Three-dimensional Ising Model Research Articles

Sheared Ising models in three dimensions

The nonequilibrium phase transition in sheared three-dimensional Ising models is investigated using Monte Carlo simulations in two different geometries corresponding to different shear normals. We demonstrate that in the high shear limit both systems undergo a strongly anisotropic phase transition at exactly known critical temperatures Tc which depend on the direction of the shear normal. Using dimensional analysis, we determine the anisotropy exponent θ = 2 as well as the correlation length exponents ν‖ = 1 and ν⊥ = 1/2. These results are verified by simulations, though considerable corrections to scaling are found. The correlation functions perpendicular to the shear direction can be calculated exactly and show Ornstein-Zernike behavior.

Application of the statistical associating fluid theory for potentials of variable range (SAFT-VR) coupled with renormalisation-group (RG) theory to model the phase equilibria and second-derivative properties of pure fluids

In earlier work, a methodology to couple SAFT-VR with an RG treatment (SAFT-VR+RG) for square-well chain and associating fluids was presented [E. Forte, F. Llovell, L.F. Vega, J.P.M. Trusler, A. Galindo, J. Chem. Phys. 134 (15) (2011) 154102]. The approach is based on a recursive procedure which begins with an initial free energy incorporating only contributions from short-wavelength density fluctuations, which are treated locally. The contribution from long-wavelength fluctuations is incorporated through the iterative procedure based on attractive interactions that incorporate the structure of the fluid following the ideas of perturbation theories and using a mapping that allows integration of the radial distribution function. The resulting SAFT-VR+RG theory provides an equation of state based on the classical SAFT-VR equation and yields the correct critical behaviour without the need for additional adjustable parameters. In this work, the application of this equation to real fluids is presented. The ability of the methodology to reproduce the phase behaviour close to and far from criticality solely through the standard SAFT-VR molecular parameters is demonstrated here. Molecular model parameters for a range of fluids, both non-associating and associating are presented, including the homologous series of n-alkanes, benzene, carbon dioxide, water, light alcohols, ammonia, hydrogen fluoride and hydrogen sulphide. These were determined by comparison with experimental data for vapour pressure and saturated liquid density. Furthermore, the models presented are used to predict second derivative properties for these compounds. Deviations compared with the original SAFT-VR equation are discussed in order to assess further the improvements introduced by the coupled theory to predict these properties. The critical exponents α, β, γ and δ have also been obtained for a number of these fluids and found to be in good agreement with the universal values of the three-dimensional Ising model.

Testing of pseudorandom sequence generators by using the three-dimensional Ising model

Testing of pseudorandom sequence generators by using the three-dimensional Ising model

Universality and critical behavior at the critical endpoint in the itinerant-electron metamagnet UCoAl

We performed nuclear-magnetic-resonance (NMR) measurements on itinerant-electron metamagnet UCoAl in order to investigate the critical behavior of the magnetism near a metamagnetic (MM) critical endpoint (CEP). We derived c-axis magnetization $M_c$ and its fluctuation $S_c$ from the measurements of Knight shift and nuclear spin-lattice relaxation rate $1/T_1$ as a function of the c-axis external field ($H_c$) and temperature ($T$). We developed contour plots of $M_c$ and $S_c$ on the $H_c$ - $T$ phase diagram, and observed the strong divergence of $S_c$ at the CEP. The critical exponents of $M_c$ and $S_c$ near the CEP are estimated, and found to be close to the universal properties of a three-dimensional (3-D) Ising model. We indicate that the critical phenomena at the itinerant-electron MM CEP in UCoAl have a common feature as a gas-liquid transition.

