Binder Parameter Research Articles

Binder Parameter in a Generalized Two-dimensional XY Model

We performed Monte Carlo simulation for a two-dimensional generalized XY model and calculated the magnetic and nematic Binder parameters. The phase diagram is re-examined based on these Binder parameters, demonstrating their power in studying generalized XY models. The Binder parameter has distinctive behaviors, resulting in different types of phase transition. More importantly, the magnetic Binder parameter gives more insights into the tricritical region where the Kosterlitz-Thouless, Ising, and 1/2 Kosterlitz-Thouless (KT) transition lines meet. It shows signatures for the intermediate region starting from the tricritical point, where the transition line is neither the same physics as the Ising transition below nor the KT transition far above the tricritical point.

Dynamical Scaling Analysis for ±J Ising Model in Three Dimensions

The ±J Ising model in three dimensions is investigated using dynamical scaling analysis for a nonequilibrium relaxation process. The spin-glass (SG) transition is analyzed under an assumed dynamical exponent z proposed for the scaling analysis of the SG transition. To estimate dynamical exponent z precisely, we introduce an extrapolation scheme for the asymptotic value of the logarithmic derivative of the dynamical Binder parameter. We obtain the phase diagram in the temperature–concentration (p–T) space, as well as the p-dependence of critical exponents γ, ν, and z. For the static exponents of the SG transition, the universality is strongly indicated numerically along the whole region of the paramagnetic (PM)–SG phase boundary. A remarkable crossover behavior is observed for estimations of critical exponents at the boundary close to the multicritical point (MCP) of PM–ferromagnetic (FM)–SG phases. In addition, the dynamical assumption for z resulting from the slow relaxation in the SG region disappears in the FM region, and the crossover between them occurs around the Nishimori Line.

Non-monotonic behavior of the Binder parameter in discrete spin systems

Abstract We study a non-monotonic behavior of the Binder parameter, which appears in discrete spin systems. We show that the Binder parameters of the Potts model are non-monotonic for q = 3 and 4, while they are monotonic for the Ising case (q = 2). Using the Fortuin–Kasteleyn graph representation, we find that the improved estimator of the Binder parameter consists of two terms with values only in the high- and low-temperature regions. The non-monotonic behavior is found to originate from the low-temperature term. With the appropriately defined order parameter, we can reduce the influence of the low-temperature term and, as a result, the non-monotonic behavior can also be reduced. We propose new definitions for the order parameter, which reduces or eliminates the non-monotonic behavior of the Binder parameter in a system for which the improved estimator of the Binder parameter is unknown.

Magnetic properties of Mn doped ZnO: A Monte Carlo simulation analysis

Using the Monte Carlo simulation method within Ising model framework in two dimension, this work presents the study of the magnetic properties of Zn(1-x)MnxO (x = 0.05, 0.1, 0.2). In the simulation process, a classical Metropolis algorithm has been used to update the distribution of spins in the matrix and the algorithm is written in Python language. The simulation is based on the Ruderman–Kittel–Kasuya–Yosida (RKKY) indirect exchange interaction of doped Mn ions in host ZnO. We have analysed the results for a finite size system and have investigated the susceptibility, Binder parameter and specific heat as a function of temperature and doping concentration. The magnetic ordering is obtained for 5 and 10% doping at low temperature region while higher doping concentration (20%) leads to dominance of anti-ferromagnetic exchange interaction. Based on these theoretical results the prediction of magnetic properties of such kind dilute magnetic semiconductor (DMS) systems is possible. We believe that present theoretical analysis of the magnetic properties of DMS will provide valuable information about the two dimensional diluted system/structure.

