Two-sided bounds on the free energy from local states in Monte Carlo simulations

We show that a precise assessment of free energy estimates in Monte Carlo simulations of lattice models is possible by using cluster variation approximations in conjunction with the local states approximations proposed by Meirovitch. The local states method (LSM) utilizes entropy expressions which recently have been shown to correspond to a converging sequence of upper bounds on the thermodynamic limit entropy density (i.e., entropy per lattice site), whereas the cluster variation method (CVM) supplies formulas that in some cases have been proven to be, and in other cases are believed to be, lower bounds. We have investigated CVM-LSM combinations numerically in Monte Carlo simulations of the two-dimensional Ising model and the two-dimensional five-states ferromagnetic Potts model. Even in the critical region the combination of upper and lower bounds enables an accurate and reliable estimation of the free energy from data of a single run. CVM entropy approximations are therefore useful in Monte Carlo simulation studies and in establishing the reliability of results from local states methods.     

Computer simulation studies of magnetic phase transitions

The advancements which have been made in the use of computer simulations to study magnetic-phase transitions and critical phenomena are reviewed. We describe how the use of a combination of sophisticated Monte Carlo simulation algorithms and reweighting (histogram) techniques have allowed the determination of the static critical behavior with unprecedented precision. The study of “dynamic” critical behavior in simple spin models by both Monte Carlo and spin dynamics methods is also reviewed. Recent estimates for dynamic critical exponents are given including those for true dynamics.     

First and second order transition of frustrated Heisenberg spin systems

Starting from the hypothesis of a second order transition we have studied modifications of the original Heisenberg antiferromagnet on a stacked triangular lattice (STA-model) by the Monte Carlo technique. The change is a local constraint restricting the spins at the corners of selected triangles to add up to zero without stopping them from moving freely (STAR-model). We have studied also the closely related dihedral and trihedral models which can be classified as Stiefel models. We have found indications of a first order transition for all three modified models instead of a universal critical behavior. This is in accordance with the renormalization group investigations but disagrees with the Monte Carlo simulations of the original STA-model favoring a new universality class. For the corresponding x-y antiferromagnet studied before, the second order nature of the transition could also not be confirmed. Received 17 May 1999 and Received in final form 30 July 1999     

Ising-XY Transition in Three-dimensional Frustrated Antiferromagnets with Collinear Spin Ordering

JETP Letters - Using Monte Carlo simulations, we study the critical behavior of two models of frustrated XY antiferromagnets with a collinear spin ordering and with an additional twofold degeneracy...     

Phase equilibria of charge-, size-, and shape-asymmetric model electrolytes

The low-temperature phase behavior of two 2:1 hard-core electrolyte models has been investigated by Monte Carlo simulations. In the first model, both bivalent cations and monovalent anions are spherical, and the charges are located at the ion's centers; in the second model, bivalent cations are modeled as rigid dimers composed of two tangent hard spheres, each carrying a positive charge at the center. It is found that the critical temperature and the critical density are strongly affected by the size asymmetry and the shape of the ions. The results presented in this work provide insights into the behavior of charged colloidal suspensions and polyelectrolytes, where large, symmetric or asymmetric ionic species carrying like charges can attract each other and give rise to thermodynamically unstable conditions.     

Anomalous dimension exponents on fractal structures for the Ising and three-state Potts model

We provide an overall picture of the magnetic critical behavior of the Ising and three-state Potts models on fractal structures. The results brought out from Monte Carlo simulations for several Hausdorff dimensions between 1 and 3 show that this behavior can be understood in the framework of weak universality. Moreover, the maxima of the susceptibility follow power laws in a very reliable way, which allows us to calculate the ratio of the exponents γ/ν and the anomalous dimension exponent η in a reliable way. At last, the evolution of these exponents with the Hausdorff dimension is discussed.     

Nonequilibrium critical poisoning in a single-species model

A simple model is introduced, which exhibits a poisining transition similar to that observed in studies of catalyc surface reactions. Steady-state behavior and critical exponents are determined via series expansion and Monte Carlo simulations. The relation between catalytic surface models and reggeon field theory is examined.     

