High-precision Monte Carlo Simulations Research Articles

Surface critical behavior of the three-dimensional O(3) model

We report results of high-precision Monte Carlo simulations of a three-dimensional lattice model in the O(3) universality class, in the presence of a surface. By a finite-size scaling analysis we have proven the existence of a special surface transition, computed the associated critical exponents, and shown the presence of an extraordinary phase with logarithmically decaying correlations.

Radial lattice quantization of 3D ϕ4 field theory

The quantum extension of classical finite elements, referred to as quantum finite elements ({\bf QFE})~\cite{Brower:2018szu,Brower:2016vsl}, is applied to the radial quantization of 3d $\phi^4$ theory on a simplicial lattice for the $\mathbb R \times \mathbb S^2$ manifold. Explicit counter terms to cancel the one- and two-loop ultraviolet defects are implemented to reach the quantum continuum theory. Using the Brower-Tamayo~\cite{Brower:1989mt} cluster Monte Carlo algorithm, numerical results support the QFE ansatz that the critical conformal field theory (CFT) is reached in the continuum with the full isometries of $\mathbb R \times \mathbb S^2$ restored. The Ricci curvature term, while technically irrelevant in the quantum theory, is shown to dramatically improve the convergence opening, the way for high precision Monte Carlo simulation to determine the CFT data: operator dimensions, trilinear OPE couplings and the central charge.

Four-point interfacial correlation functions in two dimensions. Exact results from field theory and numerical simulations

We derive exact analytic results for several four-point correlation functions for statistical models exhibiting phase separation in two-dimensions. Our theoretical results are then specialized to the Ising model on the two-dimensional strip and found to be in excellent agreement with high-precision Monte Carlo simulations.

Sample size calculation for Mann-Whitney U test with five methods

<p><strong>Background: </strong>Precise sample size estimation plays a vital role in the planning of a study specifically for medical treatment expenses that are expensive and studies that are of high risk.</p><p><strong>Methods: </strong>Among a variety of sample size calculation methods for the nonparametric Mann-Whitney U test, five potential methods are selected for evaluation in this article. The evaluation of method performance is based on the results obtained from high precision Monte Carlo simulations.</p><p><strong>Results: </strong>The sample size deviations (from the simulation ones) are performance indicators. The sum of the squared deviations over all scenarios is used as the criterion for ranking the five methods. For power comparisons, the percentage errors (relative to the simulated powers) are used. The effect size and target power both have large impacts on the minimum required sample sizes.</p><p><strong>Conclusions: </strong>Based on the ranking criterion, Shieh's method has the best performance. Noether's method always overestimates the minimum required sample sizes but not too severe.</p>

Multiparameter universality and conformal field theory for anisotropic confined systems: test by Monte Carlo simulations

Analytic predictions have been derived recently by Dohm and Wessel (2021 Phys. Rev. Lett. 126 060601) from anisotropic φ 4 theory and conformal field theory for the amplitude of the critical free energy of finite anisotropic systems in the two-dimensional Ising universality class. These predictions employ the hypothesis of multiparameter universality. We test these predictions by means of high-precision Monte Carlo (MC) simulations for of the Ising model on a square lattice with isotropic ferromagnetic couplings between nearest neighbors and with an anisotropic coupling between next-nearest neighbors along one diagonal. We find remarkable agreement between the MC data and the analytical prediction. This agreement supports the validity of multiparameter universality and invalidates two-scale-factor universality as is found to exhibit a nonuniversal dependence on the microscopic couplings of the scalar φ 4 model and the Ising model. Our results are compared with the exact result for in the three-dimensional φ 4 model with a planar anisotropy in the spherical limit. The critical Casimir amplitude is briefly discussed.

Boundary Critical Behavior of the Three-Dimensional Heisenberg Universality Class.

We study the boundary critical behavior of the three-dimensional Heisenberg universality class, in the presence of a bidimensional surface. By means of high-precision MonteCarlo simulations of an improved lattice model, where leading bulk scaling corrections are suppressed, we prove the existence of a special phase transition, with unusual exponents, and of an extraordinary phase with logarithmically decaying correlations. These findings contrast with naïve arguments on the bulk-surface phase diagram, and allow us to explain some recent puzzling results on the boundary critical behavior of quantum spin models.

Determination of electron backscattering coefficient of beryllium by a high-precision Monte Carlo simulation

We present an up-to-date Monte Carlo simulation of electron backscattering coefficient of beryllium, which is an important material in fusion reactor, at an impact energy range of electrons between 100 eV and 100 keV. The Mott’s cross section calculated with more accurate scattering potential and a relativistic dielectric functional formalism with full Penn algorithm and experimental optical data are used in the modelling of electron elastic and inelastic scatterings, respectively. This Monte Carlo simulation modelling enabled us to derive comprehensive theoretical values of backscattering coefficient, which are found significantly smaller than the previous published experimental data particularly at energies below 10 keV. Simulation of electron backscattering spectra was then carried out with and without Auger electron emission, which confirmed that Auger electrons have negligible contribution to backscattering coefficient. To validate our simulation results, we have also performed calculation for amorphous boron and carbon, while the comparison with the available experimental data shows reasonable agreement. Further simulation for carbon and water covered Be sample has revealed that the surface contamination in low vacuum conditions with several atomic/molecular layers can drastically alter the measurement values for beryllium at low energies. The low backscattering coefficient values of beryllium are partly attributed to the extremely strong forward elastic scattering for low atomic number element. Therefore, we recommend to use the here reported backscattering coefficient data of beryllium for applications.

