Three-dimensional Ising Model Research Articles

On the critical properties of the three-dimensional random Ising model

The critical behaviour of a three-dimensional weakly diluted quenched Ising model is investigated by analysing the series for the renormalization-group functions obtained in a minimal subtraction scheme. Formerly it was established (R. Folk, Yu. Holovatch, and T. Yavors'kii, Phys. Rev. B 61, 15 114 (2000)), that resummed renormalization group series possess an “optimal truncation” behaviour, provided a Chisholm-Borel resummation is applied. We apply the method of subsequent resummation and discuss the convergence properties of the series increasing the order of approximation to 5-loop level.

Numerical equation of state and other scaling functions from an improved three-dimensional Ising model

We study an improved three-dimensional Ising model with external magnetic field near the critical point by Monte Carlo simulations. From our data we determine numerically the universal scaling functions of the magnetization, that is the equation of state, of the susceptibility and of the correlation length. In order to normalize the scaling functions we calculate the critical amplitudes of the three observables on the critical line, the phase boundary and the critical isochore. The amplitudes lead to the universal ratios C +/ C −=4.756(28), R χ =1.723(13), Q c =0.326(3), and Q 2=1.201(10). We find excellent agreement of the data with the parametric representation of the asymptotic equation of state as found by field theory methods. The comparison of the susceptibility data to the corresponding scaling function shows a marginal difference in the symmetric phase, which can be explained by the slightly different value for R χ used in the parametrization. The shape of the correlation-length-scaling function is similar to the one of the susceptibility, as expected from earlier parametrizations. The peak positions of the two scaling functions are coinciding within the error bars.

Ground-state numerical study of the three-dimensional random-field Ising model

The random field Ising model in three dimensions with Gaussian random fields is studied at zero temperature for system sizes up to 60^3. For each realization of the normalized random fields, the strength of the random field, Delta and a uniform external, H is adjusted to find the finite-size critical point. The finite-size critical point is identified as the point in the H-Delta plane where three degenerate ground states have the largest discontinuities in the magnetization. The discontinuities in the magnetization and bond energy between these ground states are used to calculate the magnetization and specific heat critical exponents and both exponents are found to be near zero.

EXTREME LONG-TIME DYNAMIC MONTE CARLO SIMULATIONS FOR METASTABLE DECAY IN THE d=3 ISING FERROMAGNET

We study the extreme long-time behavior of the metastable phase of the three-dimensional Ising model with Glauber dynamics in an applied magnetic field and at a temperature below the critical temperature. For these simulations, we use the advanced simulation method of projective dynamics. The algorithm is described in detail, together with its application to the escape from the metastable state. Our results for the field dependence of the metastable lifetime are in good agreement with theoretical expectations and span more than 50 decades in time.

Inverse Monte Carlo renormalization group transformations for critical phenomena.

We introduce a computationally stable inverse Monte Carlo renormalization group transformation method that provides a number of advantages for the calculation of critical properties. We are able to simulate the fixed point of a renormalization group for arbitrarily large lattices without critical slowing down. The log-log scaling plots obtained with this method show remarkable linearity, leading to accurate estimates for critical exponents. We illustrate this method with calculations in two- and three-dimensional Ising models for a variety of renormalization group transformations.

Novel Monte Carlo algorithms and their applications

We describe a generalized scheme for the probability-changing cluster (PCC) algorithm, based on the study of the finite-size scaling property of the correlation ratio, the ratio of the correlation functions with different distances. We apply this generalized PCC algorithm to the two-dimensional 6-state clock model. We also discuss the combination of the cluster algorithm and the extended ensemble method. We derive a rigorous broad histogram relation for the bond number. A Monte Carlo dynamics based on the number of potential moves for the bond number is proposed, and applied to the three-dimensional Ising and 3-state Potts models.

Magnetic order parameter in the perovskite system CaMn 7 O 12

The magnetic ordering in the distorted perovskite system CaMn7O12 has been studied by powder neutron diffraction. The magnetic ordering in CaMn7O12 consists of two phases: α, which is ferrimagnetic and β, which is modulated. The magnetic peaks due to both phases disappear at the same temperature near 90 K. The temperature dependence of the value of the magnetic moments μ(T) in phase α can be described for 55 K<T<90 K with a power-law characteristic for critical scattering: μ(T)∝(TN-T)β with β≃0.31, in agreement with the prediction for a three-dimensional Ising model. Models of possible magnetic orderings in CaMn7O12 are discussed.

