Three-dimensional Ising Model Research Articles

Specific heat of NH4Br and NH4BrxCl1−x crystals close to the antiferroelectric transition

We calculate the specific heat CVI of NH4Br and mixed crystals of NH4BrxCl1−x using the Ising model for the transition between the disordered (D) and antiferro- (AF) electric phases in these crystalline systems. Our CVI values, which we calculated for xBr=1, 0.26, and 0.51 are in good agreement, both below and above TC, with the observed CP data from the literature. Our value of ɑ≅0.1 (T<TC and T>TC) for the critical exponent of the specific heat agrees with the value of 0.125 predicted by a three-dimensional Ising model.

Dynamics of nanoscale polar regions and critical behavior of the uniaxial relaxorSr0.61Ba0.39Nb2O6:Co

The complex dielectric susceptibility of ${\mathrm{Sr}}_{0.61}{\mathrm{Ba}}_{0.39}{\mathrm{Nb}}_{2}{\mathrm{O}}_{6}:\mathrm{Co}$ has been measured at frequencies ${10}^{1}\ensuremath{\leqslant}f\ensuremath{\leqslant}{10}^{9}\phantom{\rule{0.3em}{0ex}}\mathrm{Hz}$ and temperatures $300\ensuremath{\leqslant}T\ensuremath{\leqslant}400\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ before and after poling. The high-frequency relaxation peak observed above the ferroelectric phase transition temperature, ${T}_{c}\ensuremath{\approx}348\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, is analyzed in terms of distribution functions of the relaxation time, $g(\ensuremath{\tau})$, using Tikhonov regularization. It reveals the criticality of the largest relaxation time in agreement with activated dynamic scaling of the three-dimensional random-field Ising model. Differences of zero-field- and field-cooled $g(\ensuremath{\tau})$ are discussed.

Corrections to local scale invariance in the nonequilibrium dynamics of critical systems: Numerical evidences

Local scale invariance (LSI) has been recently proposed as a possible extension of the dynamical scaling in systems at the critical point and during phase ordering. LSI has been applied inter alia to provide predictions for the scaling properties of the response function of non-equilibrium critical systems in the aging regime following a quench from the high-temperature phase to the critical point. These predictions have been confirmed by Monte Carlo simulations and analytical results for some specific models, but they are in disagreement with field-theoretical predictions. By means of Monte Carlo simulations of the critical two- and three-dimensional Ising model with Glauber dynamics, we study the intermediate integrated response, finding deviations from the corresponding LSI predictions that are in qualitative agreement with the field-theoretical computations. This result casts some doubts on the general applicability of LSI to critical dynamics.

Hysteresis and avalanches in theT=0random-field Ising model with two-spin-flip dynamics

We study the non-equilibrium behavior of the three-dimensional Gaussian random-field Ising model at T=0 in the presence of a uniform external field using a 2-spin-flip dynamics. The deterministic, history-dependent evolution of the system is compared with the one obtained with the standard 1-spin-flip dynamics used in previous studies of the model. The change in the dynamics yields a significant suppression of coercivity, but the distribution of avalanches (in number and size) stays remarkably similar, except for the largest ones that are responsible for the jump in the saturation magnetization curve at low disorder in the thermodynamic limit. By performing a finite-size scaling study, we find strong evidence that the change in the dynamics does not modify the universality class of the disorder-induced phase transition.

Critical behavior of the three-dimensional compressible Ising antiferromagnet at constant volume: A Monte Carlo study

Extensive Monte Carlo simulations in the semi-grand-canonical ensemble are used to study the critical behavior of a three-dimensional compressible Ising model with antiferromagnetic interactions under constant volume conditions. Elastic forces between spins are introduced by the Stillinger-Weber potential and energy parameters are chosen in such a way that antiparallel spin ordering is favored, analogous to the antiferromagnetic coupling in the rigid Ising Hamiltonian. All the quantities analyzed strongly indicate that the system remains in the universality class of the standard (rigid) three-dimensional Ising model, in contrast with theoretical predictions.

Nonequilibrium critical dynamics in inhomogeneous systems

We study nonequilibrium dynamical properties of inhomogeneous systems, in particular at a free surface or at a defect plane. Thereby we consider nonconserved (model-A) dynamics of a system which is prepared in the high-temperature phase and quenched into the critical point. Using Monte Carlo simulations we measure single spin relaxation and autocorrelations, as well as manifold autocorrelations and persistence. We show that, depending on the decay of critical static correlations, the short time dynamics can be of two kinds. For slow decay of local correlations the usual domain growth process takes place with non-stationary and algebraic dynamical correlations. If, however, the local correlations decay sufficiently rapidly we have the so called cluster dissolution scenario, in which case short time dynamical correlations are stationary and have a universal stretched exponential form. This latter phenomenon takes place in the surface of the three-dimensional Ising model and should be observable in real ferromagnets.