Phase diagram for the bisected-hexagonal-lattice five-state Potts antiferromagnet.

In this paper we study the phase diagram of the five-state Potts antiferromagnet on the bisected-hexagonal lattice. This question is important since Delfino and Tartaglia recently showed that a second-order transition in a five-state Potts antiferromagnet is allowed, and the bisected-hexagonal lattice had emerged as a candidate for such a transition on numerical grounds. By using high-precision Monte Carlo simulations and two complementary analysis methods, we conclude that there is a finite-temperature first-order transition point. This one separates a paramagnetic high-temperature phase, and a low-temperature phase where five phases coexist. This phase transition is very weak in the sense that its latent heat (per edge) is two orders of magnitude smaller than that of other well-known weak first-order phase transitions.

Universality from disorder in the random-bond Blume-Capel model.

Using high-precision Monte Carlo simulations and finite-size scaling we study the effect of quenched disorder in the exchange couplings on the Blume-Capel model on the square lattice. The first-order transition for large crystal-field coupling is softened to become continuous, with a divergent correlation length. An analysis of the scaling of the correlation length as well as the susceptibility and specific heat reveals that it belongs to the universality class of the Ising model with additional logarithmic corrections which is also observed for the Ising model itself if coupled to weak disorder. While the leading scaling behavior of the disordered system is therefore identical between the second-order and first-order segments of the phase diagram of the pure model, the finite-size scaling in the ex-first-order regime is affected by strong transient effects with a crossover length scale L^{*}≈32 for the chosen parameters.

High-Precision Monte Carlo Simulation of the Ising Models on the Penrose Lattice and the Dual Penrose Lattice

We study the Ising models on the Penrose lattice and the dual Penrose lattice by means of the high-precision Monte Carlo simulation. Simulating systems up to the total system size N = 20633239, we estimate the critical temperatures on those lattices with high accuracy. For high-speed calculation, we use the generalized method of the single-GPU-based computation for the Swendsen–Wang multi-cluster algorithm of Monte Carlo simulation. As a result, we estimate the critical temperature on the Penrose lattice as Tc/J = 2.39781 ± 0.00005 and that of the dual Penrose lattice as \(T_{\text{c}}^{*}/J = 2.14987 \pm 0.00005\). Moreover, we definitely confirm the duality relation between the critical temperatures on the dual pair of quasilattices with a high degree of accuracy, \(\sinh (2J/T_{\text{c}})\sinh (2J/T_{\text{c}}^{*}) = 1.00000 \pm 0.00004\).

Improved CUDA programs for GPU computing of Swendsen–Wang multi-cluster spin flip algorithm: 2D and 3D Ising, Potts, and XY models

Improved CUDA programs for GPU computing of Swendsen–Wang multi-cluster spin flip algorithm: 2D and 3D Ising, Potts, and XY models

Application of classical nucleation theory to the formation of adhesion domains

Specific adhesion of membranes proceeds through the formation of discrete bonds, the interactions of which are strongly influenced by thermal membrane undulations. Here we study the applicability of classical nucleation theory to the formation of adhesion domains of an initially unbound membrane. By employing high precision Monte Carlo simulations for a minimal model, we obtain the free energy of an adhesion domain in the background of the unbound membrane. We find that the free energy as a function of the cluster size is well described by the corresponding expression from classical nucleation theory with the line tension as a free parameter. We determine a simple but exact relationship for the line tension of an adhesion domain that should be useful in order to provide an estimate of critical sizes and nucleation barriers in realistic situations.

Green’s Function Approach to the Bose-Hubbard Model

We use a diagrammatic hopping expansion to calculate finite-temperature Green functions of the Bose-Hubbard model which describes bosons in an optical lattice. This technique allows for a summation of subsets of diagrams, so the divergence of the Green function leads to non-perturbative results for the boundary between the superfluid and the Mott phase for finite temperatures. Whereas the first-order calculation reproduces the seminal mean-field result, the second order goes beyond and shifts the phase boundary in the immediate vicinity of the critical parameters determined by high-precision Monte-Carlo simulations of the Bose-Hubbard model. In addition, our Green’s function approach allows for calculating the excitation spectrum both for zero and finite temperature and for determining the effective masses of particles and holes.

