International Journal of Modern Physics C

The simulation results for polydisperse particle system are analyzed to study the polydispersity effect to liquid/fluid-solid transition. It is found that the transition density and the solid state are well described by the variance, σ², of radius distribution. The critical polydispersity above which fluid-solid coexisting density region vanishes is estimated to be σ² = 0.0073 ± 0.0005. A scaling plot for the solid pressure is shown.

The idea of universal finite-size-scaling functions of the Ising model is tested by Monte Carlo simulations for various lattices. Not only regular lattices such as the square lattice but quasiperiodic lattices such as the Penrose lattice are treated. We show that the finite-size-scaling functions of the order parameter for various lattices are collapsed on a single curve by choosing two nonuniversal scaling metric factors. We extend the idea of the universal finite-size-scaling functions to the order-parameter distribution function. We pay attention to the effects of boundary conditions.

Large-scale Monte Carlo simulations require high-quality random number generators to ensure correct results. The contrapositive of this statement is also true — the quality of random number generators can be tested by using them in large-scale Monte Carlo simulations. We have tested many commonly-used random number generators with high precision Monte Carlo simulations of the 2-d Ising model using the Metropolis, Swendsen-Wang, and Wolff algorithms. This work is being extended to the testing of random number generators for parallel computers. The results of these tests are presented, along with recommendations for random number generators for high-performance computers, particularly for lattice Monte Carlo simulations.

We present a status report on the ongoing analysis of the 3D Ising model with nearest-neighbor interactions using the Monte Carlo Renormalization Group (MCRG) and finite size scaling (FSS) methods on 64³, 128³, and 256³ simple cubic lattices. Our MCRG estimates are and ν = 0.625(1). The FSS results for K^c are consistent with those from MCRG but the value of ν is not. Our best estimate η = 0.025(6) covers the spread in the MCRG and FSS values. A surprise of our calculation is the estimate ω ≈ 0.7 for the correction-to-scaling exponent. We also present results for the renormalized coupling g_R along the MCRG flow and argue that the data support the validity of hyperscaling for the 3D Ising model.

With recent improvements in simulation methods and data analysis techniques, Monte Carlo simulations, used in conjunction with finite-size scaling techniques, have become the most powerful and flexible tools for studying critical phenomena. In particular, there has been a dramatic increase in resolution in studies of the three-dimensional Ising model. This paper discusses motivations for performing high-resolution studies, reviews past and current work on the model and points out some potential potholes on the road to higher resolution.

We have studied the three-dimensional Ising spin glass with a ± J distribution by Monte Carlo simulations. Using larger sizes and much better statistics than in earlier work, a finite size scaling analysis shows quite strong evidence for a finite transition temperature, T_c, with ordering below T_c. Our estimate of the transition temperature is rather lower than in earlier work, and the value of the correlation length exponent, ν, is somewhat higher. Because there may be (unknown) corrections to finite size scaling, we do not completely rule out the possibility that T_c = 0 or that T_c is finite but with no order below T_c. However, from our data, these possibilities seem less likely.

An efficient Monte Carlo algorithm for simulating hardly-relaxing systems is proposed. By using this algorithm the three-dimensional ± J Ising spin glass model is studied. The result shows that reasonable values of the critical temperature and of the critical exponents can be obtained within Monte Carlo steps much shorter than the observation time a conventional simulation usually requires.

The results of the recent numerical simulations on vector spin glasses are presented. Numerical evidence of the novel chiral-glass state, accompanied with broken spin-reflection symmetry with preserving spin-rotation symmetry, is presented. Implication to experiments on spin-glass transitions is discussed.

The present paper explains some general basic formulas concerning quantum Monte Carlo simulations, symplectic integration and other numerical calculations. A generalization of the BCH formula is given with an application to the decomposition of exponential operators in the presence of small parameters.

Although cross over between fixed points associated with different universality classes is a ubiquitious phenomena, there is little theoretical insight and less numerical evidence about exactly how cross-over phenomena occur. Here numerical evidence for the picture that cross-over phenomena are governed by the underlying structure of the associated renormalization group is given. In particular, it is shown how the cross-over is related to the effective dimension from scaling of the system. For the ferromagnetic Ising model in two and three dimensions numerical evidence will be presented that the cross-over is governed by the line of fixed points in non-integer dimensions. This allows for successful comparisons between the effective dimension from scaling for finite systems and expansions in dimension of universal quantities such as critical exponents.

Slow dynamics in the 'Griffiths phase' of the 3D ± J Ising spin glass is studied by means of Monte Carlo simulation, in which the spectra of (logarithms of) relaxation times, P (x) with x = lnτ, are analyzed in detail. At each temperature P (x) exhibits a characteristic crossover behavior at around a certain value x_c which in turn tends to diverge toward the spin-glass critical temperature T_c. The contributions to the branch x ≲ x_c are interpreted due to the ordinary critical slowing down process associated with T_c (critical dynamics). The behavior of P (x) at x ≲ x_c, on the other hand, is consistent with the picture of relaxation processes of large, rare clusters which are related to the Griffiths singularity at the Curie temperature of the corresponding pure ferromagnet (cluster dynamics).