Utilizing Simplified Viscoelastic Continuum Damage Model to characterize the fatigue behavior of Styrene-Butadiene-Styrene (SBS) modified binders

Rapid traffic growth, extreme climatic condition, and thickness reduction enthusiasm lead to an increase in the use of Styrene-Butadiene-Styrene (SBS) polymer in pavement industry. However, past studies have shown that the Superpave binder parameter for fatigue, |G*|sinδ has failed to characterize the fatigue performance of the polymer modified binders (PMBs) due to its complex structure and interaction with the binder. This study utilizes the Simplified Viscoelastic Continuum Damage (S-VECD) model with the newly developed Linear Amplitude Sweep (LAS) test to investigate the fatigue potential of PMBs with different content of SBS polymer and their correlation with mixture fatigue performance from the Four Point Flexural Beam (FPFB) test. This study also performs the Time Sweep (TS) fatigue test to verify the S-VECD prediction from the LAS test data with respect to measured fatigue life, Nf. Results show that the LAS test along with the S-VECD can successfully explain the fatigue performance of PMBs. It is found that the S-VECD predicted Nf from the LAS test data is consistent with the measured Nf from the TS test. This study also observes that there is a strong relationship between binder Nf from the LAS test and mixture fatigue endurance limit (FEL) from the FPFB test.

Finite-size scaling of Monte Carlo simulations for the fcc Ising antiferromagnet: Effects of the low-temperature phase degeneracy

The Ising antiferromagnet on a face-centered cubic (fcc) lattice with nearest-neighbor interaction only is well known to exhibit a macroscopic (exponential in the system size $L$) ground-state degeneracy. With increasing temperature, this degeneracy is expected to be lifted and the model undergoes a first-order phase transition. For a model with an exponential degeneracy in the whole low-temperature phase, it was recently found that the finite-size scaling behavior is governed by leading correction terms $\sim L^{-2}$ instead of $\sim L^{-3}$ as usual. To test the conjecture that such a transmuted behavior may effectively persist also for the fcc antiferromagnet up to some crossover system size, we have performed parallel multicanonical Monte Carlo simulations for lattices of linear size $L \le 18$ with periodic boundary conditions and determined various inverse pseudo phase transition temperatures, as well as the extremal values of the specific heat and the energetic Binder parameter. We indeed find that, for the simulated lattice sizes, the conjectured transmuted finite-size scaling ansatz fits the data better than the standard ansatz. On this basis, we extrapolate for the transition temperature an estimate of $T_0 = 1.735047(46)$.

Exact ground states of the Kaya-Berker model.

Here we study the two-dimensional Kaya-Berker model, with a site occupancy p of one sublattice, by using a polynomial-time exact ground-state algorithm. Thus, we were able to obtain T=0 results in exact equilibrium for rather large system sizes up to 777^{2} lattice sites. We obtained the sublattice magnetization and the corresponding Binder parameter. We found a critical point p_{c}=0.64(1) beyond which the sublattice magnetization vanishes. This is clearly smaller than previous results which were obtained by using nonexact approaches for much smaller systems. For each realization we also created minimum-energy domain walls from two ground-state calculations, for periodic and antiperiodic boundary conditions. The analysis of the mean and the variance of the domain-wall energy shows that there is no thermodynamic stable spin-glass phase at nonzero temperature, in contrast to previous claims about this model. For large values of p, the standard deviation of the domain-wall decreases with the system size like a power law with exponent roughly θ≃-0.1, which is different from the standard two-dimensional Ising spin glass where θ≃-0.29.

Rheological modeling of asphalt binder and asphalt mortar containing recycled asphalt material

The use of recycled materials in asphalt pavements has increased significantly over the years, with well-known environmental and economic benefits. However, the mechanism governing the interaction between the virgin asphalt binder and the aged binder contained in recycled asphalt pavement (RAP) is not entirely understood. In this paper, the effect of RAP on the rheological properties of asphalt binders and mortars was experimentally investigated and modeled with the objective of defining a relationship between the linear viscoelastic properties of binders and those of the corresponding mortars. First dynamic shear rheometer tests were performed on binders and mortars, prepared with a selected fraction of RAP, for measuring the complex modulus over a wide range of temperatures and frequencies. Then, the effective rheological properties of the combined virgin and RAP binder in the mortar were evaluated with a new approach based on a modified version of the Nielsen model, avoiding extraction and recovery. Finally, the material response was modeled using the analogical 2S2P1D model, consisting of one spring, two parabolic and one-dashpot elements combined in series and then assembled together with a second spring in parallel. A significant stiffening effect due to RAP was observed both on asphalt mortar complex modulus and on the corresponding asphalt binder parameter back-calculated with the modified Nielsen model. At the end, binder and mortar characteristic time, which is one of the 2S2P1D model parameter, was linked to the recycled material content through a simple exponential expression experimentally derived.

