method to calculate a characteristic time τ 0 proportional to τ and estimate the dynamical exponent z. In one-dimensional substrates of lengths L⩽2048, the method is applied to the Family model, the restricted solid-on-solid (RSOS) model and the ballistic deposition. Effective exponents z L converge to asymptotic values consistent with the corresponding continuum theories. For the two-dimensional Family model, the expected dynamic scaling hypothesis suggests a particular definition of τ 0 that leads to z=2, improving previous calculations based on data collapse methods. For the two-dimensional RSOS model, we obtain z≈1.6 and α<0.4, in agreement with recent large-scale simulations.

Universality class of the restricted solid-on-solid model with hopping.

We study the restricted solid-on-solid (RSOS) model with finite hopping distance l(0), using both analytical and numerical methods. Analytically, we use the hard-core bosonic field theory developed by the authors [Phys. Rev. E 62, 7642 (2000)] and derive the Villain-Lai-Das Sarma (VLD) equation for the l(0)=infinity case, which corresponds to the conserved RSOS (CRSOS) model and the Kardar-Parisi-Zhang (KPZ) equation for all finite values of l(0). Consequently, we find that the CRSOS model belongs to the VLD universality class and that the RSOS models with any finite hopping distance belong to the KPZ universality class. There is no phase transition at a certain finite hopping distance contrary to the previous result. We confirm the analytic results using the Monte Carlo simulations for several values of the finite hopping distance.

Derivation of continuum stochastic equations for discrete growth models.

We present a formalism to derive the stochastic differential equations (SDEs) for several solid-on-solid growth models. Our formalism begins with a mapping of the microscopic dynamics of growth models onto the particle systems with reactions and diffusion. We then write the master equations for these corresponding particle systems and find the SDEs for the particle densities. Finally, by connecting the particle densities with the growth heights, we derive the SDEs for the height variables. Applying this formalism to discrete growth models, we find the Edwards-Wilkinson equation for the symmetric body-centered solid-on-solid (BCSOS) model, the Kardar-Parisi-Zhang equation for the asymmetric BCSOS model and the generalized restricted solid-on-solid (RSOS) model, and the Villain-Lai-Das Sarma equation for the conserved RSOS model. In addition to the consistent forms of equations for growth models, we also obtain the coefficients associated with the SDEs.

Surface roughening: Kinetics, adsorbate-induced effects, and manifestation in catalytic reactions

Using the Monte Carlo (MC) technique, we have analyzed the kinetics of roughening of a clean surface at T&amp;gt;TR (TR is the roughening temperature), the kinetics of relaxation from a roughened to a flat surface at T&amp;lt;TR, the effect of adsorption on surface roughening, and the influence of adsorbate-induced roughening on the kinetics of the catalytic 2A+B2→2AB reaction. (i) For a clean surface, the results were obtained by employing the standard restricted-solid-on-solid (RSOS) model with surface diffusion occurring via jumps to nearest-neighbor and next-nearest-neighbor sites. The kinetics of surface roughening at T&amp;gt;TR and of surface relaxation at T&amp;lt;TR are found to be logarithmic in time. In particular, the mean-square height difference at T&amp;gt;TR increases with time as w2∝ln t. (ii) Complementing the RSOS model by terms describing adsorption, we demonstrate that adsorbate-induced surface roughening may occur with significant lowering of TR, due, respectively, to indirect next-nearest-neighbor adsorbate–substrate repulsion, attractive nearest- neighbor adsorbate–substrate lateral interaction, and/or repulsive adsorbate-adsorbate lateral interaction. In all these cases, the ratio of TRnear adsorbate saturation and for the clean surface, respectively, is given by an analytical expression that is a simple combination of the corresponding interactions. The coverage dependence of TR in MC simulations is nonlinear and slightly different compared to the mean-field-type prediction. (iii) The effect of adsorbate-induced roughening on a catalytic reaction can be dramatic even at weak roughening, with qualitative and quantitative changes in the kinetics. For example, the absolute conversion rate at maximum rate, and the reactant mixture corresponding to this maximum can be very different on the flat and roughened surface. The behavior of the adsorbate coverages near the points corresponding to the kinetic phase transitions can change character qualitatively.

Numerical Examination of the Universality of the Equilibrium Crystal Shape in Some SOS Models

By means of the Monte Carlo twist method, we examine the universality of the following critical properties of the equilibrium crystal shape in the discrete Gaussian (DG) model and restricted solid-on-solid (RSOS) model: c1) a finite curvature jump at the roughening temperature T R , c2) the profile of a facet with a singularlity at the facet edge for T < T R , and c3) the radius of the facet near T R proportional to \(\exp(-a/ \sqrt{1-T/T_{R}})\). We obtain the following results with the help of some results obtained analytically. The DG and RSOS models have T R =1.44 ±0.02 and 1.60 ±0.05, respectively. c2) is universal for these models. For the DG model c1) well agrees with the exact result of the body-centered SOS model, whereas for the RSOS model it is almost equal to twice the latter. Finite-size-scaling for c3) is excellent for the DG model, while it is consistent but inconclusive for the RSOS model.