Universal amplitude ratios in the Ising model in three dimensions

We use a high-precision Monte Carlo simulation to determine the universal specific-heat amplituderatio A + /A − in the three-dimensional Ising model via the impact angleϕ of complex temperature zeros. We also measure the correlation-length critical exponentν from finite-size scaling and the specific-heat exponentα through hyperscaling. Extrapolations to the thermodynamic limit yieldϕ = 59.2(1.0)°,A + /A − = 0.56(3),ν = 0.630 48(32) andα = 0.1086(10). These results are compatible with some previous estimates from a variety of sources andrule out recently conjectured exact values.

Phase behaviour and dynamics in primitive models of molecular ionic liquids

The phase behaviour and dynamics of molecular ionic liquids are studied using primitive models and extensive computer simulations. The models account for size disparity between cation and anion, charge location on the cation, and cation-shape anisotropy, which are all prominent features of important materials such as room-temperature ionic liquids. The vapour-liquid phase diagrams are determined using high-precision Monte Carlo simulations, setting the scene for in-depth studies of ion dynamics in the liquid state. Molecular dynamics simulations are used to explore the structure, single-particle translational and rotational autocorrelation functions, cation orientational autocorrelations, self diffusion, viscosity, and frequency-dependent conductivity. The results reveal some of the molecular-scale mechanisms for charge transport, involving molecular translation, rotation, and association.

Limits of the ratios of tracer diffusivities for diffusion by vacancy pairs: Application to compound semiconductors

In this paper, the vacancy-pair diffusion mechanism in cubic materials is revisited with emphasis on the limits of the ratio of the tracer diffusion coefficients to demonstrate the operation of this mechanism. The vacancy-pair mechanism puts very strict upper and lower bounds on this ratio. Extensive high precision Monte Carlo simulation is used to calculate tracer correlation factor, ratios of tracer diffusivities, and percolation effects (in mixed compounds). The three major cubic lattices—NaCl, CsCl, and zinc blende structures—are analyzed, and correct limits for the ratio of the tracer diffusivities are found to be 5.6, 11.2, and 2.6, respectively. For the case of the CsCl structure correlation factors are redetermined using the matrix method to obtain good agreement with the Monte Carlo results. Self-diffusion in the compound semiconductors PbSe (NaCl structure) InSb, GaSb, CdTe, and HgCdTe (zinc blende structure) is reanalyzed. It is shown that the vacancy-pair mechanism is unlikely to contribute to self-diffusion in CdTe and HgCdTe but it remains a valid possibility for self-diffusion in PbSe, InSb, and GaSb.

DOMAIN GROWTH IN COMPRESSIBLE 2D ISING MODELS UNDERGOING PHASE SEPARATION

High precision Monte Carlo simulations are used to study domain growth in a compressible two-dimensional spin-exchange Ising model with continuous particle positions and zero total magnetization. For mismatched systems, we measure significant deviations from the theoretically expected late-time domain growth, R(t) = A + Btn with n = 1/3. For a compressible model with no mismatch, we measure only a slight deviation from n = 1/3. Our results indicate that our current understanding of domain growth is incomplete.

High precision Monte Carlo simulations of interfaces in the three-dimensional Ising model: a comparison with the Nambu-Goto effective string model

Motivated by the recent progress in the effective string description of the interquark potential in lattice gauge theory, we study interfaces with periodic boundary conditions in the three-dimensional Ising model. Our Monte Carlo results for the associated free energy are compared with the next-to-leading order (NLO) approximation of the Nambu-Goto string model. We find clear evidence for the validity of the effective string model at the level of the NLO truncation.

Cluster growth and dynamic scaling in a two-lane driven diffusive system

Using high precision Monte Carlo simulations and a mean-field theory, we explore coarsening phenomena in a simple driven diffusive system. The model is reminiscent of vehicular traffic on a two-lane ring road. At sufficiently high density, the system develops jams (clusters) which coarsen with time. A key parameter is the passing probability, $\gamma$. For small values of $\gamma$, the growing clusters display dynamic scaling, with a growth exponent of 2/3. For larger values of $\gamma$, the growth exponent must be adjusted, suggesting the ordered (jammed) state is not a genuine phase but rather a finite size effect.

Potts correlators and the static three-quark potential

We discuss the two- and three-point correlators in the two-dimensional three-state Pottsmodel in the high temperature phase of the model. By using the form factor approach andperturbed conformal field theory methods we are able to describe both the large distanceand the short distance behaviours of the correlators. We compare our predictions with a setof high precision Monte Carlo simulations (performed on the triangular lattice realizationof the model) finding a complete agreement in both regimes. In particular we use thetwo-point correlators to fix the various non-universal constants involved in thecomparison (whose determination is one of the results of our analysis) and then usethese constants to compare numerical results and theoretical predictions for thethree-point correlator with no free parameter. Our results can be used to shed somelight on the behaviour of the three-quark correlator in the confining phase of the(2 + 1)-dimensional SU(3) lattice gauge theory which is related by dimensional reduction to the three-spin correlatorin the high temperature phase of the three-state Potts model. The picture which emerges isthat of a smooth crossover between a Δ-type law at short distances and aY-type law at large distances.