Determination of critical linear lattice size for the four dimensional Ising model on the Creutz cellular automaton

The four-dimensional Ising model is simulated on the Creutz cellular automaton (CCA) near the infinite-lattice critical temperature for the lattice with the linear dimension 4 ⩽ L ⩽ 22. The temperature dependence of Binder parameter (gL) are analyzed for the lattice with the linear dimension 4 ⩽ L ⩽ 22. In this study conducted highly detailed, two different types of behavior were determined as a result of varying linear lattice dimension. The infinite lattice critical temperatures are obtained to be Tc = 6.6845 ± 0.0005 in interval 4 ⩽ L ⩽ 12 and Tc = 6.6807 ± 0.0024 in interval 14 ⩽ L ⩽ 22. The finite and infinite lattice critical exponents for the order parameter, the magnetic susceptibility and the specific heat are computed from the results of simulations by using finite-size scaling relations. Critical linear lattice size have been identified as L = 14.

Erratum: “Spin and Chiral Orderings of Frustrated Quantum Spin Chains” [J. Phys. Soc. Jpn. 68, 3185 (1999)

In p. 3186, below Eq. (3), it was incorrectly stated that by the exact-diagonalization (ED) calculation for the open chains with even N [8 N 20 (S = 1/2) and 8 N 16 (S = 1)], the ground state was found to belong to the subspace of S total 1⁄4 0 with even parity.1) In fact, for the S = 1/2 XY and Heisenberg open chains, the ground state belongs to the subspace of S total 1⁄4 0 with even (odd) parity when N = 4n (4n + 2), where n is an integer. For the S = 1 case, the ground state for a wide parameter region belongs to the subspace of S total 1⁄4 0 with even parity, while the ground state with odd parity appears in some parameter regions, whose position is confined within a range 0:3 . j . 0:6 and changes depending on the system size N and on whether XY or Heisenberg. This correction has no effect on the other parts of the paper. The data of the Binder parameter shown in Fig. 1 were obtained by the ED calculation for the whole subspace of S total 1⁄4 0 including both the even and odd parity states, and therefore, they are not affected by the correction. As for the density-matrix renormalization-group (DMRG) calculation for the S = 1 XY chains, we note that the appearance of the odd-parity ground state found by the ED is a boundary effect in small systems with frustration, and the infinite-system DMRG for the subspace of S total 1⁄4 0 with even parity should capture the ground-state properties in thermodynamic limit correctly. Indeed, we have newly performed the DMRG calculation for the whole subspace of S total 1⁄4 0 and obtained the same results as the previous ones within a numerical accuracy. As to the data shown in Fig. 2, for example, the differences between the previous and newly-obtained results averaged on the distance r are at most 1:2 10 3 (chiral correlation), 1:5 10 4 (string correlation), and 4:8 10 4 (spin correlation), except for the chiral and string correlations at the transition point j 1⁄4 jc1 1⁄4 0:473 for which the averaged differences are 1:2 10 2 or less (chiral) and 1:1 10 3 (string). These differences have no effect on our argument on the behaviors of the correlation functions, and therefore, our conclusions in the paper remain the same. We also correct a misprint in Fig. 2(a): The distance r of the chiral correlation C ðrÞ was shifted by one due to a simple error in the data processing. Therefore, the label for the x-axis should read r + 1 instead of r. Finally, we note that, although the ED data of the Binder parameter in Fig. 1 suggested the absence of the chiral order in the S = 1/2 XY chains, it has been revealed by the DMRG that the chiral order indeed appears in the S = 1/2 chains with easy-plane anisotropy. The DMRG results and the reason why the ED analysis missed the chiral order in the S = 1/2 chains have already been reported in Ref. 2.

