Dynamics of the Random Ising Model with Long-Range Interaction

Critical dynamics of the random Ising model with long-range interaction decaying as r-(d σ) where d is the dimensionality) is studied by the theoretic renormalization-group approach. The system is released to an evolution within a model A dynamics. Asymptotic scaling laws are studied in a frame of the expansion in = 2σ - d. In dimensions d ＜ 2σ. the dynamic exponent z is calculated to the second order in at the random fixed point.``     

Rigorous bounds of the Lyapunov exponents of the one-dimensional random Ising model

We find analytic upper and lower bounds of the Lyapunov exponents of the product of random matrices related to the one-dimensional disordered Ising model, using a deterministic map which transforms the original system into a new one with smaller average couplings and magnetic fields. The iteration of the map gives bounds which estimate the Lyapunov exponents with increasing accuracy. We prove, in fact, that both the upper and the lower bounds converge to the Lyapunov exponents in the limit of infinite iterations of the map. A formal expression of the Lyapunov exponents is thus obtained in terms of the limit of a sequence. Our results allow us to introduce a new numerical procedure for the computation of the Lyapunov exponents which has a precision higher than Monte Carlo simulations.     

Upper bounds on the critical temperature for various ising models

Upper bounds are obtained for spin ±1 systems. In the case of only nearestneighbor interactions on, for example, the square lattice we obtain _cJ>0.3592. The method's strength is seen when considering systems with longer-range interactions. For example, we obtain _cJ>0.360 compared to the previous best bound of _c J 0.345 for the one-dimensional lattice with 1/r ² interactions. The method relies upon an identity between correlation functions and then the use of correlation inequalities to obtain the final bounds.     

二维伊辛模型相变临界现象的元胞自动机模拟

用元胞自动机模型模拟二维伊辛模型的相变临界现象,得出了相变图和时空演化图;当不存在外场时,数值模拟得到的临界温度与理论值精确吻合。     

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.     

Higher-order susceptibilities of the regular and the random Ising model on the Cayley tree. I

Explicit expressions for the fourth-order susceptibility (⁴), the fourth derivative of thebulk free energy with respect to the external field, are given for the regular and the random-bond Ising model on the Cayley tree in the thermodynamic limit, at zero external field. The fourth-order susceptibility for the regular system diverges at temperature T _c ⁽⁴⁾ = 2k _B ^–1 J/ln{1+2/[(z–1)^3/4–1]}, confirming a result obtained by Müller-Hartmann and Zittartz [Phys. Rev. Lett. 33:893 (1974)]; Herez is the coordination number of the lattice,J is the exchange integral, andk _B is the Boltzmann constant. The temperatures at which ⁽⁴⁾ and the ordinary susceptibility (²) diverge are given also for the random-bond and the random-site Ising model and for diluted Ising models.     

Study of Ising model on the rectangular-triangular lattice

The Ising model is studied on a new type of lattice which is named the rectangular-triangular lattice. The critical temperature for the ferromagnetic lattice is calculated exactly and it is shown that the antiferromagnetic lattice does not order at any temperature. Ground state properties are investigated and some features of frustration on the antiferromagnetic Ising lattice outlined.     

Corrections to the critical temperature in 2D Ising systems with Kac potentials

We consider ad=2 Ising system with a Kac potential whose mean-field critical temperature is 1. Calling >0 the Kac parameter, we prove that there existsc ^*>0 so that the true inverse critical temperature _cr() > 1 +by ² log ^-1, for anyb<c ^* and correspondingly small. We also show that if 0 andbc ^*, suitably, then the correlation functions (normalized and rescaled) converge to those of a non-Gaussian Euclidean field theory.     

The direct correlation function of a one-dimensional Ising model

The strictly finite range of the direct correlation function for a homogeneous nearest neughbor Ising chain is shown to persist in the presence of arbitrary site-dependent coupling constants and an arbitrary external field. A method is developed to examine the range of the direct correlation function for many-neighbor interactions. It is found from numerical examples that, in general, third-neighbor and higher interactions induce long-range direct correlations, as does the presence of a field in the second-neighbor case.     