Sweeny and Gliozzi dynamics for simulations of Potts models in the Fortuin-Kasteleyn representation.

We compare the correlation times of the Sweeny and Gliozzi dynamics for two-dimensional Ising and three-state Potts models, and the three-dimensional Ising model for the simulations in the percolation representation. The results are also compared with Swendsen-Wang and Wolff cluster dynamics. It is found that Sweeny and Gliozzi dynamics have essentially the same dynamical critical behavior. Contrary to Gliozzi's claim [Phys. Rev. E 66, 016115 (2002)], the Gliozzi dynamics has critical slowing down comparable to that of other cluster methods. For the two-dimensional Ising model, both Sweeny and Gliozzi dynamics give good fits to logarithmic size dependences for the correlation times; for two-dimensional three-state Potts model, their dynamical critical exponent z is 0.49+/-0.01; the three-dimensional Ising model has z=0.37+/-0.02.

Overlap distribution of the three-dimensional Ising model.

We study the Parisi overlap probability density P(L)(q) for the three-dimensional Ising ferromagnet by means of Monte Carlo (MC) simulations. At the critical point, P(L)(q) is peaked around q=0 in contrast with the double peaked magnetic probability density. We give particular attention to the tails of the overlap distribution at the critical point, which we control over up to 500 orders of magnitude by using the multioverlap MC algorithm. Below the critical temperature, interface tension estimates from the overlap probability density are given and their approach to the infinite volume limit appears to be smoother than for estimates from the magnetization.

Effect of shape on the critical nucleus size in a three-dimensional Ising model: Energetic and kinetic approaches

The initial growth rate and the excitation energy of spin clusters of different shapes (cubic, stair-like, pyramidal) on a three-dimensional cubic lattice were calculated, assuming Ising-type interactions between spins and the Glauber-type spin dynamics. Using energetic and kinetic approaches, the critical cluster size was obtained and compared with existing Monte Carlo “experiments.” It was shown that in most cases the kinetic approach fits better the “experimental” results than the energetic one. A shape independent kinetic criterium for a critical cluster size was also formulated on the basis of an analysis of all possible configurations of a cluster with a given number of spins.

MICROCANONICAL DETERMINATION OF THE ORDER PARAMETER CRITICAL EXPONENT

A highly efficient Monte Carlo method for the calculation of the density of states of classical spin systems is presented. As an application, we investigate the density of states ΩN(E, M) of two- and three-dimensional Ising models with N spins as a function of energy E and magnetization M. For a fixed energy lower than a critical value Ec,N the density of states exhibits two sharp maxima at M = ± Msp(E) which define the microcanonical spontaneous magnetization. An analysis of the form Msp(E) ∝ (Ec, ∞ - E)βε yields very good results for the critical exponent βε, thus demonstrating that critical exponents can be determined by analyzing directly the density of states of finite systems.

Critical structure factor in using systems.

We perform a large-scale Monte Carlo simulation of the three-dimensional Ising model on simple cubic lattices of size L(3) with L=128 and 256. We determine the corresponding structure factor (Fourier transform of the two-point function) and compare it with several approximations and with experimental results. We also compute the turbidity as a function of the momentum of the incoming radiation, focusing in particular on the deviations from the Ornstein-Zernike expression of Puglielli and Ford.

25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple-cubic lattice.

25th-order high-temperature series are computed for a general nearest-neighbor three-dimensional Ising model with arbitrary potential on the simple cubic lattice. In particular, we consider three improved potentials characterized by suppressed leading scaling corrections. Critical exponents are extracted from high-temperature series specialized to improved potentials, obtaining gamma=1.2373(2), nu=0.63012(16), alpha=0.1096(5), eta=0.036 39(15), beta=0.326 53(10), and delta=4.78 93(8). Moreover, biased analyses of the 25th-order series of the standard Ising model provide the estimate Delta=0.52(3) for the exponent associated with the leading scaling corrections. By the same technique, we study the small-magnetization expansion of the Helmholtz free energy. The results are then applied to the construction of parametric representations of the critical equation of state, using a systematic approach based on a global stationarity condition. Accurate estimates of several universal amplitude ratios are also presented.