Exact ground states of one-dimensional long-range random-field Ising magnets

We investigate the one-dimensional long-range random-field Ising magnet with Gaussian distribution of the random fields. In this model, a ferromagnetic bond between two spins is placed with a probability $p\ensuremath{\sim}{r}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\sigma}}$, where $r$ is the distance between these spins and $\ensuremath{\sigma}$ is a parameter to control the effective dimension of the model. Exact ground states at zero temperature are calculated for system sizes up to $L={2}^{19}$ via graph theoretical algorithms for four different values of $\ensuremath{\sigma}\ensuremath{\in}{0.25,0.4,0.5,1.0}$ while varying the strength $h$ of the random fields. For each of these values several independent physical observables are calculated, i.e., magnetization, Binder parameter, susceptibility, and a specific-heat-like quantity. The ferromagnet-paramagnet transitions at critical values ${h}_{c}(\ensuremath{\sigma})$ as well as the corresponding critical exponents are obtained. The results agree well with theory, and interestingly we find for $\ensuremath{\sigma}=1/2$ the data is compatible with a critical random-field strength ${h}_{c}>0$.

Monte Carlo simulations of the three-dimensionalXYspin glass focusing on chiral and spin order

The ordering of the three-dimensional isotropic {\it XY} spin glass with the nearest-neighbor random Gaussian coupling is studied by extensive Monte Carlo simulations. To investigate the ordering of the spin and the chirality, we compute several independent physical quantities including the glass order parameter, the Binder parameter, the correlation-length ratio, the overlap distribution and the non-self-averageness parameter, {\it etc}, for both the spin-glass (SG) and the chiral-glass (CG) degrees of freedom. Evidence of the spin-chirality decoupling, {\it i.e.}, the CG and the SG order occurring at two separated temperatures, $0<T_{SG}<T_{CG}$, is obtained from the glass order parameter, which is fully corroborated by the Binder parameter. By contrast, the CG correlation-length ratio yields a rather pathological and inconsistent result in the range of sizes we studied, which may originate from the finite-size effect associated with a significant short-length drop-off of the spatial CG correlations. Finite-size-scaling analysis yields the CG exponents $\nu_{CG}=1.36^{+0.15}_{-0.37}$ and $\eta_{CG}=0.26^{+0.29}_{-0.26}$, and the SG exponents $\nu_{SG}=1.22^{+0.26}_{-0.06}$ and $\eta_{SG}=-0.54^{+0.24}_{-0.52}$. The obtained exponents are close to those of the Heisenberg SG, but are largely different from those of the Ising SG. The chiral overlap distribution and the chiral Binder parameter exhibit the feature of a continuous one-step replica-symmetry breaking (1RSB), consistently with the previous reports. Such a 1RSB feature is again in common with that of the Heisenberg SG, but is different from the Ising one, which may be the cause of the difference in the CG critical properties from the Ising SG ones despite of a common $Z_2$ symmetry.

Critical behavior of the fidelity susceptibility for the [formula omitted] transverse-field Ising model

The overlap (inner product) between the ground-state eigenvectors with proximate interaction parameters, the so-called fidelity, plays a significant role in the quantum-information theory. In this paper, the critical behavior of the fidelity susceptibility is investigated for a two-dimensional transverse-field (quantum) Ising model by means of the numerical diagonalization method. In order to treat a variety of system sizes N=12,14,…,32, we adopt the screw-boundary condition. Finite-size artifacts (scaling corrections) of the fidelity susceptibility appear to be suppressed, as compared to those of the Binder parameter. As a result, we estimate the fidelity susceptibility critical exponent as αF=0.715(20).

