Evidence for spinodal singularities in high-dimensional nearest-neighbor ising models

Conventional theories of nucleation predict that the metastable state has an average lifetime which monotonically decreases as the system is quenched further from the condensation point. However, theories based on the coarsegrained Ginzburg-Landau free energy functional seem to indicate that for systems above six dimensions there is a sharp spinodal dividing the metastable and unstable regimes where the lifetime of the metastable state diverges. Monte Carlo simulations are used to investigate this discrepency. Both nucleation rates and bulk susceptibility measurements seem to support the prediction of the Ginzburg-Landau theories.     

Properties of metastable ising models evolving under the Swendsen-Wang dynamics

The behavior of the metastable nearest neighbor Ising model governed by Swendsen-Wang dynamics (SW) is investigated ind=2. The results are compared to those obtained in standard Metropolis dynamics. Both the SW and Metropolis systems are observed to decay from the metastable state via the formation of nucleating droplets. Nucleation rates are measured and found to agree with those predicted by classical nucleation theory. The growth rates of the droplets are observed to differ between the two dynamics. In addition, the dynamic critical exponentz is measured in a mean-field (Curie-Weiss) metastable Ising model at the spinodal. It is found that for SW dynamics,z=2. Since this is the same value as that obtained in the Metropolis case, this result shows that SW does not change the dynamical universality class at the spinodal.     

动力学伊辛模型在振荡外场中的成核

研究了二维的动力学伊辛模型在一个具有偏向的振荡外场中的成核过程,主要关注成核时间与外场振荡周期ω的关系．随着ω的变化,成核时间出现最小值,最小的成核时间对应的平均临界核的大小也是最小的,这表明存在一个最佳的振荡频率,相比较于一个确定的外场,它更有利于成核．同时还研究了外场的初始相位的影响．     

Metastability and exponential approach to equilibrium for low-temperature stochastic Ising models

We present a proof of the exponential convergence to equilibrium of single-spin-flip stochastic dynamics for the two-dimensional Ising ferromagnet in the low-temperature case with not too small external magnetic fieldh uniformly in the volume and in the boundary conditions.     

Percolation of the minority spins in high-dimensional Ising models

We present some new results on the region in the-h plane where the + spins percolate for the nearest neighbor Ising model. In particular, it is shown that in high enough dimensionsd there is percolation of the minority spins at inverse temperatures< ₊ with some ₊>_c, for which ₊/gb_c1/2log(cd),c a constant.On leave from Rutgers University.     

Nucleation and metastability in three-dimensional Ising models

We present Monte Carlo experiments on nucleation theory in the nearest-neighbor three-dimensional Ising model and in Ising models with long-range interactions. For the nearest-neighbor model, our results are compatible with the classical nucleation theory (CNT) for low temperatures, while for the long-range model a breakdown of the CNT was observed near the mean-field spinodal. A new droplet model and a zeroth-order theory of droplet growth are also presented.Supported in part by grants from ARO, ONR, and NSF.     

The role of dimensionality in the kinetic Ising model of spinodal decomposition: Evidence from zero-temperature quenches

We study a three-dimensional Ising lattice gas model with spin-exchange dynamics quenched from infinite to zero temperature. We consider a wide range of values of the binary composition (i.e., magnetization) and annealed vacancy concentration. We find that, as in two dimensions, the system freezes in a configuration very far from equilibrium, and that the interface energy per bond in the frozen state, which is very large, in all cases takes very nearly the same values as in two dimensions. We discuss the implications of these results regarding the irrelevance of dimensionality in this problem.     

Metastability in the Two-Dimensional Ising Model with Free Boundary Conditions

We investigate metastability in the two dimensional Ising model in a square with free boundary conditions at low temperatures. Starting with all spins down in a small positive magnetic field, we show that the exit from this metastable phase occurs via the nucleation of a critical droplet in one of the four corners of the system. We compute the lifetime of the metastable phase analytically in the limit T 0, h 0 and via Monte Carlo simulations at fixed values of T and h and find good agreement. This system models the effects of boundary domains in magnetic storage systems exiting from a metastable phase when a small external field is applied.     

Mean-Field Theory for Percolation Models of the Ising Type

The q=2 random cluster model is studied in the context of two mean-field models: the Bethe lattice and the complete graph. For these systems, the critical exponents that are defined in terms of finite clusters have some anomalous values as the critical point is approached from the high-density side, which vindicates the results of earlier studies. In particular, the exponent ~ which characterizes the divergence of the average size of finite clusters is 1/2, and ~, the exponent associated with the length scale of finite clusters, is 1/4. The full collection of exponents indicates an upper critical dimension of 6. The standard mean field exponents of the Ising system are also present in this model (=1/2, =1), which implies, in particular, the presence of two diverging length-scales. Furthermore, the finite cluster exponents are stable to the addition of disorder, which, near the upper critical dimension, may have interesting implications concerning the generality of the disordered system/correlation length bounds.     

