Fourth-order Binder Cumulants Research Articles

A comparative study of critical temperature estimation of atomic fluid and chain molecules using fourth-order Binder cumulant and simplified scaling laws

Grand-canonical transition-matrix Monte Carlo simulation with histogram reweighting and finite-size scaling technique are used to calculate fourth-order Binder cumulant of order parameter along the vapour–liquid coexistence line to calculate the critical temperature of bulk and confined square-well fluid in slit pore of two pore sizes. Further, this approach is utilised to estimate the critical temperatures of relatively more complex fluids such as n-alkanes confined in graphite and mica slit pores of different slit widths. The estimated critical temperatures are compared with the critical temperature obtained for the same systems using simplified form of the scaling law. This investigation reveals that critical temperatures of simple and complex fluids in bulk state and under confinement, estimated using the scaling law, are within reasonable accuracy with that obtained using more accurate and rigorous approach of fourth-order Binder's cumulant.

Phase transitions in two-dimensional ferromagnetic Potts model with q = 3 on a triangular lattice

Phase transitions in the two-dimensional ferromagnetic Potts model with the number of spin states q = 3 on a triangular lattice were studied using Monte-Carlo method. The systems of linear size L = 20–120 were considered. Using the method of the fourth-order Binder cumulants, it was demonstrated that the second order phase transition is observed in the two-dimensional ferromagnetic Potts model. The static critical exponents for the heat capacity α, susceptibility γ, magnetization β and correlation radius ν were calculated based on the finite-dimensional scaling theory.

Phase transitions in the two-dimensional ferro- and antiferromagnetic potts models on a triangular lattice

The phase transitions in the two-dimensional ferro- and antiferromagnetic Potts models with q = 3 states of spin on a triangular lattice are studied using cluster algorithms and the classical Monte Carlo method. Systems with linear sizes L = 20–120 are considered. The method of fourth-order Binder cumulants and histogram analysis are used to discover that a second-order phase transition occurs in the ferromagnetic Potts model and a first-order phase transition takes place in the antiferromagnetic Potts model. The static critical indices of heat capacity (α), magnetic susceptibility (γ), magnetization (β), and correlation radius index (ν) are calculated for the ferromagnetic Potts model using the finite-size scaling theory.

Influence of quenched nonmagnetic impurities on phase transitions in a three-dimensional diluted Potts model with spin state number q = 4

The influence of quenched nonmagnetic impurities on phase transitions and critical phenomena in the 3D Potts model with the spin state number q = 4 is studied using the Monte Carlo method. Systems with the linear size L = 20–32 and spin concentrations p = 1.00, 0.90, 0.65 are considered. The fourth order Binder cumulant method is used to demonstrate that in the strongly diluted regime, a phase transition of the second kind is observed in this model for the spin concentration p = 0.65, and a phase transition of the first kind is observed for the pure (p = 1.00) and weakly diluted (p = 0.90) models. The theory of finite-dimensional scaling is used to calculate the static critical parameters of heat capacity α, susceptibility γ, magnetization β, and correlation radius ν.

Phase transitions in a three-dimensional diluted Potts model with 4 spin states

A Monte–Carlo method is used to study phase transitions and critical phenomena in a three-dimensional Potts model with 4 spin states and nonmagnetic impurities. Systems with linear sizes L=20–32 and spin concentrations p=1.00, 0.90, and 0.65 are studied. A fourth order Binder cumulant method is used to show that this model yields a second order phase transition for strong dilution with a spin concentration p=0.65, while the pure model (p=1.00) and the model with weak dilution (p=0.90) yield a first order phase transition. The static critical indices for the heat capacity α, susceptibility γ, magnetization β, and correlation radius ν are calculated using a finite-dimensional scaling theory.

Monte Carlo investigation of phase transitions in the frustrated Heisenberg model on a triangular lattice

The critical properties and phase transitions of the three-dimensional frustrated antiferromagnetic Heisenberg model on a triangular lattice have been investigated using the Monte Carlo method with a replica algorithm. The critical temperature has been determined and the character of the phase transitions has been analyzed using the method of fourth-order Binder cumulants. A second-order phase transition has been found in the three-dimensional frustrated Heisenberg model on a triangular lattice. The static magnetic and chiral critical exponents of the heat capacity α, the susceptibility γ and γk, the magnetization β and βk, the correlation length ν and νk, as well as the Fisher exponents η and ηk, have been calculated in terms of the finite-size scaling theory. It has been demonstrated that the three-dimensional frustrated antiferromagnetic Heisenberg model on a triangular lattice forms a new universality class of the critical behavior.

Features of the phase transitions in the three-dimensional diluted potts model with number of spin states q = 3

The effect of frozen disorder, implemented in the form of nonmagnetic impurities, on phase transitions in the three-dimensional Potts model with number of states q = 3 has was investigated by the Monte Carlo method, using the Wolf single-cluster algorithm. Systems with linear sizes L = 20–44 were considered at spin concentrations p = 1.00–0.65. The method of fourth-order Binder cumulants was used to demonstrate that a first-order phase transition is observed for the pure model (p = 1.00) and a second-order phase transition occurs at concentrations p = 0.90, 0.80, 0.70, and 0.65.

Investigation of the Critical Properties in the 3D Site-Diluted Potts Model

The effect of quenched nonmagnetic impurities on phase transitions in the three-dimensional Potts model with the number of spin states q=3 is studied using the Wolff single-cluster algorithm of the Monte Carlo method. By the method of fourth-order Binder cumulants, it is demonstrated that the second-order phase transition occurs in the model under study at spin concentrations p=0.9, 0.8, 0.7, and 0.65, while the first-order phase transition is observed in the pure model (p=1.0). The static critical exponents (CEs) α (heat capacity), γ (susceptibility), β (magnetization), and ν (correlation length) are calculated based on the finite-size scaling theory.

