Villain Model Research Articles

Massless phases for the Villain model in d ≥ 3

Massless phases for the Villain model in d ≥ 3

Quantitative bounds on vortex fluctuations in 2d$2d$ Coulomb gas and maximum of the integer‐valued Gaussian free field

AbstractIn this paper, we study the influence of the vortices on the fluctuations of systems such as the Coulomb gas, the Villain model, or the integer‐valued Gaussian free field (GFF). In the case of the Villain model, we prove that the fluctuations induced by the vortices are at least of the same order of magnitude as the ones produced by the spin wave. We obtain the following quantitative upper bound on the two‐point correlation in when The proof is entirely nonperturbative. Furthermore, it provides a new and algorithmically efficient way of sampling the Coulomb gas. For the Coulomb gas, we obtain the following lower bound on its fluctuations at high inverse temperature: This estimate coincides with the predictions based on Renormalization group (RG) analysis by José et al. [Phys. Rev. B 16 (1977), no. 3, 1217] and suggests that the Coulomb potential at inverse temperature should scale like a GFF of inverse temperature of order .Finally, we transfer the above vortex fluctuations via a duality identity to the integer‐valued GFF by showing that its maximum deviates in a quantitative way from the maximum of a usual GFF. More precisely, we show that with high probability when where is an integer‐valued GFF in the box at inverse temperature . Applications to the free energies of the Coulomb gas, the Villain model, and the integer‐valued GFF are also considered.

Villain model with long-range couplings

The nearest-neighbor Villain, or periodic Gaussian, model is a useful tool to understand the physics of the topological defects of the two-dimensional nearest-neighbor XY model, as the two models share the same symmetries and are in the same universality class. The long-range counterpart of the two-dimensional XY has been recently shown to exhibit a non-trivial critical behavior, with a complex phase diagram including a range of values of the power-law exponent of the couplings decay, σ, in which there are a magnetized, a disordered and a critical phase [1]. Here we address the issue of whether the critical behavior of the two-dimensional XY model with long-range couplings can be described by the Villain counterpart of the model. After introducing a suitable generalization of the Villain model with long-range couplings, we derive a set of renormalization-group equations for the vortex-vortex potential, which differs from the one of the long-range XY model, signaling that the decoupling of spin-waves and topological defects is no longer justified in this regime. The main results are that for σ < 2 the two models no longer share the same universality class. Remarkably, within a large region of its the phase diagram, the Villain model is found to behave similarly to the one-dimensional Ising model with 1/r2 interactions.

Finite temperature off-diagonal long-range order for interacting bosons

Characterizing the scaling with the total particle number ($N$) of the largest eigenvalue of the one--body density matrix ($\lambda_0$), provides informations on the occurrence of the off-diagonal long-range order (ODLRO) according to the Penrose-Onsager criterion. Setting $\lambda_0\sim N^{\mathcal{C}_0}$, then $\mathcal{C}_0=1$ corresponds to ODLRO. The intermediate case, $0<\mathcal{C}_0<1$, corresponds for translational invariant systems to the power-law decaying of (non-connected) correlation functions and it can be seen as identifying quasi-long-range order. The goal of the present paper is to characterize the ODLRO properties encoded in $\mathcal{C}_0$ [and in the corresponding quantities $\mathcal{C}_{k \neq 0}$ for excited natural orbitals] exhibited by homogeneous interacting bosonic systems at finite temperature for different dimensions. We show that $\mathcal{C}_{k \neq 0}=0$ in the thermodynamic limit. In $1D$ it is $\mathcal{C}_0=0$ for non-vanishing temperature, while in $3D$ $\mathcal{C}_0=1$ ($\mathcal{C}_0=0$) for temperatures smaller (larger) than the Bose-Einstein critical temperature. We then focus our attention to $D=2$, studying the $XY$ and the Villain models, and the weakly interacting Bose gas. The universal value of $\mathcal{C}_0$ near the Berezinskii--Kosterlitz--Thouless temperature $T_{BKT}$ is $7/8$. The dependence of $\mathcal{C}_0$ on temperatures between $T=0$ (at which $\mathcal{C}_0=1$) and $T_{BKT}$ is studied in the different models. An estimate for the (non-perturbative) parameter $\xi$ entering the equation of state of the $2D$ Bose gases, is obtained using low temperature expansions and compared with the Monte Carlo result. We finally discuss a double jump behaviour for $\mathcal{C}_0$, and correspondingly of the anomalous dimension $\eta$, right below $T_{BKT}$ in the limit of vanishing interactions.