Phase diagram and universality of the Lennard-Jones gas-liquid system

The gas-liquid phase transition of the three-dimensional Lennard-Jones particles system is studied by molecular dynamics simulations. The gas and liquid densities in the coexisting state are determined with high accuracy. The critical point is determined by the block density analysis of the Binder parameter with the aid of the law of rectilinear diameter. From the critical behavior of the gas-liquid coexisting density, the critical exponent of the order parameter is estimated to be β = 0.3285(7). Surface tension is estimated from interface broadening behavior due to capillary waves. From the critical behavior of the surface tension, the critical exponent of the correlation length is estimated to be ν = 0.63(4). The obtained values of β and ν are consistent with those of the Ising universality class.

Rheological analysis of multi-stress creep recovery (MSCR) test

It has been known for some time that the current AASHTO M 320-05 specification does not address the true rutting potential of modified asphalt binders. The multi-stress creep recovery (MSCR) test was proposed by the United States Federal Highway Administration to replace the AASHTO M 320-05 high-temperature specification parameter and various SHRP+test methods. The new parameter of non-recoverable compliance, Jnr, is currently being considered as a replacement for the Superpave high-temperature binder parameter of |G*|/sin δ (ω = 10 rad/s) (AASHTO TP 70-09). An investigation of the suggested new parameter in capturing the rutting potential and a rheological analysis of the MSCR test method were carried out for conventional and modified asphalt binders and are reported in this paper. The laboratory wheel tracking test was conducted to study the rutting potential of asphalt mixes prepared with the same asphalt binders. Correlation of the new parameter, Jnr, with rutting in asphalt mixes is briefly discussed. A linear viscoelastic compliance model was developed to describe the MSCR test and is also discussed.

Finite-size scaling relations for a four-dimensional Ising model on Creutz cellular automatons

The four-dimensional Ising model is simulated on Creutz cellular automatons using finite lattices with linear dimensions 4 ≤ L ≤ 8. The temperature variations and finite-size scaling plots of the specific heat and the Binder parameter verify the theoretically predicted expression near the infinite lattice critical temperature for 7, 14, and 21 independent simulations. Approximate values for the critical temperature of the infinite lattice of Tc(∞) = 6.6965(35), 6.6961(30), 6.6960(12), 6.6800(3), 6.6801(2), 6.6802(1) and 6.6925(22) (without the logarithmic factor), 6.6921(22) (without the logarithmic factor), 6.6909(2) (without the logarithmic factor), 6.6822(13) (with the logarithmic factor), 6.6819(11) (with the logarithmic factor), and 6.6808(8) (with the logarithmic factor) are obtained from the intersection points of the specific heat curves, the Binder parameter curves, and straight line fits of specific heat maxima for 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the results, 6.6802(1) and 6.6808(8), are in very good agreement with the results of a series expansion of Tc(∞), 6.6817(15) and 6.6802(2), the dynamic Monte Carlo value Tc(∞) = 6.6803(1), the cluster Monte Carlo value Tc(∞) = 6.680(1), and the Monte Carlo value using the Metropolis-Wolff cluster algorithm Tc(∞) = 6.6802632 ± 5 · 10−5. The average values calculated for the critical exponent of the specific heat are α =− 0.0402(15), − 0.0393(12), − 0.0391(11) with 7, 14, and 21 independent simulations, respectively. As the number of independent simulations increases, the result, α =− 0.0391(11), agrees with the series expansions result, α =− 0.12 ± 0.03 and the Monte Carlo result using the Metropolis-Wolff cluster algorithm, α ≥ 0 ± 0.04. However, α =− 0.0391(11) is inconsistent with the renormalization group prediction of α = 0.

The Finite-Size Scaling Study of the Specific Heat and the Binder Parameter of the Two-Dimensional Ising Model for the Fractals Obtained by Using the Model of Diffusion-Limited Aggregation