Non-Equilibrium Critical Dynamics of Ferromagnets with Long Range Correlated Defects

The simultaneous effect of non-equilibrium initial states and correlation betweendefects of the structure on the evolution of anisotropic disordered systems at the critical pointwas analyzed. The field theory description of the non-equilibrium critical behavior of three-dimensional disordered systems with the long-range correlated defects was given and the dy-namical critical exponent of the short-time evolution was calculated in the two-loop approxima-tion without the use of the "-expansion. The values of the dynamical critical exponent obtainedby using various methods for summing asymptotic series were compared with the results ofthe computer simulation of the non-equilibrium critical behavior of the three-dimensional dis-ordered Ising model in the short-time regime.

Nonequilibrium phase transitions and stationary-state solutions of a three-dimensional random-field Ising model under a time-dependent periodic external field

Nonequilibrium behavior and dynamic phase-transition properties of a kinetic Ising model under the influence of periodically oscillating random fields have been analyzed within the framework of effective-field theory based on a decoupling approximation. A dynamic equation of motion has been solved for a simple-cubic lattice (q=6) by utilizing a Glauber-type stochastic process. Amplitude of the sinusoidally oscillating magnetic field is randomly distributed on the lattice sites according to bimodal and trimodal distribution functions. For a bimodal type of amplitude distribution, it is found in the high-frequency regime that the dynamic phase diagrams of the system in the temperature versus field amplitude plane resemble the corresponding phase diagrams of the pure kinetic Ising model. Our numerical results indicate that for a bimodal distribution, both in the low- and high-frequency regimes, the dynamic phase diagrams always exhibit a coexistence region in which the stationary state (ferro or para) of the system is completely dependent on the initial conditions, whereas for a trimodal distribution, the coexistence region disappears depending on the values of the system parameters.

Dynamical Monte Carlo studies of the three-dimensional bimodal random-field Isingmodel

The nature of the three-dimensional random-field Ising model with a bimodal probabilitydistribution is investigated using finite-time scaling combined with the Monte Carlorenormalization group method, in the presence of a linearly varying temperature. Our resultssupport the existence of a first-order phase transition for this model, obtained throughreducing the influence of finite-size effects. The critical exponents are estimated to beν = 0.74(3), β = 0.27(1), α = − 0.023(5), andγ = 1.48(3) with corrections to the scaling. The Rushbrooke scaling law is satisfied with theseexponents, and in turn identifies the asymptotic critical behavior.

Magnetic properties and critical behavior of disordered Fe1−xRuxalloys: A Monte Carlo approach

We study the critical behavior of a quenched random-exchange Ising model with competing interactions on a bcc lattice. This model was introduced in the study of the magnetic behavior of Fe(1-x)Ru(x) alloys for ruthenium concentrations x=0%, x=4%, x=6%, and x=8%. Our study is carried out within a Monte Carlo approach, with the aid of a re-weighting multiple histogram technique. By means of a finite-size scaling analysis of several thermodynamic quantities, taking into account up to the leading irrelevant scaling field term, we find estimates of the critical exponents α, β, γ, and ν, and of the critical temperatures of the model. Our results for x=0% are in excellent agreement with those for the three-dimensional pure Ising model in the literature. We also show that our critical exponent estimates for the disordered cases are consistent with those reported for the transition line between paramagnetic and ferromagnetic phases of both randomly dilute and ±J Ising models. We compare the behavior of the magnetization as a function of temperature with that obtained by Paduani and Branco (2008), qualitatively confirming the mean-field result. However, the comparison of the critical temperatures obtained in this work with experimental measurements suggest that the model (initially obtained in a mean-field approach) needs to be modified.

Interface-roughening phase diagram of the three-dimensional Ising model for all interaction anisotropies from hard-spin mean-field theory

The roughening phase diagram of the d=3 Ising model with uniaxially anisotropic interactions is calculated for the entire range of anisotropy, from decoupled planes to the isotropic model to the solid-on-solid model, using hard-spin mean-field theory. The phase diagram contains the line of ordering phase transitions and, at lower temperatures, the line of roughening phase transitions, where the interface between ordered domains roughens. Upon increasing the anisotropy, roughening transition temperatures settle after the isotropic case, whereas the ordering transition temperature increases to infinity. The calculation is repeated for the d=2 Ising model for the full range of anisotropy, yielding no roughening transition.