Statistical investigation on the dynamic behaviors of maximal spatial persistence in saturated surfaces of discrete growth models

The dynamic scaling behaviors of the maximal spatial persistence τmax in saturated surface of (1+1)-dimensional discrete growth models, including Wolf–Villain model, Family model, Ballistic Deposition model and Restricted Solid-on-Solid model, are investigated by means of Kinetic Monte Carlo method. Linear relationships are found for the mean value <τmax(L)>≈A⋅L, and the root-mean square deviation (i.e. the second moment) σ(L)≈B⋅L (where σ=(〈τmax2〉−〈τmax〉2)1∕2), respectively, versus the system size L. The distribution of the maximal spatial persistence is also analyzed, and data collapsing for different system sizes to a single curve implying good scaling behavior. It is shown that the distribution of maximal spatial persistence does not satisfy the commonly used extreme statistical distribution, namely Weibull and Gumbel distribution, but can better conform to Asym2Sig distribution.

Lee\u2013Yang Property and Gaussian Multiplicative Chaos

The Lee–Yang property of certain moment generating functions having only pure imaginary zeros is valid for Ising type models with one-component spins and XY models with two-component spins. Villain models and complex Gaussian multiplicative chaos are two-component systems analogous to XY models and related to Gaussian free fields. Although the Lee–Yang property is known to be valid generally in the first case, we show that is not so in the second. Our proof is based on two theorems of general interest relating the Lee–Yang property to distribution tail behavior.

An electric-field representation of the harmonic XY model

The two-dimensional harmonic XY (HXY) model is a spin model in which the classical spins interact via a piecewise parabolic potential. We argue that the HXY model should be regarded as the canonical classical lattice spin model of phase fluctuations in two-dimensional condensates, as it is the simplest model that guarantees the modular symmetry of the experimental systems. Here we formulate a lattice electric-field representation of the HXY model and contrast this with an analogous representation of the Villain model and the two-dimensional Coulomb gas with a purely rotational auxiliary field. We find that the HXY model is a spin-model analogue of a lattice electric-field model of the Coulomb gas with an auxiliary field, but with a temperature-dependent vacuum (electric) permittivity that encodes the coupling of the spin vortices to their background spin-wave medium. The spin vortices map to the Coulomb charges, while the spin-wave fluctuations correspond to auxiliary-field fluctuations. The coupling explains the striking differences in the high-temperature asymptotes of the specific heats of the HXY model and the Coulomb gas with an auxiliary field. Our results elucidate the propagation of effective long-range interactions throughout the HXY model (whose interactions are purely local) by the lattice electric fields. They also imply that global spin-twist excitations (topological-sector fluctuations) generated by local spin dynamics are ergodically excluded in the low-temperature phase. We discuss the relevance of these results to condensate physics.

Critical Properties of the Two-Dimensional Villain Model with Ferromagnetic Impurities

This work investigates the square lattice in the presence of an-ti-ferromagnetic exchange considering the two-dimensional Ising Villain model. A computational scheme is constructed to evaluate the degeneracy of ground states at zero temperature employing the Pfaffian method through perturbation theory. The entropy and canonical spin glass of the considered model are precisely obtained. The critical point between the Villain phase and canonical spin glass phase with the line fit of entropy at the thermodynamic limit is investigated. It is also shown that the transition point of the Villain model lies in the low concentration regime.

Schramm–Loewner evolution theory of the asymptotic behaviors of (2+1)-dimensional Wolf–Villain model

Abstract Although extensive analytical and numerical work has focus on investigating the (2+1)-dimensional Wolf–Villain (WV) model, some problems concerning its asymptotical behaviors, such as the universality class to which it belongs remain controversial. The Schramm–Loewner evolution (SLE) theory is an attractive approach to describe the fluctuation phenomena and random processes, and has also been applied to the analysis of the stochastic growth of surfaces. In this work, we applied SLE theory to the analysis of the saturated surface to conduct in-depth research, with a new perspective, on the asymptotical behaviors of the (2+1)-dimensional WV model. On the basis of the analysis of the saturated surface contour lines, we determine that the diffusion coefficient calculated for (2+1)-dimensional WV model is κ = 2.91 ± 0.01 , and that for Family model is κ = 2.88 ± 0.01 . Accordingly we can conclude from the view of SLE theory, that the (2+1)-dimensional WV model, similar to the Family model, also belongs to the Edwards–Wilkinson universality class.

Spin Glass Phase Exists in the Random Weak Disorder for the Villain Model

In this work we have studied non random Villain model by introducing simple defects to calculate degeneracies of the first excited states using Pfaffian approach through a perturbation theory. The distributions of excitations of the ground states are displayed graphically. The results are indicated that spin glass occurs in the weak disorder for the Villain model. At the concentration of defect bonds p=0.03, the distribution behaves in the same manner as for p=0.5 for different sizes of lattice. The latest result of the spin glass is presented in this paper.

