Two-dimensional Ferromagnetic Ising Model Research Articles

Competitive dynamics in a three-dimensional Ising model

We consider a three-dimensional ferromagnetic Ising model on a cubic lattice in contact with a heat bath at temperature T. The states of the system evolve in time according to two stochastic processes: the one-spin-flip Glauber dynamics where the order parameter is not conserved, and the two-spin-exchange Kawasaki kinetics, which conserves the order parameter. The former process mimics an input of energy into the system. Monte Carlo simulations were employed to determine the phase diagram for the stationary states of the model, and the corresponding critical exponents. Similarly to the observed for the related two-dimensional ferromagnetic Ising model, the phase diagram obtained exhibits the phenomenon of self-organization. Although the stationary states are mainly ferromagnetic at low temperatures, an antiferromagnetic phase appears for extremely high values of the flux of energy. Unlike the ferromagnetic case, the region of the phase diagram occupied by the antiferromagnetic phase is now larger. The determined critical exponents for this nonequilibrium model are in agreement with the well-known accepted values for the three-dimensional equilibrium Ising model.

Extrapolation-CAM theory for critical exponents

By intentionally underestimating the rate of convergence of exact-diagonalization values for the mass or energy gaps of finite systems, we form families of sequences of gap estimates. The gap estimates cross zero with generically nonzero linear terms in their Taylor expansions, so that $\nu = 1$ for each member of these sequences of estimates. Thus, the Coherent Anomaly Method can be used to determine $\nu$. Our freedom in deciding exactly how to underestimate the convergence allows us to choose the sequence that displays the clearest coherent anomaly. We demonstrate this approach on the two-dimensional ferromagnetic Ising model, for which $\nu = 1$. We also use it on the three-dimensional ferromagnetic Ising model, finding $\nu \approx 0.629$, in good agreement with other estimates.

Higher-order-magnetization-cumulant universality of the two-dimensional Ising model

We study the higher-order cumulant universality of the order-parameter distribution at criticality for the two-dimensional nearest-neighbor ferromagnetic Ising model with Monte Carlo methods. These cumulants are interesting because they are a quantitative measure of the shape of the critical order parameter distribution. This shape has been predicted to be universal. Up to now, only the fourth-order cumulant has been studied in any detail. The set of higher-order cumulants as a whole is more sensitive to the details of the distribution and hence is a stronger test. It cannot be sampled efficiently with standard Monte Carlo sampling. To the best of our knowledge, no definite numerical results have been reported in the literature. Using the umbrella-sampling technique, we are able to obtain accurate estimates for the fourth-, sixth-, eighth-, and tenth-order cumulants for both the square and triangular lattices of different lattice sizes. These results support universality.

Constrained variational problem with applications to the Ising model

We continue our study of the behavior of the two-dimensional nearest neighbor ferromagnetic Ising model under an external magnetic fieldh, initiated in our earlier work. We strengthen further a result previously proven by Martirosyan at low enough temperature, which roughly states that for finite systems with (−)-boundary conditions under a positive external field, the boundary effect dominates in the system if the linear size of the system is of orderB/h withB small enough, while ifB is large enough, then the external field dominates in the system. In our earlier work this result was extended to every subcritical value of the temperature. Here for every subcritical value of the temperature we show the existence of a critical valueB 0 (T) which separates the two regimes specified above. We also find the asymptotic shape of the region occupied by the (+)-phase in the second regime, which turns out to be a “squeezed Wulff shape”. The main step in our study is the solution of the variational problem of finding the curve minimizing the Wulff functional, which curve is constrained to the unit square. Other tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field, extended to all temperatures below the critical one.

Complete analyticity for 2D Ising completed

We study the behavior of the two-dimensional nearest neighbor ferromagnetic Ising model under an external magnetic fieldh. We extend to every subcritical value of the temperature a result previously proven by Martirosyan at low enough temperature, and which roughly states that for finite systems with — boundary conditions under a positive external field, the boundary effect dominates in the bulk if the linear size of the system is of orderB/h withB small enough, while ifB is large enough, then the external field dominates in the bulk. As a consequence we are able to complete the proof that “complete analyticity for nice sets” holds for every value of the temperature and external field in the interior of the uniqueness region in the phase diagram of the model. The main tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field, and recently extended to all temperatures below the critical one by Ioffe.

Surface tension and universality in the Ising model

We report on a new numerical computation of the surface tension between domains of opposite magnetisation in the Ising model in two and three dimensions. The method is quite general and can be applied to any statistical-mechanics model. It is based on a partition of the lattice into two halves, which are slowly driven from one magnetization state to the other. The free energy of the interface is the result of the difference between the total free energies of the homogeneous and of the mixed phases. We first test the method in the two-dimensional (ferromagnetic) Ising model by comparing with the Onsager solution. We then compare the results in three dimensions with previous Monte Carlo simulations of thermal equilibrium and of decays via nucleation. The three-dimensional results are also used in the computation of the universal amplitude combination that involves the correlation length and the surface tension. Near the critical temperature, the numerical results agree well with measurements performed in the laboratory.

Behavior of droplets for a class of Glauber dynamics at very low temperature

We consider a class of Glauber dynamics for the two-dimensional nearest neighbor ferromagnetic Ising model in which the rate with which each spin flips depends only on the increment in energy caused by its flip in a monotonic non-increasing fashion. We extend to this class results previously shown for the particular case of Metropolis dynamics [NS]. We show that for fixed volume and external field 0<h<1, at very low temperature small rectangular droplets of sping +1 in a sea of spins −1 tend to shrink, while large droplets tend to grow and cover the whole system. The threshold between the two behaviors is sharply defined, the critical length being 2/h. An example is given which shows that without the assumption of monotonicity of the rates this result may be false. We use the result on critical droplets to show that starting from the configuration with all spins down, the systems in the class that we consider evolve in a metastable fashion until the configuration with all spins up is reached. For similar systems in higher dimensions we show that under analogous conditions on the rates, small enough droplets are likely to shrink, while large enough droplets are likely to grow.