GPU-based single-cluster algorithm for the simulation of the Ising model

We present the GPU calculation with the common unified device architecture (CUDA) for the Wolff single-cluster algorithm of the Ising model. Proposing an algorithm for a quasi-block synchronization, we realize the Wolff single-cluster Monte Carlo simulation with CUDA. We perform parallel computations for the newly added spins in the growing cluster. As a result, the GPU calculation speed for the two-dimensional Ising model at the critical temperature with the linear size L = 4096 is 5.60 times as fast as the calculation speed on a current CPU core. For the three-dimensional Ising model with the linear size L = 256, the GPU calculation speed is 7.90 times as fast as the CPU calculation speed. The idea of quasi-block synchronization can be used not only in the cluster algorithm but also in many fields where the synchronization of all threads is required.

Universal amplitude ratios in the Ising model in three dimensions

We use a high-precision Monte Carlo simulation to determine the universal specific-heat amplituderatio A + /A − in the three-dimensional Ising model via the impact angleϕ of complex temperature zeros. We also measure the correlation-length critical exponentν from finite-size scaling and the specific-heat exponentα through hyperscaling. Extrapolations to the thermodynamic limit yieldϕ = 59.2(1.0)°,A + /A − = 0.56(3),ν = 0.630 48(32) andα = 0.1086(10). These results are compatible with some previous estimates from a variety of sources andrule out recently conjectured exact values.

Parallel-tempering cluster algorithm for computer simulations of critical phenomena

In finite-size scaling analyses of Monte Carlo simulations of second-order phase transitions one often needs an extended temperature range around the critical point. By combining the parallel-tempering algorithm with cluster updates and an adaptive routine to find the temperature window of interest, we introduce a flexible and powerful method for systematic investigations of critical phenomena. As a result, we gain one to two orders of magnitude in the performance for two- and three-dimensional Ising models in comparison with the recently proposed Wang-Landau recursion for cluster algorithms based on the multibondic algorithm, which is already a great improvement over the standard multicanonical variant.

Critical free energy and Casimir forces in rectangular geometries

We study the critical behavior of the free energy and the thermodynamic Casimir force in a L(∥)(d-1) × L block geometry in 2<d<4 dimensions with aspect ratio ρ=L/L(∥) on the basis of the O(n) symmetric ϕ4 lattice model with periodic boundary conditions and with isotropic short-range interactions. Exact results are derived in the large-n limit describing the geometric crossover from film (ρ=0) over cubic (ρ=1) to cylindrical (ρ=∞) geometries. For n=1, three perturbation approaches in the minimal renormalization scheme at fixed d are presented that cover both the central finite-size regime near T(c) for 1/4≲ρ≲3 and the region well above and below T(c). At bulk T(c), we predict the critical Casimir force in the vertical (L) direction to be negative (attractive) for a slab (ρ<1), positive (repulsive) for a rod (ρ>1), and zero for a cube (ρ=1). Our results for finite-size scaling functions agree well with Monte Carlo data for the three-dimensional Ising model by Hasenbusch for ρ=1 and by Vasilyev et al. for ρ=1/6 above, at, and below T(c).

Progress in numerical simulations of systems with a θ-vacuum like term: The two- and three-dimensional Ising model within an imaginary magnetic field

Using the approach developed in Azcoiti et al. (2003) [1], we succeeded in reconstructing the behaviour of the antiferromagnetic Ising model with imaginary magnetic field iθ for two and three dimensions in the low temperature regime. A mean-field calculation, expected to work well for high dimensions, is also carried out, and the mean-field results coincide qualitatively with those of the two- and three-dimensional Ising model. The mean-field analysis reveals also a phase structure more complex than the one expected for QCD with a topological θ-term.