Optimization of field-dependent nonperturbative renormalization group flows

We investigate the influence of the momentum cutoff function on the field-dependent nonperturbative renormalization group flows for the three-dimensional Ising model, up to the second order of the derivative expansion. We show that, even when dealing with the full functional dependence of the renormalization functions, the accuracy of the critical exponents can be simply optimized, through the principle of minimal sensitivity, which yields $\ensuremath{\nu}=0.628$ and $\ensuremath{\eta}=0.044$.

Monte Carlo studies of critical behaviour of systems with long-range correlated disoder

Monte Carlo simulations of the short-time dynamic behaviour are reported for three-dimensional Ising model and XY-model with long-range spatially correlated disorder at criticality, in the case corresponding to linear defects. The static and dynamic critical exponents are computed with the use of the corrections to scaling. The obtained values of the exponents are in a good agreement with results of the field-theoretic description of the critical behaviour of this model in the two-loop approximation and with our results of Monte Carlo simulations of three-dimensional Ising model in equilibrium state.

Random Ising model in three dimensions: theory, experiment and simulation - a difficult coexistence

We discuss different approaches for studying the influence of disorder in the three-dimensional Ising model. From the theoretical point of view, renormalisation group calculations provide quite accurate results. Experiments carried out on crystalline mixtures of compounds lead to measurements as accurate as three digits on the values of critical exponents. Numerically, extensive Monte Carlo simulations then pretend to be of comparable accuracy. Life becomes complicated when details are compared between the three approaches.

Spanning avalanches in the three-dimensional Gaussian random-field Ising model with metastable dynamics: Field dependence and geometrical properties

Spanning avalanches in the 3D Gaussian Random Field Ising Model (3D-GRFIM) with metastable dynamics at $T=0$ have been studied. Statistical analysis of the field values for which avalanches occur has enabled a Finite-Size Scaling (FSS) study of the avalanche density to be performed. Furthermore, a direct measurement of the geometrical properties of the avalanches has confirmed an earlier hypothesis that several types of spanning avalanches with two different fractal dimensions coexist at the critical point. We finally compare the phase diagram of the 3D-GRFIM with metastable dynamics with the same model in equilibrium at $T=0$.

Profile and Width of Rough Interfaces

In the context of Landau theory and its field theoretical refinements, interfaces between coexisting phases are described by intrinsic profiles. These intrinsic interface profiles, however, are neither directly accessible by experiment nor by computer simulation as they are broadened by long-wavelength capillary waves. In this paper we study the separation of the small scale intrinsic structure from the large scale capillary wave fluctuations in the Monte Carlo simulated three-dimensional Ising model. To this purpose, a blocking procedure is applied, using the block size as a variable cutoff, and a translationally invariant method to determine the interface position of strongly fluctuating profiles on small length scales is introduced. While the capillary wave picture is confirmed on large length scales and its limit of validity is estimated, an intrinsic regime is, contrary to expectations, not observed.

Microscopic approach to critical phenomena at interfaces: An application to complete wetting in the Ising model

We study how the formalism of the hierarchical reference theory (HRT) can be extended to inhomogeneous systems. HRT is a liquid-state theory which implements the basic ideas of the Wilson momentum-shell renormalization group (RG) to microscopic Hamiltonians. In the case of homogeneous systems, HRT provides accurate results even in the critical region, where it reproduces scaling and nonclassical critical exponents. We applied the HRT to study wetting critical phenomena in a planar geometry. Our formalism avoids the explicit definition of effective surface Hamiltonians but leads, close to the wetting transition, to the same renormalization group equation already studied by RG techiques. However, HRT also provides information on the nonuniversal quantities because it does not require any preliminary coarse graining procedure. A simple approximation to the infinite HRT set of equations is discussed. The HRT evolution equation for the surface free energy is numerically integrated in a semi-infinite three-dimensional Ising model and the complete wetting phase transition is analyzed. A renormalization of the adsorption critical amplitude and of the wetting parameter is observed. Our results are compared to available Monte Carlo simulations.

Monte Carlo study of the bulk magnetic properties of magnetite

Within the framework of a three-dimensional Ising model with nearest magnetic neighbor interactions, and using a Monte Carlo–Metropolis algorithm for equilibrium and energy minimization, we compute the canonical ensemble averages for magnetization, magnetic susceptibility and specific heat of stoichiometric magnetite Fe 3O 4. In the model, atoms are distributed on an inverse spinel structure for which interactions arising from antiferromagnetic Fe 3+ A–Fe 3+ A, Fe 3+ A–Fe 3+ B, Fe 3+ A–Fe 2+ B couplings and ferromagnetic Fe 3+ B–Fe 3+ B, Fe 3+ B–Fe 2+ B, Fe 2+ B–Fe 2+ B bonds are taken into account. Labels A and B refer to tetrahedral and octahedral sites, respectively. On the basis of finite-size scaling theory, our best estimates of critical exponents for the correlation length, specific heat, magnetization and susceptibility are ν = 0.65 ( 2 ) , α = 0.23 ( 2 ) , β = 0.35 ( 1 ) and γ = 1.13 ( 2 ) , respectively. Simulation results as a function of temperature also allow obtaining information of the contributions per magnetic ion to the total magnetization and to the susceptibility in a differentiated fashion.