Villain's fractal growth of poly[1-[4-(3-carboxy-4-hydroxyphenylazo) benzenesulfonamido]-1,2-ethanediyl, sodium salt] J-aggregates onto layer-by-layer films and its effect on film absorbance spectrum

Morphology of poly(allylamine hydrochloride) and poly[1-[4-(3-carboxy-4-hydroxyphenylazo) benzenesulfonamido]-1,2-ethanediyl, sodium salt] (PAZO) layer-by-layer (LBL) films is shown to influence the orientation of PAZO chromophores with respect to solid support surface, which in turn is related with observed red-shifts changes of the chromophore absorbance peak position relative to that of solution spectrum, as the bilayers are being deposited. For the first bilayers, an increase of red shift values is observed, while roughness and grain radius are kept practically constant; after the 5th bilayer, the red-shift values decrease, while grain sizes increase and the number of grains decreases. This behavior is consistent with adsorption of coiled PAZO molecules, treated as pseudo-particles, with the chromophores head-to-head oriented-J aggregates. These aggregates adsorb perpendicularly to the substrate surface for the first layers and, as roughness and grain radius increase, the adsorption of the J aggregates takes place parallel to the solid support surface, which gives rise to a decrease in the red shift value. Moreover, the adsorption of these pseudo-particles follows a fractal growth characterized by a scaling exponent of α = 0.80 ± 0.02 and a temporal growth exponent of β = 0.17 ± 0.02. These values suggest a layer growth according with Villain model, which accounts for the interactions between deposited particles and the surface. This is in accordance with the electrostatic forces driving LbL film formation and accounts for the observed morphology behavior for the different number of layers.

CONTROLLED RANDOMNESS AND FRUSTRATION OF THE MANY-BODY INTERACTIONS

On the Ising spin systems constructed with four-body interactions, there appears the "spin-pair ordered phase" or "spin-pair glass phase" in a certain temperature. Especially, the random four-body models include the frustration and the spin-pair glass phase appears in those models. This frustration may play an important role of spin-pair glass transitions, similarly to the case of ordinary spin-glass transitions. In this study, we clarify the way to discriminate quantitatively "frustrated unit cells" from "nonfrustrated ones" on the random four-body models. Then we introduce a parameter to control the frustration in the Ising spin systems with random four-body interactions. This parameter enables us to analyze the frustration continuously in many-body models. Thus we have analyzed the phase transitions and obtained phase diagrams using the frustration parameter. These interesting phase diagrams show that there appear the spin-pair ordered phases even on the completely frustrated models. This result is essentially different from the random two-body models, namely spin-glass models (there appears no ferromagnetic phase in the fully frustrated spin-glass models, which correspond to the Villain models).

Nanostructured films from phthalocyanine and carbon nanotubes: Surface morphology and electrical characterization

We report on the investigation of the surface morphology and DC conductivity of nanostructured layer-by-layer (LbL) films from nickel tetrasulfonated phthalocyanine (NiTsPc) alternated with either multi-walled carbon nanotubes (MWNTs/NiTsPc) or multi-walled carbon nanotubes dispersed in chitosan (MWNTs+Ch/NiTsPc). We have explored the surface morphology of the films by using fractal concepts and dynamic scale laws. The MWNTs/NiTsPc LbL films were found to have a fractal dimension of ca. 2, indicating a quasi Euclidean surface. MWNTs+Ch/NiTsPc LbL films are described by the Lai–Das Sarma–Villain (LDV) model, which predicts the deposition of particles and their subsequent relaxation. An increase in the wetting contact angle of MWNTs+Ch/NiTsPc LbL films was observed, as compared with MWNTs/NiTsPc LbL films, which presented an increase in the fractal dimension of the first system. Room temperature conductivities were found be ca. 0.45S/cm for MWNTs/NiTsPc and 1.35S/cm for MWNTs+Ch/NiTsPc.

First-order transition in theXYmodel on a fully frustrated simple cubic lattice

We study the nature of the phase transition in the fully frustrated simple cubic lattice with the XY spin model. This system is the Villain's model generalized in three dimensions. The ground state is very particular with a 12-fold degeneracy. Previous studies have shown unusual critical properties. With the powerful Wang-Landau flat-histogram Monte Carlo method, we carry out in this work intensive simulations with very large lattice sizes. We show that the phase transition is clearly of first order, putting an end to the uncertainty which has lasted for more than 20 years.

Mound morphology of the 2+1 -dimensional Wolf–Villain model caused by the step-edge diffusion effect

The mound morphology of the 2+1-dimensional Wolf–Villain model is studied by numerical simulation. The diffusion rule of this model has an intrinsic mechanism, i.e., the step-edge diffusion, to create a local uphill particle current, which leads to the formation of the mound. In the simulation, a noise reduction technique is employed to enhance the local uphill particle current. Our results for the dynamic exponent 1 / z and the roughness exponent α obtained from the surface width show a dependence on the strength of the step-edge diffusion. On the other hand, λ ( t ) , which describes the separation of the mounds, grows as a function of time in a power-law form in the regime where the coalescence of mounds occurs, λ ( t ) ∼ t n , with n ≈ 0.23 – 0.25 for a wide range of the deposition conditions under the step-edge diffusion effect. For m = 1 , a noise reduction factor of unity, the behavior of λ ( t ) in the saturated regime is also simulated. We find that the evolution behavior of λ ( t ) in the whole process can be described by the standard Family–Vicsek scaling.