Effective-field-theory analysis of the three-dimensional random-field Ising model on isometric lattices

An Ising model with quenched random magnetic fields is examined for single-Gaussian, bimodal, and double-Gaussian random-field distributions by introducing an effective-field approximation that takes into account the correlations between different spins that emerge when expanding the identities. Random-field distribution shape dependencies of the phase diagrams and magnetization curves are investigated for simple cubic, body-centered-cubic, and face-centered-cubic lattices. The conditions for the occurrence of reentrant behavior and tricritical points on the system are also discussed in detail.

Critical scaling to infinite temperature

Three-dimensional Ising model ferromagnets on different lattices with nearest-neighbor interactions, and on simple-cubic lattices with equivalent interactions out to further neighbors, are studied numerically. The susceptibility data for all these systems are analyzed using the critical renormalization-group theory formalism over the entire temperature range above ${T}_{c}$ with an appropriate choice of scaling variable and scaling expressions. Representative experimental data on a metallic ferromagnet (Ni) and an elementary fluid (Xe) are interpreted in the same manner so as to estimate effective coordination numbers.

Aspect-ratio dependence of thermodynamic Casimir forces

We consider the three-dimensional Ising model in a L(⊥)×L(∥)×L(∥) cuboid geometry with a finite aspect ratio ρ=L(⊥)/L(∥) and periodic boundary conditions along all directions. For this model the finite-size scaling functions of the excess free energy and thermodynamic Casimir force are evaluated numerically by means of Monte Carlo simulations. The Monte Carlo results compare well with recent field theoretical results for the Ising universality class at temperatures above and slightly below the bulk critical temperature T(c). Furthermore, the excess free energy and Casimir force scaling functions of the two-dimensional Ising model are calculated exactly for arbitrary ρ and compared to the three-dimensional case. We give a general argument that the Casimir force vanishes at the critical point for ρ=1 and becomes repulsive in periodic systems for ρ>1.

Real-space renormalization group for the transverse-field Ising model in two and three dimensions

The two- and three-dimensional transverse-field Ising models with ferromagnetic exchange interactions are analyzed by means of the real-space renormalization-group method. The basic strategy is a generalization of a method developed for the one-dimensional case, which exploits the exact invariance of the model under renormalization and is known to give the exact values of the critical point and critical exponent ν. The resulting values of the critical exponent ν in two and three dimensions are in good agreement with those for the classical Ising model in three and four dimensions. To the best of our knowledge, this is the first example in which a real-space renormalization group on (2+1)- and (3+1)-dimensional Bravais lattices yields accurate estimates of the critical exponents.

Zero-temperature freezing in the three-dimensional kinetic Ising model

We investigate the relaxation of the Ising-Glauber model on a periodic cubic lattice after a quench to zero temperature. In contrast to the conventional picture from phase-ordering kinetics, we find the following: (i) domains at long time are highly interpenetrating and topologically complex, with average genus growing algebraically with system size; (ii) the long-time state is almost never static, but rather contains "blinker" spins that can flip ad infinitum with no energy cost. (iii) The energy relaxation is extremely slow, with a characteristic time that grows exponentially with system size.

Three-dimensional (3D) Ising universality in magnets and critical indices at fluid–fluid phase transition

Experimental data for critical exponents in some magnetic materials are compared with recent theoretical results on the three-dimensional (3D) Ising model, as derived by one of us (ZDZ) based on two conjectures [Z.D. Zhang, Conjectures on the exact solution of three-dimensional (3D) simple orthorhombic Ising lattices, Phil. Mag. 87 (2007), pp. 5309–5419]. It is found that critical exponents in some bulk magnetic materials indeed form a 3D Ising universality. Our attention is then focused on the critical indices at fluid–fluid phase transition. We suggest to use Zhang's exponent β = 3/8 to fit the experimental data over the wider asymptotic region near the critical point of a fluid–fluid phase transition. The 3D Ising universality should exist for critical indices in a certain class of magnets and at fluid–fluid phase transition.