The pattern of escape from metastability of a stochastic ising model

We study further the metastable behavior of Metropolis dynamics for the two-dimensional nearest neighbor ferromagnetic Ising model, with positive and small external field, in the limit as the temperature vanishes (see [NS]). We focus on the typical features of the escape (nucleation) from the (metastable) configuration with all spins −1, to the (stable) configuration with all spins +1. Using the reversibility of the process as the main tool, we prove (for the discrete time version of the model) that the first step of a typical escaping path is the time reverse of a typical time evolution of a shrinking subcritical rectangular droplet, which is one slice smaller than a critical droplet. This subcritical droplet then evolves in a time of order 1 to a critical droplet, which finally grows with features described in [NS].

Thermal conductivity of a kinetic ising model

Using a novel extension of the microcanonical Monte Carlo algorithm, we have simulated the behavior of a two-dimensional nearest-neighbor ferromagnetic Ising model in the presence of a temperature gradient. The technique consists of setting the temperatures of boundary spins, while allowing "demons" associated with the other sites to control heat transfer. We demonstrate that our system is in local thermodynamic equilibrium, and compute the thermal conductivity as a function of temperature.

Thermodynamics at less than the correlation volume.

Consider an infinite thermodynamic system A, an open subsystem ${A}_{V}$, with fixed volume V, of A, and an open subsystem ${A}_{V\mathcal{'}}$, with fixed volume V', of ${A}_{V}$. Evidence is presented to support the following thermodynamic fluctuation rule: At any time, the probability of finding the thermodynamic state of ${A}_{V\mathcal{'}}$ in some range, given the thermodynamic state of ${A}_{V}$, is independent of the thermodynamic state of A, even if the volume $V--- is less than the correlation volume of $A---. This rule was confirmed by Monte Carlo simulation of the two-dimensional ferromagnetic Ising model. Both thermodynamic properties and the local correlation function were examined. Another rule, which states that, for a given state of ${A}_{V}$, the fluctuation probability distribution for the thermodynamic state of ${A}_{V\mathcal{'}}$ depends on the volumes only as \ensuremath{\Delta}t=(V'${)}^{\mathrm{\ensuremath{-}}1}$-${V}^{\mathrm{\ensuremath{-}}1}$ was also tested. It works well at volumes larger than the correlation volume, but was shown to break down at volumes less than the correlation volume. In addition, an easily justified rule stating that the average value of a density of a globally conserved additive quantity of a subsystem equals that of the infinite system was tested and found to work well. These rules are part of an effort to extend thermodynamic fluctuation theory to volumes less than the correlation volume.

Monte Carlo study of the domain kinetics of the Ising model with random coupling constants.

Monte Carlo simulations are used to study the domain kinetics of the two-dimensional ferromagnetic Ising model with random coupling constants which have a Gaussian distribution with a mean value J and a width \ensuremath{\Delta}J. When this system is quenched to low temperature, the evolution of initially circular domains is observed. We find that the relation between the decay time t and size L of the domain is given by L(T)=${\mathrm{Ct}}^{a}$, where a is an exponent less than 1/2 varying with \ensuremath{\Delta}J and the quenching temperature T, or 1/2 for the pure system. The larger the \ensuremath{\Delta}J and the lower the temperature T is, the smaller the exponent becomes.

Breakdown of self-similar scaling in the two-dimensional random-field Ising model: A Monte Carlo study

We have simulated the two-dimensional ferromagnetic Ising model in a random magnetic field with spin-flip dynamics. After the system is deeply quenched into the unstable region of the phase diagram, we observe novel dynamical behavior, during an early-to-intermediate time regime, for the average size of the growing domains, R\ifmmode\bar\else\textasciimacron\fi{}. We find that self-similar scaling for the structure factor [that is, scaling to a single time-dependent length such as R\ifmmode\bar\else\textasciimacron\fi{}(t)] breaks down for long times. Our results may have relevance to the problem of island-growth kinetics in some chemisorbed systems.

Thermodynamics near the correlation volume.

I present and test three simple thermodynamic fluctuation rules which may in some cases hold for subsystems of infinite systems with volumes less than the correlation volume. Tests at volumes near the correlation volume are made in the two-dimensional square ferromagnetic Ising model by Monte Carlo simulation. Fluctuations into the metastable and spinodal regions are discussed. Aside from difficulties apparently resulting from the small volumes used in the simulations, the rules are found to work well.

Domain Growth in the Ising Model in a Random Magnetic Field

Simulations of the two-dimensional ferromagnetic Ising model in a random magnetic field with spin-flip dynamics are reported. After the system is deeply quenched into the unstable region of the phase diagram, novel dynamical behavior for the average size of the growing domains is observed.

On Limiting Gibbs States of the Two-Dimensional Ising Models

We consider limiting Gibbs states in the two-dimensional ferromagnetic Ising model at sufficiently low temperatures. We prove that every limiting Gibbs state corresponding to a boundary condition such that N^+/N^– &lt; θ &lt; 3/5 on every boundary is μ^– , where N^+ is the number of up-spins on the boundary and N^- is that of down-spins. We also prove that for each θ &gt; 3/5 , there exists a boundary condition such that 3/5 &lt; N^+/N^– ≤ θ on every boundary, and the limiting Gibbs state corresponding to this boundary condition is μ^+ .