We review our recent studies on the dynamical correlations in MC simulations from the view point of the statistical dependence. Attentions are paid to the reduction of the statistical degrees of freedom for correlated data. Possible biases on several cumulants, such as the susceptibility and the Binder number due to finite MC length are discussed. A new method for calculating the equilibrium relaxation time from the analysis of the statistical dependence is presented. We apply it to the critical dynamics of the Ising model to estimate the dynamical critical exponent accurately.

We present results from a computational study of a class of 2D two-temperature non-equilibrium Ising models. In these systems the dynamics is a local competition of two equilibrium dynamics at different temperatures. We analyzed non-equilibrium versions of Metropolis, heat bath/Glauber and Swendsen-Wang dynamics and found strong evidence that some of these dynamics have the same critical exponents and belong to the same universality class as the equilibrium 2D Ising model.

Spin dynamics simulations have been used to study dynamic critical behavior of classical Heisenberg magnets. The temporal evolutions of the spin configurations were determined numerically from coupled equations of motion by a fourth-order predictor corrector method, with initial spin configurations generated by Monte-Carlo simulations. The neutron scattering function, S (q, ω), was calculated from the space and time displaced spin-spin correlation function and the dynamic critical exponent z was extracted using dynamic finite size scaling theory. For both ferromagnetic and antiferromagnetic models we find good agreement with theoretical predictions and experimental results.

Motivated by several physical systems, driven models with three states (i.e., lattice gases with two species of particles) are considered. We study a non-interacting system in which the two species are driven in opposite directions. As the overall particle density is varied, a transition occurs, from a disordered phase with high particle current, to ordered, locked phases with minimal current. Further, there are several locked states, with the frequency of their occurrence depending on the aspect ratio of the system. Many of the novel phenomena can be understood in terms of a simple continuum theory. Variations and generalizations, as well as the physical systems they model, are discussed.

We make a hybrid Monte Carlo spin dynamics simulation of a dilute system of the classical Heisenberg spins coupled via the Ruderman-Kittel-Kasuya-Yoshida interaction in a wide range of spin concentration x. We find the critical concentration of the ferromagnetism x_c ~0.15. For x < x_c, the data predict the spin glass phase transition at a finite temperature T_g. For x ≳ x_c, those predict the reentrance to a mixed phase of the ferromagnetism and a transverse spin glass at a low temperature T_m. The obtained phase diagram is seen to be quite similar to those of dilute alloys Au_{1 − x}Fe_x and Pt_{1 − x}Mn_x.

A new technique for the negative sign problem in the quantum Monte Carlo method using the Suzuki-Trotter decomposition is introduced. In order to reduce the cancellation between between samples with positive and negative weights, we make use of the transfer matrix method, which has been named the Transfer-Matrix Monte Carlo method. Applications to the Heisenberg antiferromagnet on the ∆-chain and on the kagome lattice, and also to the Kondo lattice system also are given.

The recent generalization of the Fortuin-Kasteleyn percolation representation and the Swendsen-Wang (SW) cluster algorithm is reviewed. The new representation yields cluster algorithm for the XYZ model with arbitrary magnitude of spins, with arbitrary anisotropies, and in any dimensions. The new algorithm often reduces computational autocorrelation times by several orders of magnitude. By using the new algorithm together with methods of Bayesian statistical inference, we obtained the dynamical structure factor S (k, ω) for the S =1/2 XY model in two dimensions.

The three-dimensional spin-1/2 Heisenberg model on a simple cubic lattice is studied for ferromagnetic and antiferromagnetic interactions using the Decoupled Cell Method for quantum Monte Carlo. Results for the relaxation time τ_L are determined for both ferromagnetic and antiferromagnetic systems and found to be similar to those found for the classical (s → ∞) Heisenberg model. The scaling of τ_L with system size is used to extract the dynamical critical exponent z for the two systems. The values of z = 1.98 ± 0.12 for the ferromagnet and z = 1.94 ± 0.09 for the antiferromagnet are in good agreement with theoretical predictions and previous Monte Carlo studies of the classical Heisenberg model.

A Monte Carlo algorithm for the random Boson Hubbard model is reported. The analytic expression of the matrix elements is presented, and the ergodicity of the Monte Carlo flips is discussed. The results in one dimension supports a previously proposed perturbational scaling argument.

We investigate the effect of an impurity bond on low-lying excited states of the antiferromagnetic Ising-like chain in terms of domain-wall excitations, using an analytical method and a numerical method. The fundamental difference between the chains with even and odd numbers of spins is discussed.