Ising Field Theory in a Magnetic Field: Analytic Properties of the Free Energy

We study the analytic properties of the scaling function associated with the 2D Ising model free energy in the critical domain TT _c , H0. The analysis is based on numerical data obtained through the Truncated Free Fermion Space Approach. We determine the discontinuities across the Yang–Lee and Langer branch cuts. We confirm the standard analyticity assumptions and propose extended analyticity; roughly speaking, the latter states that the Yang–Lee branching point is the nearest singularity under Langer's branch cut. We support the extended analyticity by evaluating numerically the associated extended dispersion relation.     

Kinetic Ising model in a time-dependent oscillating external magnetic field: effective-field theory

Recently,Shi et al.[2008 Phys.Lett.A 372 5922] have studied the dynamical response of the kinetic Ising model in the presence of a sinusoidal oscillating field and presented the dynamic phase diagrams by using an effective-field theory(EFT) and a mean-field theory(MFT).The MFT results are in conflict with those of the earlier work of Tom’e and de Oliveira,[1990 Phys.Rev.A 41 4251].We calculate the dynamic phase diagrams and find that our results are similar to those of the earlier work of Tom’e and de Oliveira;hence the dynamic phase diagrams calculated by Shi et al.are incomplete within both theories,except the low values of frequencies for the MFT calculation.We also investigate the influence of external field frequency(ω) and static external field amplitude(h0) for both MFT and EFT calculations.We find that the behaviour of the system strongly depends on the values of ω and h0.     

A discrete variational problem related to Ising droplets at low temperatures

We consider a variational problem on thed-dimensional latticeZ ^d which has applications in the study of the meatastable behavior of the stochastic Ising model. The problem, an isoperimetric one, is to find what is the smallest area a finite subset ofZ ^d can have restricted to three classes of subsets ofZ ^d . If is one of these subsets, we define its volume as the number of points in it and its area as the number of pairs of points inZ ^d which are neighbors and such that only one of them belongs to .     

Droplet dynamics for asymmetric Ising model

Nucleation from a metastable state is studied for an anisotropic Ising model at very low temperatures. It turns out that the critical nucleus as well as configurations on a typical path to it differ from the Wulff shape of an equilibrium droplet.     

A geometric interpretation of the roughening transition in Ising models

Progress in the area of the Ising model roughening transition has previously been limited by the lack of a good definition for the interface separating the pure phases. In the present work, a graphical definition is introduced and it is shown that roughening occurs precisely when this interface fluctuates to infinity.     

The distribution of clusters for the Ising model

We rigorously prove that the probabilityP _n for the origin to belong to a cluster of exactlyn positive spins in thev-dimensional Ising model behaves as exp(–n^{(v – 1)/v}) in various regions, including in particular the low-temperature positive and negative phases in zero magnetic field.     

Diagrammatic expansion and metastability in the random-field Ising model

Within the perturbation diagrammatic expansion we discuss the origin of differences in determinations of the lower critical dimension of the random-field Ising model and show that below four dimensions metastability and hysteresis occur. We also explain the occurrence of a quasicritical d=2 behavior at weak random fields, which is responsible for local stability of the ordered state above two dimensions.     

Fast algorithm for the simulation of Ising models

Using a new microcanonical algorithm efficiently vectorized on a Cray XMP, we reach a simulation speed of 1.5 nsec per update of one spin, three times faster than the best previous method known to us. Data for the nonlinear relaxation with conserved energy are presented for the two-dimensional Ising model.     

Ising Models on Hyperbolic Graphs II

We consider Ising models with ferromagnetic interactions and zero external magnetic field on the hyperbolic graph (v, f), where v is the number of neighbors of each vertex and f is the number of sides of each face. Let T _c be the critical temperature and T _c =supTT _c: ^f=( ⁺+ ^–)/2, where ^f is the free boundary condition (b.c.) Gibbs state, ⁺ is the plus b.c. Gibbs state and ^– is the minus b.c. Gibbs state. We prove that if the hyperbolic graph is self-dual (i.e., v=f) or if v is sufficiently large (how large depends on f, e.g., v35 suffices for any f3 and v17 suffices for any f17) then 0<T _c <T _c, in contrast with that T _c =T _c for Ising models on the hypercubic lattice Z ^d with d2, a result due to Lebowitz.⁽²²⁾ While whenever T<T _c , ^f=( ⁺+ ^–)/2. The last result is an improvement in comparison with the analogous statement in refs. 28 and 33, in which it was only proved that ^f=( ⁺+ ^–)/2 when TT _c and it remains to show in both papers that ^f =( ⁺+ ^–)/2 whenever T<T _c . Therefore T _c and T _c divide [0, ] into three intervals: [0, T _c ), (T _c , T _c), and (T _c, ] in which ^{+ –} but ^f =( ⁺+ ^–)/2, ^{+ –} and ^f ( ⁺+ ^–)/2, and ⁺= ^–, respectively.     

Ising models on the lattice Sierpinski gasket

Ferromagnetic Ising models on the lattice Sierpinski gasket are considered. We prove the Dobrushin-Shlosmann mixing condition and discuss corresponding properties of the stochastic Ising models.     

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.