The two-dimensional Ising model with nearest-neighbour pair interactions is simulated on the Creutz cellular automaton by using the finite-size lattices with the linear dimensions L = 80, 120, 160, and 200. The temperature variations and the finite-size scaling plots of the specific heat and the Binder parameter verify the theoretically predicted expression near the infinite lattice critical temperature. The approximate values for the critical temperature of the infinite lattice Tc = 2.287(6), Tc = 2.269(3), and Tc =2.271(1) are obtained from the intersection points of specific heat curves, Binder parameter curves, and the straight line fit of specific heat maxima, respectively. These results are in agreement with the theoretical value (Tc =2.269) within the error limits. The values obtained for the critical exponent of the specific heat, α = 0.04(25) and α = 0.03(1), are in agreement with α = 0 predicted by the theory. The values for the Binder parameter by using the finite-size lattices with the linear dimension L = 80, 120, 160, and 200 at Tc = 2.269(3) are calculated as gL(Tc) = −1.833(5), gL(Tc) = −1.834(3), gL(Tc) = −1.832(2), and gL(Tc) = −1.833(2), respectively. The value of the infinite lattice for the Binder parameter, gL(Tc) = −1.834(11), is obtained from the straight line fit of gL(Tc) = −1.833(5), gL(Tc) = −1.834(3), gL(Tc) = −1.832(2), and gL(Tc) = −1.833(2) versus L = 80, 120, 160, and 200, respectively

Monte Carlo simulations of the directional-ordering transition in the two-dimensional classical and quantum compass model

A comprehensive study of the two-dimensional (2D) compass model on the square lattice is performed for classical and quantum spin degrees of freedom using Monte Carlo and quantum Monte Carlo methods. We employ state-of-the-art implementations using Metropolis, stochastic series expansion, and parallel tempering techniques to obtain the critical ordering temperatures and critical exponents. In a preinvestigation we reconsider the classical compass model where we study and contrast the finite-size scaling behavior of ordinary periodic boundary conditions against annealed boundary conditions. It is shown that periodic boundary conditions suffer from extreme finite-size effects which might be caused by closed-loop excitations on the torus. These excitations also appear to have severe effects on the Binder parameter. On this footing we report on a systematic Monte Carlo study of the quantum compass model. Our numerical results are at odds with recent literature on the subject which we trace back to neglecting the strong finite-size effects on periodic lattices. The critical temperatures are obtained as ${T}_{c}=0.1464(2)J$ and ${T}_{c}=0.055(1)J$ for the classical and quantum versions, respectively, and our data support a transition in the 2D Ising universality class for both cases.

Near Critical Coexistence for an AUA Model of Thiophenes

Near critical coexistence has been determined by means of parallel tempering coupled with grand canonical Monte Carlo simulations which were later recombined by using histogram reweighting techniques. The data collected during the simulations is not only useful to determine accurately the critical point but also to provide estimates for the coexistence density jump between the phases in equilibrium. A recently introduced algorithm by Kim [Kim Y.C. (2005) Phys. Rev. 71, 051501; Kim Y.C. (2005) Comput. Phys. Commun. 169, 295], based on the scaling of the positions of the different minima found for the Binder parameter, has been applied to the case of a realistic model of thiophene consisting of different Lennard Jones sites. Contrary to the case of the Hard Core Square Well (HCSW) and Restricted Primitive Model (RPM) systems, significant corrections to scaling are found in this case. By readapting the algorithm we are able to calculate the coexistence in the critical region.

THE FINITE-SIZE SCALING STUDY OF THE SPECIFIC HEAT AND THE BINDER PARAMETER FOR THE FIVE-DIMENSIONAL ISING MODEL

The five-dimensional Ising model with nearest-neighbor pair interactions is simulated on the Creutz cellular automaton by using finite-size lattices with the linear dimensions L=4, 6, 8, 10, 12, 14, and 16. The temperature variations and the finite-size scaling plots of the specific heat and Binder parameter verify the theoretically-predicted expression near the infinite-lattice critical temperature. The approximate values for the critical temperature of the infinite-lattice, T c =8.8063, T c =8.7825 and T c =8.7572, are obtained from the intersection points of specific heat curves, Binder parameter curves and the straight line fit of specific heat maxima, respectively. These results are in agreement with the more precise value of T c =8.7787. The value obtained for the critical exponent of the specific heat, i.e. α=0.009, is also in agreement with α=0 predicted by the theory.