Intrinsic ferromagnetism in CoBr2 nanolayers: a DFT+U and Monte Carlo study

The intrinsic ferromagnetism of CoBr₂ bulk was investigated using DFT (density functional theory) combined with the full potential linear augmented plane wave method and Monte Carlo simulations. The ground state of CoBr₂ exhibits ferromagnetic behavior and a semiconductor character. We used the generalized gradient approximation (GGA) and GGA+U (Hubbard correction) approximations to determinate the magnetic moment. The magnetic moment reached the experimental value and was in good agreement with the other theoretical values. The value obtained was used as an input to a Monte Carlo study to calculate the thermal magnetization and magnetic hysteresis cycles. Ferromagnetic behavior was observed and was found to be due to an positive exchange interaction. These results lead us to believe that this material could be a promising spintronic material.     

Coulomb and liquid dimer models in three dimensions

We study classical hard-core dimer models on three-dimensional lattices using analytical approaches and Monte Carlo simulations. On the bipartite cubic lattice, a local gauge field generalization of the height representation used on the square lattice predicts that the dimers are in a critical Coulomb phase with algebraic, dipolar correlations, in excellent agreement with our large-scale Monte Carlo simulations. The nonbipartite fcc and Fisher lattices lack such a representation, and we find that these models have both confined and exponentially deconfined but no critical phases. We conjecture that extended critical phases are realized only on bipartite lattices, even in higher dimensions.     

Dynamic critical behavior in models of ferromagnetic gadolinium

A numerical technique combining Monte Carlo and molecular dynamics simulations is used for the first time to examine the complex critical dynamics of models of ferromagnetic gadolinium in which both strong exchange interactions and relativistic effects of several different types are taken into account. A finite-size scaling technique is used to calculate the corresponding dynamic critical exponents. The role played by isotropic dipole-dipole interaction in the critical behavior of gadolinium is evaluated. The results obtained provide an explanation for the anomalous dynamic critical behavior of gadolinium.     

An Analysis of Polynomial Chaos Approximations for Modeling Single-Fluid-Phase Flow in Porous Medium Systems

We examine a variety of polynomial-chaos-motivated approximations to a stochastic form of a steady state groundwater flow model. We consider approaches for truncating the infinite dimensional problem and producing decoupled systems. We discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature, it would be necessary to find new multivariate cubature rules. Finally, we use the acceleration of Monte Carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than Monte Carlo for the relatively small domains considered in this work. A curse of dimensionality in the series expansion of the log-normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work, other than the Monte Carlo method.     

Spin-glass transition of the three-dimensional Heisenberg spin glass

It is shown, by means of Monte Carlo simulation and finite size scaling analysis, that the Heisenberg spin glass undergoes a finite-temperature phase transition in three dimensions. There is a single critical temperature, at which both a spin glass and a chiral glass ordering develop. The Monte Carlo algorithm, adapted from lattice gauge theory simulations, makes it possible to thermalize lattices of size L = 32, larger than in any previous spin-glass simulation in three dimensions. High accuracy is reached thanks to the use of the Marenostrum supercomputer. The large range of system sizes studied allows us to consider scaling corrections.     

Stochastic FDTD accuracy improvement through correlation coefficient estimation

This paper introduces a new scheme to improve the accuracy of the stochastic finite difference time domain (S-FDTD) method. S-FDTD, reported recently by Smith and Furse, calculates the variations in the electromagnetic fields caused by variability or uncertainty in the electrical properties of the materials in the model. The accuracy of the S-FDTD method is controlled by the approximations for correlation coefficients between the electrical properties of the materials in the model and the fields propagating in them. In this paper, new approximations for these correlation coefficients are obtained using Monte Carlo method with a small number of runs, terming them as Monte Carlo correlation coefficients (MC-CC). Numerical results for two bioelectromagnetic simulation examples demonstrate that MC-CC can improve the accuracy of the S-FDTD method and yield more accurate results than previous approximations.     

Numerical Calculations for the Surface on a Semi-Infinite Ising Model with a Random Surface Field

We study the critical behavior of the surface on a semi-infinite simple cubic lattice Ising model with a bimodal random surface field by large cell mean-field renormaliza tion group method and Monte Carlo simulations. Our results show that the surface ferromagnetic phase exists in the weak random field range above the bulk critical temperature. The surface. specific heat is not divergence and the susceptibility show a cusp singularity at the surface ferromagnetic-paramagnetic transition for a relatively large and om field.     