Spin vortices and vacancies: Interactions and pinning on a square lattice

The study gives a decisive answer to the recently risen question about the type and origin of interaction between spin vortices and spin vacancies in two-dimensional (2D) spin models. The approach is based on the low-temperature approximation of the 2D $XY$ model known as the Villain model and does not involve any additional approximations, thus preserving the lattice structure. The exact form of the Hamiltonian describing a system of topological charges and a vacant site supports the attractive type of interaction between the vacancy and the charges. The quantitative difference between the characteristics of the vortex behavior in the 2D $XY$ and Villain models due to the different energy of the vortex cores in the two models is pointed out. This leads to a conclusion that the interaction between a vortex and a spin vacancy and between a vortex and the antivortex differs quantitatively for small separations in the two mentioned models.

Extensive numerical study of the anomalous dynamic scaling of the Wolf–Villain model

In order to study the microscopic physical mechanisms of roughness surfaces exhibiting the anomalous scaling behavior, the Wolf–Villain model in 1 + 1 and 2 + 1 dimensions is investigated by the kinetic Monte-Carlo simulation on long time and large length scale (the growth time and the system size are respectively extended to t = 2 29 , L = 400 000 for 1 + 1 dimensions, and t = 2 21 , L × L = 512 × 512 for 2 + 1 dimensions). In the 2+1-dimensional simulations, the noise reduction technique is employed so as to eliminate the crossover effects in the growth process. Our calculations show that the Wolf–Villain model in 1 + 1 dimensions very probably exhibits intrinsic anomalous scaling behavior in the time and length simulation range of this paper, and the 2 + 1 -dimensional Wolf–Villain model leads to a pyramidal mounded morphology. Some properties of the mounded pattern in the 2 + 1 -dimensional Wolf–Villain model are discussed in the final part of this presentation.

Scaling law analysis of paraffin thin films on different surfaces

The dynamics of paraffin deposit formation on different surfaces was analyzed based on scaling laws. Carbon-based films were deposited onto silicon (Si) and stainless steel substrates from methane (CH4) gas using radio frequency plasma enhanced chemical vapor deposition. The different substrates were characterized with respect to their surface energy by contact angle measurements, surface roughness, and morphology. Paraffin thin films were obtained by the casting technique and were subsequently characterized by an atomic force microscope in noncontact mode. The results indicate that the morphology of paraffin deposits is strongly influenced by substrates used. Scaling laws analysis for coated substrates present two distinct dynamics: a local roughness exponent (αlocal) associated to short-range surface correlations and a global roughness exponent (αglobal) associated to long-range surface correlations. The local dynamics is described by the Wolf–Villain model, and a global dynamics is described by the Kardar–Parisi–Zhang model. A local correlation length (Llocal) defines the transition between the local and global dynamics with Llocal approximately 700nm in accordance with the spacing of planes measured from atomic force micrographs. For uncoated substrates, the growth dynamics is related to Edwards–Wilkinson model.

Duality argument for chiral-nematic phase of planar spins

A duality argument for the recently discovered chiral-nematic phase of the XY model in a triangular lattice is presented. We show that a new Ising variable naturally emerges in mapping the antiferromagnetic J1-J2 classical XY spin Hamiltonian onto an appropriate Villain model on a triangular lattice. The new variable is the chirality degree of freedom, which exists in addition to the usual vortex variables, in the dual picture. Elementary excitations and the associated phase transition of the Ising degrees of freedom are discussed in some detail.

Nonequilibrium dynamics of fully frustrated Ising models atT = 0

We consider two fully frustrated Ising models: the antiferromagnetic triangular model in a field ofstrength h = HTkB, as well as the Villain model on the square lattice. After a quench from a disordered initial state toT = 0 we studythe nonequilibrium dynamics of both models by Monte Carlo simulations. In a finite system of linearsize, L, we define and measure sample-dependent ‘first passage time’,tr, which isthe number of Monte Carlo steps until the energy is relaxed to the ground state value. The distribution oftr, in particular its meanvalue, ⟨tr(L)⟩, is shown to obeythe scaling relation, ⟨tr(L)⟩∼L2ln(L/L0), for both models. Scaling of the autocorrelation function of the antiferromagnetictriangular model is shown to involve logarithmic corrections, both atH = 0 andat the field-induced Kosterlitz–Thouless transition: however, the autocorrelation exponent is foundto be H-dependent.