Critical dynamics of the three-dimensional Ising model: A Monte Carlo study

Extensive Monte Carlo simulations have been performed to analyze the dynamical behavior of the three-dimensional Ising model with local dynamics. We have studied the equilibrium correlation functions and the power spectral densities of odd and even observables. The exponential relaxation times have been calculated in the asymptotic one-exponential time region. We find that the critical exponentz=2.09 ±0.02 characterizes the algebraic divergence with lattice size for all observables. The influence of scaling corrections has been analyzed. We have determined integrated relaxation times as well. Their dynamical exponentz _int agrees withz for correlations of the magnetization and its absolute value, but it is different for energy correlations. We have applied a scaling method to analyze the behavior of the correlation functions. This method verifies excellent scaling behavior and yields a dynamical exponentz _scal which perfectly agrees withz.     

A Comparison Between Broad Histogram and Multicanonical Methods

We discuss the conceptual differences between the broad histogram (BHM) and reweighting methods in general, and particularly the so-called multicanonical (MUCA) approaches. The main difference is that BHM is based on microcanonical, fixed-energy averages which depend only on the good statistics taken inside each energy level. The detailed distribution of visits among different energy levels, determined by the particular dynamic rule one adopts, is irrelevant. Contrary to MUCA, where the results are extracted from the dynamic rule itself, within BHM any microcanonical dynamics could be adopted. As a numerical test, we have used both BHM and MUCA in order to obtain the spectral energy degeneracy of the Ising model in 4×4×4 and 32×32 lattices, for which exact results are known. We discuss why BHM gives more accurate results than MUCA, even using the same Markovian sequence of states. In addition, such an advantage increases for larger systems.     

Interface motion in a two-dimensional Ising model with a field

We determine by Monte Carlo simulations the width of an interface between the stable phase and the metastable phase in a two-dimensional Ising model with a magnetic field, in the case of nonconversed order parameter (Glauber dynamics). At zero temperature, the width increases ast with–1/3, as predicted by earlier theories. As temperature increases, the value of the effective exponent that we measure decreases toward the value 1/4, which is the value in the absence of magnetic field.     

Energy of Long-Lifetime Configurations in Zero-Temperature Dynamics

Using Monte Carlo method with zero-temperature dynamics, we investigate energy evolution of Ising spin configuration on a square lattice. The energies of some configurations exhibit long duration before those configurations reach the final state -- ground state or frozen stripe state. For ground-state dynamical realization, the duration occurs when the energy per spin is 4/L, where L is the lattice size. For stripe-state dynamical realization, the energy is slightly higher than 2/L when the duration appears in the last evolution stage. In addition, it is found that the average energy per spin in final state is approximately 2/3L.     

Critical behaviour of the ferromagnetic Ising model on a triangular lattice

We use a new updated algorithm scheme to investigate the critical behaviour of the two-dimensional ferromagnetic Ising model on a triangular lattice with the nearest neighbour interactions. The transition is examined by generating accurate data for lattices with L= 8, 10, 12, 15, 20, 25, 30, 40 and 50. The updated spin algorithm we employ has the advantages of both a Metropolis algorithm and a single-update method. Our study indicates that the transition is continuous at T_c= 3.6403({2}). A convincing finite-size scaling analysis of the model yields υ=0.9995(21), β / υ = 0.12400({17}), γ / υ = 1.75223(22), γ '/υ=1.7555(22), α/υ= 0.00077(420) (scaling) and α / υ = 0.0010(42) (hyperscaling). The present scheme yields more accurate estimates for all the critical exponents than the Monte Carlo method, and our estimates are shown to be in excellent agreement with their predicted values.     

A new multispin coding algorithm for Monte Carlo simulation of the Ising model

We present a new algorithm for Monte Carlo simulation of the Ising model. The usual serial architecture of a computer is exploited in a novel way, enabling parallel but independent calculations to be carried out on as many spins as there are bits in a computer word in each fundamental move. The algorithm enjoys a substantial increase in execution speed over more usual multispin coding algorithms. By its very nature, the algorithm constitutes a design for a special-purpose processor.     

Fast vectorized algorithm for the Monte Carlo simulation of the random field Ising model

An algorithm for the simulation of the 3-dimensional random field Ising model with a binary distribution of the random fields is presented. It uses multi-spin coding and simulates 64 physically different systems simultaneously. On one processor of a Cray YMP it reaches a speed of 184 million spin updates per second. For smaller field strength we present a version of the algorithm that can perform 242 million spin updates per second on the same machine.     

Comment on “A Comparison Between Broad Histogram and Multicanonical Methods”

A recent paper by A. R. Lima, P. M. C. de Oliveira, and T. J. P. Penna [J. Stat. Phys. 99:691 (2000)], seems to contain at least two mistakes which deserve comment, one concerning the numerical data, the other being of a conceptual kind.     

Critical behavior of the specific heat for pure and site-diluted simple cubic Ising systems

The exponent of the specific heatC is determined for the pure and the site-diluted simple cubic Ising model (concentrationx=0, 0.2, 0.4 of nonmagnetic sites) by a finite-size scaling analysis of the peak value C_max(L) for systems of linear dimensionsL=8, 16, 32, and 64. The C_max values are obtained by the Ferrenberg-Swendsen algorithm, using Monte Carlo data from a fully-vectorized multi-spin coding program. We obtain =0.11 for x=0 and a crossover to a negative value upon dilution, with =–0.029(4) both forx=0.2 andx=0.4.     

Surface Critical Behavior of Two-Dimensional Dilute Ising Models

Ising models with nearest neighbor ferromagnetic random couplings on a square lattice with a (1, 1) surface are studied, using Monte Carlo techniques and a star-triangle transformation method. In particular, the critical exponent of the surface magnetization is found to be close to that of the perfect model, Β₁ = 1/2. The crossover from surface to bulk critical properties is discussed.     