A Monte Carlo Sampling Scheme for the Ising Model

In this paper we describe a Monte Carlo sampling scheme for the Ising model and similar discrete-state models. The scheme does not involve any particular method of state generation but rather focuses on a new way of measuring and using the Monte Carlo data. We show how to reconstruct the entropy S of the model, from which, e.g., the free energy can be obtained. Furthermore we discuss how this scheme allows us to more or less completely remove the effects of critical fluctuations near the critical temperature and likewise how it reduces critical slowing down. This makes it possible to use simple state generation methods like the Metropolis algorithm also for large lattices.     

Universality aspects of the 2d random-bond Ising and 3d Blume-Capel models

We report on large-scale Wang-Landau Monte Carlo simulations of the critical behavior of two spin models in two- (2d) and three-dimensions (3d), namely the 2d random-bond Ising model and the pure 3d Blume-Capel model at zero crystal-field coupling. The numerical data we obtain and the relevant finite-size scaling analysis provide clear answers regarding the universality aspects of both models. In particular, for the random-bond case of the 2d Ising model the theoretically predicted strong universality’s hypothesis is verified, whereas for the second-order regime of the Blume-Capel model, the expected d = 3 Ising universality is verified. Our study is facilitated by the combined use of the Wang-Landau algorithm and the critical energy subspace scheme, indicating that the proposed scheme is able to provide accurate results on the critical behavior of complex spin systems.     

Lattice-gas model driven by Hubbard electrons

Self-consistent Monte Carlo simulations are undertaken for a lattice-gas model which is driven by the free energy of electrons described by a Hubbard model with electronic hopping restricted to ions at nearest-neighbor sites. Our previous work, an independent-electron tight-binding lattice-gas model (bcc or fcc), is modified to introduce two effects: the disorder of the dense system and, more importantly, the role of the electronic correlation. The first effect is achieved using an fcc lattice, but restricted so an occupied site can have no more than eight, instead of twelve, occupied nearest-neighbor sites. To treat correlations, the electronic intra-atomic repulsion is, at first, included via the Gutzwiller approximation at finite temperature; this approach is simple enough to be solved for all cases in the large, disordered systems used in our Monte Carlo simulations but can still give a good qualitative representation of the main effects of the electronic correlations. Then, the exact treatment of the Hubbard model for clusters with up to six atoms is integrated into the calculation. We obtain vapor-liquid coexistence curves and then, approximations to the electronic conductivities and paramagnetic susceptibilities at coexistence conditions. This simple model is, in part, motivated by experiments on the alkali-metal fluids.     

Monte Carlo simulation of quantum statistical lattice models

In this article we review recent developments in computational methods for quantum statistical lattice problems. We begin by giving the necessary mathematical basis, the generalized Trotter formula, and discuss the computational tools, exact summations and Monte Carlo simulation, that will be used to examine explicit examples. To illustrate the general strategy, the method is applied to an analytically solvable, non-trivial, model: the one-dimensional Ising model in a transverse field. Next it is shown how to generalized Trotter formula most naturally leads to different path-integral representations of the partition function by considering one-dimensional fermion lattice models. We show how to analyze the different representations and discuss Monte Carlo simulation results for one-dimensional fermions. Then Monte Carlo work on one- and two-dimensional spin-$\text{1}\text{2}$ models based upon the Trotter formula approach is reviewed and the more dedicated Handscomb Monte Carlo method is discussed. We consider electron-phonon models and discuss Monte Carlo simulation data on the Molecular Crystal Model in one, two and three dimensions and related one-dimensional polaron models. Exact numerical results are presented for free fermions and free bosons in the canonical ensemble. We address the main problem of Monte Carlo simulations of fermions in more than one dimension: the cancellation of large contributions. Free bosons on a lattice are compared with bosons in a box and the effects of finite size on Bose-Einstein condensation are discussed.     

Histogram Monte Carlo position-space renormalization group: Applications to the site percolation

We study site percolation on the square lattice and show that, when augmented with histogram Monte Carlo simulations for large lattices, the cell-to-cell renormalization group approach can be used to determine the critical probability accurately. Unlike the cell-to-site method and an alternate renormalization group approach proposed recently by Sahimi and Rassamdana, both of which rely onab initio numerical inputs, the cell-to-cell scheme is free of prior knowledge and thus can be applied more widely.