Probability-Changing Cluster Algorithm: Study of Three-Dimensional Ising Model and Percolation Problem

We present a detailed description of the idea and procedure for the newly proposed Monte Carlo algorithm of tuning the critical point automatically, which is called the probability-changing cluster (PCC) algorithm [Y. Tomita and Y. Okabe, Phys. Rev. Lett. {\bf 86} (2001) 572]. Using the PCC algorithm, we investigate the three-dimensional Ising model and the bond percolation problem. We employ a refined finite-size scaling analysis to make estimates of critical point and exponents. With much less efforts, we obtain the results which are consistent with the previous calculations. We argue several directions for the application of the PCC algorithm.

Antisymmetric and other subleading corrections to scaling in the local potential approximation

For systems in the universality class of the three-dimensional Ising model we compute the critical exponents in the local potential approximation (LPA), that is, in the framework of the Wegner–Houghton equation. We are mostly interested in antisymmetric corrections to scaling, which are relatively poorly studied. We find the exponent for the leading antisymmetric correction to scaling ω A ≈1.691 in the LPA. This high value implies that such corrections cannot explain asymmetries observed in some Monte Carlo simulations.

Application of new Monte Carlo algorithms to random spin systems

We explain the idea of the probability-changing cluster (PCC) algorithm, which is an extended version of the Swendsen–Wang algorithm. With this algorithm, we can tune the critical point automatically. We show the effectiveness of the PCC algorithm for the case of the three-dimensional (3D) Ising model. We also apply this new algorithm to the study of the 3D diluted Ising model. Since we tune the critical point of each random sample automatically with the PCC algorithm, we can investigate the sample-dependent critical temperature and the sample average of physical quantities at each critical temperature, systematically. We have also applied another newly proposed algorithm, the Wang–Landau algorithm, to the study of the spin glass problem.

Crossover effects in the bond-diluted Ising model in three dimensions

We investigate by Monte Carlo simulations the critical properties of the three-dimensional bond-diluted Ising model. The phase diagram is determined by locating the maxima of the magnetic susceptibility and is compared to mean-field and effective-medium approximations. The calculation of the size-dependent effective critical exponents shows the competition between the different fixed points of the model as a function of the bond dilution.

Phase transitions at surfaces, edges, and corners

Results of large-scale Monte Carlo simulations of three-dimensional Ising models with edges and corners are reviewed. At the ordinary transition, angle dependent critical exponents are observed, whereas at the surface transition edge and corner critical exponents are non-universal and depend on the details of the model. The results obtained at the surface transition are compared to exact findings on critical two-dimensional Ising models with different types of defect lines.

Nonlinear susceptibilities of a weakly disordered uniaxial ferromagnet in the critical region

For the three-dimensional random Ising model, the effective sextic coupling constant v_6 and the nonlinear susceptibilities of the fourth (chi_4) and sixth (chi_6) orders are calculated at criticality. These quantities are shown to differ markedly from their counterparts for pure uniaxial magnets. In particular, the ratio v_6/v_{4}^2 entering the equation of state of the random Ising model turns out to be equal to 0.87, while in pure magnets v_6/v_{4}^2 = 1.65. The universal susceptibility ratios m^3 chi_4/chi^2 and m^6 chi_6/chi^3 (m - the inverse correlation length) are found to differ by factors 1.6 and 2.7, respectively, for random and uniform Ising models. These big differences of the universal quantities can be measured both in physical and computer experiments, and such measurements may be considered as a tool for an identification of the random critical behavior.

`Fragile' glass-forming behaviour ofa three-dimensional Ising ferromagnet with uniformfrustration

By using Monte Carlo simulation, we study the dynamics around equilibrium of a uniform frustrated three-dimensionalIsing model with nearest-neighbour ferromagnetic interactions and small, but long-range (coulombic) antiferromagnetic interactions. This model exhibits a strong slowing down of the relaxations for temperatures below the critical temperature of the unfrustrated model. The characteristics ofthis slowing down - super-Arrhenius activated temperaturedependence of the relaxation time and non-exponential decayof the correlation function - are similar to those observed in fragile glass-forming liquids.