Investigation of the influence of quenched nonmagnetic impurities on phase transitions in the three-dimensional Potts model

The influence of quenched nonmagnetic impurities on phase transitions in the three-dimensional Potts model with the number of spin states q = 3 is investigated using the Wolff single-cluster algorithm of the Monte Carlo method. The systems with linear sizes L = 20–44 at the spin concentrations p = 1.0, 0.9, 0.8, and 0.7 are analyzed. It is demonstrated with the use of the method of fourth-order Binder cumulants that the second-order phase transition occurs in the model under consideration at the spin concentrations p = 0.9, 0.8, and 0.7 and that the first-order phase transition is observed in the pure model (p = 1.0). The static critical exponents α (heat capacity), γ (susceptibility), β (magnetization), and ν (correlation length) are calculated in the framework of the finite-size scaling theory. The problem regarding the universality classes of the critical behavior of weakly diluted systems is discussed.

The Ising model in a Bak–Tang–Wiesenfeld sandpile

We study the spin-1 Ising model with non-local constraints imposed by the Bak–Tang–Wiesenfeld sandpile model of self-organized criticality (SOC). The model is constructed as if the sandpile is being built on a (honeycomb) lattice with Ising interactions. In this way we combine two models that exhibit power-law decay of correlation functions characterized by different exponents. We discuss the model properties through an order parameter and the mean energy per node, as well as the temperature dependence of their fourth-order Binder cumulants. We find: (i) a thermodynamic phase transition at a finite T c between paramagnetic and antiferromagnetic phases, and (ii) that above T c the correlation functions decay in a way typical of SOC. The usual thermodynamic criticality of the two-dimensional Ising model is not affected by SOC constraints (the specific heat critical exponent α ≈ 0 ), nor are SOC-induced correlations affected by the interactions of the Ising model. Even though the constraints imposed by the SOC model induce long-range correlations, as if at standard (thermodynamic) criticality, these SOC-induced correlations have no impact on the thermodynamic functions.

Monte Carlo study of the metamagnet Ising model in a random and uniform field

Monte Carlo simulation has been used to determine the phase diagram of a metamagnet Ising model in the presence of a random and uniform magnetic field. The model consists of a spin-1/2 metamagnet in which the nearest neighbor and next nearest neighbor spin interactions are antiferromagnetic (J1 0), respectively. We used a bimodal probability distribution for the random magnetic field. We have calculated the staggered magnetization and the fourth-order Binder cumulants in order to obtain the critical points. The phase diagram in the uniform field versus temperature plane presents continuous and first-order transition lines. The phase transition lines, together with the critical and tricritical points, have been obtained for several random field values.

Critical behavior of ferromagnetic Ising thin films

In the present work, we study the magnetic properties and critical behavior of simple cubic ferromagnetic thin films. We simulate L× L× d films with semifree boundary conditions on the basis of the Monte Carlo method and the Ising model with nearest neighbor interactions. A Metropolis dynamics was implemented to carry out the energy minimization process. For different film thickness, in the nanometer range, we compute the temperature dependence of the magnetization, the magnetic susceptibility and the fourth order Binder's cumulant. Bulk and surface contributions of these quantities are computed in a differentiated fashion. Additionally, according to finite size scaling theory, we estimate the critical exponents for the correlation length, magnetic susceptibility, and magnetization. Results reveal a strong dependence of critical temperature and critical exponents on the film thickness. The obtained critical exponents are finally compared to those reported in literature for thin films.

Entropic sampling via Wang-Landau random walks in dominant energy subspaces

Dominant energy subspaces of statistical systems are defined with the help of restrictive conditions on various characteristics of the energy distribution, such as the probability density and the fourth order Binder's cumulant. Our analysis generalizes the ideas of the critical minimum energy subspace (CRMES) technique, applied previously to study the specific heat's finite-size scaling. Here, we illustrate alternatives that are useful for the analysis of further finite-size anomalies and the behavior of the corresponding dominant subspaces is presented for the two-dimensional (2D) Baxter-Wu and the 2D and 3D Ising models. In order to show that a CRMES technique is adequate for the study of magnetic anomalies, we study and test simple methods which provide the means for an accurate determination of the energy-order-parameter (E,M) histograms via Wang-Landau random walks. The 2D Ising model is used as a test case and it is shown that high-level Wang-Landau sampling schemes yield excellent estimates for all magnetic properties. Our estimates compare very well with those of the traditional Metropolis method. The relevant dominant energy subspaces and dominant magnetization subspaces scale as expected with exponents alpha/nu and gamma/nu, respectively. Using the Metropolis method we examine the time evolution of the corresponding dominant magnetization subspaces and we uncover the reasons behind the inadequacy of the Metropolis method to produce a reliable estimation scheme for the tail regime of the order-parameter distribution.

Monte Carlo study of the critical behavior of the two-dimensional biquadratic planar rotator model

We present an analysis of the critical properties of the two-dimensional rigid rotor model with a biquadratic interaction $H=\ensuremath{-}J{\ensuremath{\sum}}_{i,j}{S}_{i}\ensuremath{\cdot}{S}_{j}+D{\ensuremath{\sum}}_{i,j}({S}_{i}\ensuremath{\cdot}{S}_{j}{)}^{2}.$ We use a self-consistent harmonic approximation and a Monte Carlo calculation to study the Hamiltonian. The results for the specific heat, susceptibility, fourth order Binder's cumulant and helicity modulus obtained by Monte Carlo suggest that the model has a critical BKT line $\ensuremath{-}\ensuremath{\infty}<D/J<{D}_{c}/J,$ where ${D}_{c}/J=0.96(9).citation$