A mean-field equation of motion for the dynamic Ising model

A mean-field type of approximation is used to derive two differential equations, one approximately representing the average behavior of the Ising model with Glauber (spin-flip) stochastic dynamics, and the other doing the same for Kawasaki (spin-exchange) dynamics. The proposed new equations are compared with the Cahn-Allen and Cahn-Hilliard equations representing the same systems and with information about the exact behavior of the microscopic models.     

On the mean-field Ising model in a random external field

We use a method developed by van Hemmen to obtain the free energy of the mean-field Ising model in a random external magnetic field. Some results of previous mean-field calculations are confirmed and generalized. The tricritical point in the global phase diagram is discussed in detail. We also consider different probability distributions of the random fields and provide some proofs regarding the conditions for the existence of a tricritical point.     

Rigorous results concerning a generalized mean-field theory of dilute disordered Ising magnets

A nonlinear integral equation describing the distribution of the internal random field in a disordered Ising system under conditions appropriate to describe a spin-glass is studied.Using the principle of contraction mapping it is shown that the equation has one and only one solution over a finite range of temperatures, corresponding to a nontrivial transition to a spin-glass type state.     

Magnetic hysteresis in a molecular Ising ferrimagnet: Glauber dynamics approach

We use agent-based modeling to investigate the effect of conservatism and partisanship on the efficiency with which large populations solve the density classification task – a paradigmatic problem for information aggregation and consensus building. We find that conservative agents enhance the populations’ ability to efficiently solve the density classification task despite large levels of noise in the system. In contrast, we find that the presence of even a small fraction of partisans holding the minority position will result in deadlock or a consensus on an incorrect answer. Our results provide a possible explanation for the emergence of conservatism and suggest that even low levels of partisanship can lead to significant social costs. Electronic supplementary material  Supplementary Online Material     

Ising model on the generalized Bruhat-Tits tree

The partition function and the correlation functions of the Ising model on the generalized Bruhat-Tits tree are calculated. We computed also the averages of these correlation functions when the corresponding vertices are attached to the boundary of the generalized Bruhat-Tits tree.     

Ising model on the generalized Bruhat-Tits tree

The partition function and the correlation functions of the Ising model on the generalized Bruhat-Tits tree are calculated. We computed also the averages of these correlation functions when the corresponding vertices are attached to the boundary of the generalized Bruhat-Tits tree.     

Dynamic hysteresis loops of the spin-2 bilayer Ising model

Based on the mean-field theory and Glauber-type stochastic dynamics, the dynamic hysteresis loops (DHLs) of the spin-2 Ising model are studied on the bilayer square lattice. The DHLs are given for different values of temperature, crystal-field, exchange interaction and oscillating field frequency. It is found that the physical parameters have a strong effect on the shape and number of the DHLs. The results are compared with some theoretical and experimental works and found in a qualitatively good agreement.     

Dynamic hysteresis behaviors in the kinetic Ising system on triangular lattice

We studied dynamic hysteresis behaviors of the spin-1 Blume-Capel (BC) model in a triangular lattice by means of the effective-field theory (EFT) with correlations and using Glauber-type stochastic dynamics. The effects of the exchange interaction (J), crystal field (D), temperature (T) and oscillating frequency (w) on the hysteresis behaviors of the BC model in a triangular lattice are investigated in detail. Results are compared with some other dynamic studies and quantitatively good agreement is found.     

Magnetization of the Ising model on the generalized checkerboard lattice

We consider the Ising model on the generalized checkerboard lattice. Using a recent result by Baxter and Choy, we derive exact expressions for the magnetization of nodal spins at two values of the magnetic field,H=0 andH=i1/2kT. Our results are given in terms of Boltzmann weights of a unit cell of the checkerboard lattice without specifying its cell structures.     

Nucleation and hysteresis in Ising model: classical theory versus computer simulation

We have studied the nucleation in the nearest neighbour ferromagnetic Ising model, in different (d) dimensions, by extensive Monte-Carlo simulation using the heat-bath dynamics. The nucleation time () has been studied as a function of the magnetic field (h) for various system sizes in different dimensions (d=2,3,4). The logarithm of the nucleation time is found to be proportional to the power (-(d-1)) of the magnetic field (h) in d dimensions. The size dependent crossover from coalescence to nucleation regime is observed in all dimensions. The distribution of metastable lifetimes are studied in both regions. The numerical results are compared and found to be consistent with the classical theoretical predictions. In two dimensions, we have also studied the dynamical response to a sinusoidally oscillating magnetic field. The reversal time is studied as a function of the inverse of the coercive field. The applicability of the classical nucleation theory to study the hysteresis and coercivity has been discussed. Received: 21 January 1998 / Accepted: 17 March 1998     

Heterogenous mean-field analysis of a generalized voter-like model on networks

In this paper, we investigate the transport properties of electrons on the surface of topological insulators with tuneable external ferromagnetic barriers. It is confirmed that the transport properties of electrons can be greatly affected by the magnitude (or the direction) of the magnetisation and the gate voltage. Some electronically or magnetically controlled devices may be constructed based on our research.     