Critical exponents at a reentrant isotropic–calamitic nematic phase transition

The critical exponents of the reentrant isotropic (I RE)–calamitic nematic (N C) and the N C–isotropic (I) phase transitions found in a lyotropic mixture of potassium laurate, decanol, and water are experimentally determined. Using optical birefringence measurements, along the entire range of the nematic phase, it was found that while the usual high temperature N C–I phase transition presents the well-known discontinuity, typical of a first order phase transition, with a critical exponent in the same class of universality of the three-dimensional Ising model, within the interval ( 0.36 ± 0.05 ), the low temperature N C–I RE phase transition presents strong indications of a second order phase transition, with a critical exponent in the interval ( 0.16 ± 0.01 ). This experimental achievement indicates that the high temperature and the low temperature (reentrant) nematic–isotropic phase transitions do not belongs to the same universal class.

Dynamics of Nucleation in the Ising Model

Reactive pathways to nucleation in a three-dimensional Ising model at 60% of the critical temperature are studied using transition path sampling of single spin flip Monte Carlo dynamics. Analysis of the transition state ensemble (TSE) indicates that the critical nuclei are rough and anisotropic. The TSE, projected onto the free energy surface characterized by cluster size, N, and surface area, S, indicates the significance of other variables in addition to these two traditional reaction coordinates for nucleation. The transmission coefficient, κ, along N is κ ≈ 0.35, and this reduction of the transmission coefficient from unity is explained in terms of the stochastic nature of the dynamic model.

Critical indices for systems of different space dimensionality

Equations for calculation of correlation length critical indices and the other critical indices for systems of different space dimensionality have been derived on the basis of the fluctuation theory of phase transition (FTPT) and inequalities for values of the critical indices. The values of the critical indices for two-, three-, four-spatial dimensionalities have been calculated by using these equations. The results obtained have a good agreement with classical theory of critical phenomena for d=4, three-dimensional Ising model ( d=3), and satisfy the Onsager solution for two-dimensional Ising model ( d=2).

Computer simulations at the fixed point using an inverse renormalization group transformation

Following Brandt and Ron's suggestion of inverting the renormalization group transformation used in Monte Carlo renormalization, it is shown that efficient computer simulations of the fixed point of the transformation can be carried out on very large systems without critical slowing down. We illustrate the new method with calculations of critical exponents for the two- and three-dimensional Ising models, based on several different transformations.

Reconstruction of the free energy in the metastable region using the path ensemble

By quenching into the metastable region of the three-dimensional Ising model, we investigate the paths that the magnetization (energy) takes as a function of time. We accumulate the magnetization (energy) paths into time-dependent distributions from which we reconstruct the free energy as a function of the magnetic field, temperature, and system size. From the reconstructed free energy, we obtain the free-energy barrier that is associated with the transition from a metastable state to the stable equilibrium state. Although mean-field theory predicts a sharp transition between the metastable and the unstable region where the free-energy barrier is zero, the results for the nearest-neighbor Ising model show that the free-energy barrier does not go to zero.

Aging phenomena in critical semi-infinite systems

Nonequilibrium surface autocorrelation and autoresponse functions are studied numerically in semi-infinite critical systems in the dynamical scaling regime. Dynamical critical behavior is examined for a nonconserved order parameter in semi-infinite two- and three-dimensional Ising models as well as in the Hilhorst--van Leeuwen model. The latter model permits a systematic study of surface aging phenomena, as the surface critical exponents change continuously as function of a model parameter. The scaling behavior of surface two-time quantities is investigated and scaling functions are confronted with predictions coming from the theory of local scale invariance. Furthermore, surface fluctuation-dissipation ratios are computed and their asymptotic values are shown to depend on the values of surface critical exponents.

The influence of chromium impurity centers on the critical properties of a weakly polar ferroelectric Li2Ge7O15

The Cr3+ EPR spectra of Li2Ge7O15 (LGO) crystals are analyzed in the temperature range of the ferroelectric phase transition. The temperature dependence of the local order parameter is determined from the measured splittings of the EPR lines in the polar phase. The experimental critical exponent of the order parameter β=0.31 in the range from the phase transition temperature TC to (TC-T) ∼ 40 K corresponds to the critical exponent of the three-dimensional Ising model. Analysis of the available data demonstrates that, away from the phase transition temperature TC, the macroscopic and local properties of LGO crystals are characterized by a crossover from the fluctuation behavior to the classical behavior described in terms of the mean-field theory. The temperature dependence of the local order parameter for LGO: Cr crystals does not exhibit a crossover from the Ising behavior (β=0.31) to the classical behavior (β=0.5). This is explained by the defect nature of Cr3+ impurity centers, which weaken the spatial correlations in the LGO host crystal. The specific features of the critical properties of LGO: Cr3+ crystals are discussed within a microscopic model of structural phase transitions.