Monte Carlo simulation of the compensation and critical behaviors of a ferrimagnetic core/shell nanoparticle Ising model

A Monte Carlo simulation has been used to study the magnetic properties and the critical behaviors of a single spherical nanoparticle, consisting of a ferromagnetic core of spins surrounded by a ferromagnetic shell of S=±1, 0 or , spins with antiferromagnetic interface coupling, located on a simple cubic lattice. A number of characteristic phenomena has been found. In particular, the effects of the shell coupling and the interface coupling on both the critical and compensation temperatures are investigated. We have found that, for appropriate values of the system parameters, two compensation temperatures may occur in the present system.     

Dynamical analysis of low-temperature monte carlo cluster algorithms

We present results on the Swendsen-Wang dynamics for the Ising ferromagnet in the low-temperature case without external field in the thermodynamic limit. We discuss in particular the rate of convergence to the equilibrium Gibbs state in finite and infinite volume, the absence of ergodicity in the infinite volume, and the long-time behavior of the probability distribution of the dynamics for various starting configurations. Our results are purely dynamical in nature in the sense that we never use the reversibility of the process with respect to the Gibbs state, and they apply to a stochastic particle system withnon- Gibbsian invariant measure.     

Statistical and systematic errors in Monte Carlo sampling

We have studied the statistical and systematic errors which arise in Monte Carlo simulations and how the magnitude of these errors depends on the size of the system being examined when a fixed amount of computer time is used. We find that, depending on the degree of self-averaging exhibited by the quantities measured, the statistical errors can increase, decrease, or stay the same as the system size is increased. The systematic underestimation of response functions due to the finite number of measurements made is also studied. We develop a scaling formalism to describe the size dependence of these errors, as well as their dependence on the bin length (size of the statistical sample), both at and away from a phase transition. The formalism is tested using simulations of thed=3 Ising model at the infinite-lattice transition temperature. We show that for a 96×96×96 system noticeable systematic errors (systematic underestimation of response functions) are still present for total run lengths of 10⁶ Monte Carlo steps/site (MCS) with measurements taken at regular intervals of 10 MCS.This paper is dedicated to Jerry Percus on the occasion of his 65th birthday.     

A fast algorithm for the Cyber 205 to simulate the 3D Ising model

We describe a computer program that performs the Metropolis algorithm for the 3D Ising model at a peak speed of 98 million spin updates per second on a 2-pipe CDC Cyber 205. This speed is achieved using the special vector capabilities of the Cyber 205 and multispin coding techniques.     

Two-Dimensional Dilute Ising Models: Critical Behavior near Defect Lines

We consider two-dimensional Ising models with randomly distributed ferromagnetic bonds and study the local critical behavior at defect lines by extensive Monte Carlo simulations. Both for ladder- and chain-type defects, nonuniversal critical behavior is observed: the critical exponent of the defect magnetization is found to be a continuous function of the strength of the defect coupling. Analyzing corresponding stability conditions, we obtain new evidence that the critical exponent of the bulk correlation length of the random Ising model does not depend on dilution, i.e., =1.     

Finite-size effects in surface tension. I. Fluctuating interfaces

We consider a two-dimensional Ising cylinder of circumferenceM and heightN, with a floating interface introduced by the appropriate boundary conditions. An exact analysis of the finite-size effects in surface tension is given and the scaling function for all temperatures is calculated. The results are compared with the Monte Carlo data of Mon and Jasnow.On leave from: Department of Theoretical Chemistry, Oxford University, Oxford, OX1 3UB, England.     

关于状态分立系统的改进的Metropolis方法

对状态分立系统的平衡态统计问题给出了一个改进的Metropolis方法,并在伊辛模型上与传统方法进行了对比,得到了一致的结果,效率提高了约25%.针对伊辛模型的抽样过程给出了其马尔科夫过程所对应的Master方程.将所提出的方法应用于伊辛模型,相当于对其Master方程进行了模拟.作为推广,一大类临界动力学模型的Master方程都可以用本方法进行模拟,以用于其暂态过程的研究.     

On the Swendsen-Wang dynamics. I. Exponential convergence to equilibrium

We present rigorous results on the exponential convergence to equilibrium for the Swendsen-Wang stochastic dynamics for thed-dimensional Ising ferromagnet with external magnetic fieldh in the thermodynamic limit. We consider various situations, mainly in the low-temperature regime, in which boundary conditions are homogeneous and parallel or opposite to the external field. In the latter case we relate directly the tunneling from the metastable phase to the stable one with the exponential convergence to equilibrium.     

Monte Carlo technique for very large ising models

Rebbi's multispin coding technique is improved and applied to the kinetic Ising model with size 600*600*600. We give the central part of our computer program (for a CDC Cyber 76), which will be helpful also in a simulation of smaller systems, and describe the other tricks necessary to go to large lattices. The magnetizationM atT=1.4*T _c is found to decay asymptotically as exp(-t/2.90) ift is measured in Monte Carlo steps per spin, and M(t = 0) = 1 initially.     

Evidence for spinodal singularities in high-dimensional nearest-neighbor ising models

Conventional theories of nucleation predict that the metastable state has an average lifetime which monotonically decreases as the system is quenched further from the condensation point. However, theories based on the coarsegrained Ginzburg-Landau free energy functional seem to indicate that for systems above six dimensions there is a sharp spinodal dividing the metastable and unstable regimes where the lifetime of the metastable state diverges. Monte Carlo simulations are used to investigate this discrepency. Both nucleation rates and bulk susceptibility measurements seem to support the prediction of the Ginzburg-Landau theories.