Scaling properties of driven interfaces: Symmetries, conservation laws, and the role of constraints

One-dimensional models of surface dynamics are studied analytically and by numerical simulation. In all cases the number of particles that form the deposit is conserved. We find that an initially flat surface, in general, roughens as a function of time and can be characterized by a width \ensuremath{\xi} which obeys the scaling form \ensuremath{\xi}(L,t)=${\mathit{L}}^{\mathrm{\ensuremath{\chi}}}$f(t/${\mathit{L}}^{\mathit{z}}$) for a deposit on a substrate of linear dimension L. Restricted solid-on-solid (RSOS) -type models in which the microscopic dynamics obeys detailed balance are shown to be in the ``free-field'' universality class with exponents z=4 and \ensuremath{\chi}=1/2. On the other hand, if detailed balance is broken, several universality classes exist. As the ``maximum-height-difference'' constraint in a RSOS model is varied, one can observe a phase transition between a flat phase (z=2, \ensuremath{\chi}=0) and a grooved phase characterized by a steady-state exponent \ensuremath{\chi}=1 but with no scaling in the relaxation process. At the transition point, a nontrivially rough surface emerges with exponents (z\ensuremath{\approxeq}3.67, \ensuremath{\chi}\ensuremath{\approxeq}0.33) close to those of the conserved Kardar-Parisi-Zhang equation. We propose a phenomenological model that may account for these observations.

Surface phase transitions: Roughening, preroughening, orientational roughening, and reconstruction in an isotropic frustrated Ising model.

We study the roughening, the preroughening, and the orientational roughening transitions for interfaces between phases in the isotropic competing Ising Hamiltonian H=-J${\mathrm{\ensuremath{\Sigma}}}_{\mathit{nm}}$${\mathit{s}}_{\mathit{i}}$${\mathit{s}}_{\mathit{j}}$-2M ${\mathrm{\ensuremath{\Sigma}}}_{\mathit{lnnn}}$${\mathit{s}}_{\mathit{i}}$${\mathit{s}}_{\mathit{j}}$. The symbols, NN, DNN, and LNNN represent nearest-neighbor, diagonal-nearest-neighbor, and linear-next-nearest-neighbor pairs, respectively, with Jg0 and -\ensuremath{\infty}M\ensuremath{\infty}. By performing low-temperature expansions in the three-dimensional cubic lattice, it is found that the roughening transition curve passes through the region in which paramagnetic, ferromagnetic, and modulated phases merge, and is parallel to the phase boundary of ferromagnetic and modulated phases. It is also found that crossover behavior occurs in the roughening transition from the surface tension dominant regime to the bending-energy dominant regime. We examine the preroughening transition for the case of the restricted solid-on-solid (RSOS) model. The preroughening transition line is found to be distinct from the roughening transition line. This confirms the recent prediction by den Nijs [Phys. Rev. Lett. 64, 435 (1990)]. Within the RSOS model, we find a multistate point for reconstructed surface structures. This is analogous to the multistate point in the bulk phase diagram. The orientational roughening transition is also examined. However, it was not possible to obtain the orientational roughening transition line at a temperature higher than that of the translational roughening transition, presumably due to the limited order of the low-temperature series. The results deduced from the low-temperature expansions are compared to the sine-Gordon renormalization-group calculations based on the SOS and RSOS models. Finally we discuss the implications of the surface structure, and the connections with bulk structure.

Scattering Amplitudes in Offcritical Models and RSOS Integrable Models

In the present paper we study scaling field theories related to the class of exactly solvable noncritical models of statistical mechanics which are known as RSOS (restricted solid-on-solid) or IRF (interaction round a face) models. These models are related to simple Lie algebras and they generalize the integrable version of RSOS model proposed by Andrews, Baxter and Forrester. 1> The models related to gl(n) were found by Jimbo, Miwa and Okado,2> the models related to other classical Lie algebras were obtained by the same authors in Ref. 3) and by Pasquier in Ref. 4) for Dn-case with a special coupling constant. Critical properties of JMO model were found in Ref. 5) where the authors obtain the explicit formulas for magnetization in the model and compute its critical behavior. It was established there that the critical behavior of these models was described by coset conformal field theories. 6>.7> Therefore it is natural to expect that corre­ sponding scaling field theories will be integrable perturbations of these conformal field theories. s> In this paper we describe the thermodynamics of the noncritical magnetic chain. Then we find the ground state, excitations over the ground state and the scattering amplitudes of excitations. By getting the scaling limit over the ground state of the model we compute the central charges of corresponding field theories, the dimensions of perturbations, the spectrum and scattering amplitudes of excitations. Under certain assumptions we generalize these results for any Lie algebra of A-D-E type. In certain simplest cases this gives known S-matrices like Es, E1, Es scalar S-matrices.9> In certain other models matrix elements of the S-matrix are given by Pasquier's solutions 4> of the Yang-Baxter equation. We do not discuss here the relation of RSOS models to integrable models of vertex type. 23>' 25> Partially the relation was explained in Refs. 4) and 26). The relation between the spectrum of row-to-row transfer matrices in RSOS and in vertex models will be given in a separate publication. For some of the scaling models the same answer was obtained by Bernard and