Hysteresis loop signatures of phase transitions in a mean-field model of disordered Ising magnet

The multiplicity of long-lived states in frustrated disordered magnets makes the task to experimentally deter-mine which of them has the lowest free energy (and thus what thermodynamic phase the sample is in) seem rather hopeless. Nevertheless here we show in the framework of Landau-type phenomenological model that signatures of the mean-field equilibrium phase transitions in such highly nonequilibrium systems may be found in the evolution of the hysteresis loop form. Thus the sequence of transitions from spin-glass to mixed phase and to ferromagnetic one results in the changes from inclined hysteresis loop to that with the developing vertical sides and to one with the perfectly vertical sides. Such relation between loop form and the location of global minimum may hold beyond the mean-field approximation and can be useful in the real experiments and Monte-Carlo simulations of the problems involving rugged potential landscape.     

Heterogenous mean-field analysis of a generalized voter-like model on networks

We propose a generalized framework for the study of voter models in complex networks at the heterogeneous mean-field (HMF) level that (i) yields a unified picture for existing copy/invasion processes and (ii) allows for the introduction of further heterogeneity through degree-selectivity rules. In the context of the HMF approximation, our model is capable of providing straightforward estimates for central quantities such as the exit probability and the consensus/fixation time, based on the statistical properties of the complex network alone. The HMF approach has the advantage of being readily applicable also in those cases in which exact solutions are difficult to work out. Finally, the unified formalism allows one to understand previously proposed voter-like processes as simple limits of the generalized model.     

Three-spin correlation of the Ising model on the generalized checkerboard lattice

The Ising model on the generalized checkerboard lattice is studied and the three-spin correlation function is obtained for the three nodal spins surrounding a unit cell of the checkerboard lattice. As an application of this result, the spontaneous magnetization of the internal spin within a unit cell is calculated.     

Transitions between modulated phases in an Ising model with third-neighbour interactions (mean-field approximation)

We study order–disorder transitions in a three-dimensional Ising lattice in which all the spins belonging to the same xy plane have the same degree of disorder, so that the structure can be effectively reduced to a chain of layers. The layers interact with each other up to third neighbours. Employing the mean-field approximation, we find the different configurations that undergo the transition to total disorder in terms of the interaction constants and work out a diagram displaying the possible sequences of modulated phases that can be found when the temperature goes from 0 to the order–disorder transition point. At intermediate temperatures the average values of the spins of the layers for periodic structures are found by solving an equation system. Substitution of these values into the expression of the free energy allows one to determine the most stable structure for each set of interaction constants and for each temperature. The model predicts a transition between two modulated structures with the same wavelength but different unit cells, for suitable values of the interaction constants. The formalism is also applied to substances like UNi₂Si₂, with only a partial agreement with experiment.     

Magnetic phase diagrams of axial third-nearest-neighbour Ising model

The magnetic phase diagrams of axial third-nearest-neighbour Ising (A3NNI) model under an external field have been studied by means of the molecular field approximation. The phase boundaries among paramagnetic, (anti)ferromagnetic and various modulated phases are determined by analysing the frequency-dependent susceptibility, or by solving the coupled equations for magnetisations for spins up to 17, iteratively. The resultant phase diagram is found to be very consistent with the exact results for the ground state spin ordering and to be far improved than existing finite temperature phase diagrams for the same model.     

Metastable states and T=0 hysteresis in the random-field Ising model on random graphs

We study the ferromagnetic random-field Ising model on random graphs of fixed connectivity z (Bethe lattice) in the presence of an external magnetic field H. We compute the number of single-spin-flip stable configurations with a given magnetization m and study the connection between the distribution of these metastable states in the H-m plane (focusing on the region where the number is exponentially large) and the shape of the saturation hysteresis loop obtained by cycling the field between - and + at T=0. The annealed complexity A(m,H) is calculated for z=2,3,4 and the quenched complexity Q(m,H) for z=2. We prove explicitly for z=2 that the contour Q(m,H)=0 coincides with the saturation loop. On the other hand, we show that A(m,H) is irrelevant for describing, even qualitatively, the observable hysteresis properties of the system.