The one-dimensional kinetic Ising model: A series expansion study

Exact power series expansions (through eight terms) in the time are derived for relaxation in the one-dimensional Ising model with nearest-neighbor interactions for a general rate parameter where the activation energy is a variable fraction of the energy required to break nearest-neighbor bonds. It is found that the qualitative nature of the relaxation is very dependent on this parameter, varying from nearly simple exponential decay (as with Glauber dynamics) for an intermediate value of this parameter, to an initial rate of change that is either much slower or faster than a simple exponential at the extremes of the range of variation of the parameter. The rate equations for the limit of rapid internal diffusion (internal equilibration) are integrated for several special values of the rate parameter. In general the internal equilibration approximation is not a good representation of the relaxation except when the relaxation is similar to Glauber dynamics.     

Detect genuine multipartite entanglement in the one-dimensional transverse-field Ising model

Recently Seevinck and Uffink argued that genuine multipartite entanglement (GME) had not been established in the experiments designed to confirm GME. In this paper, we use the Bell-type inequalities introduced by Seevinck and Svetlichny [M. Seevinck, G. Svetlichny, Phys. Rev. Lett. 89 (2002) 060401] to investigate the GME problem in the one-dimensional transverse-field Ising model. We show explicitly that the ground states of this model violate the inequality when the external transverse magnetic field is weak, which indicate that the ground states in this model with weak magnetic field are fully entangled. Since this model can be simulated with nuclear magnetic resonance, our results provide a fresh approach to experimental test of GME.     

Upper bounds on the critical temperature for the two-dimensional Blume-Emery-Griffiths model

We obtain rigorous upper bounds for the critical temperature associated with second-order phase transitions of the two-dimensional spin-1 BEG model for real values ofK andD coupling constants and forJ0. We use some correlation equalities and inequalities to show the exponential decay of the two-point function characterizing the disordered phase.     

Upper bounds for Ising model correlation functions

A Griffiths correlation inequality for Ising ferromagnets is refined and is used to obtain improved upper bounds for critical temperatures. It is shown that, for non-negative external fields, the mean field magnetization is an upper bound for the magnetization of Ising ferromagnets.On leave (1970–71) from Northwestern University, Evanston, Illinois 60201. Supported at IAS by a grant from the Alfred P. Sloan Foundation.     

Critical Behavior of Ising Model with Long Range Correlated Quenched Impurities

The theoretic renormalization-group approach is applied to the study of the critical behavior of the ddimensional Ising model with long-range correlated quenched impurities, which has a power-like correlations r-(d-ρ).The asymptotic scaling law is studied in the framework of the expansion in ε = 4 - d. In d ＜ 4, the dynamic exponent z .is calculated up to the second order in ρ with ρ = O(ε1/2). The shape function is obtained in one-loop calculation.When d = 4, the logarithmic corrections to the critical behavior are found. The finite size effect on the order parameter relaxation rate is also studied.     

The dynamic critical exponent of the three-dimensional Ising model

We measure the dynamic exponent of the three-dimensional Ising model using a damage spreading Monte Carlo approach as described by MacIsaac and Jan. We simulate systems fromL=5 toL=60 at the critical temperature,T _c =4.5115. We report a dynamic exponent,z=2.35±0.05, a value much larger than the consensus value of 2.02, whereas if we assume logarithmic corrections, we find thatz=2.05±0.05.     

采用正方元胞的Ising模型实空间重正化群解

在二维正方形晶格上,将元胞取为4格点正方形,采用3种不同的规则定义块自旋状态,进行了重正化群计算,得出了更为精确的结果;解决了元胞内格点数为偶数的重正化群计算问题.     

Phase diagrams of Ising models on Husimi trees II. Pair Wand multisite interaction systems

We continue an earlier study of multisite interaction Ising spin models on Husimi trees. In particular, attention is given to systems with both a nearestneighbor pair interaction and three-site interactions. We use our calculations of the phase diagrams of the systems on Husimi trees as approximations of systems with the same interactions but on a regular lattice, e.g., the triangle lattice. Specific models where exact results are available are used as test cases. All of the work involves computation of quantities, such as the magnetization, by iterative processes. Hence we are dealing with a discrete map and for certain values of the interaction strengths we obtain for the magnetization diagram results involving period doubling, chaos, period-three windows, etc., all phenomena of recent interest in connection with dynamical systems and now associated with certain Ising spin systems.     

Absence of phase transitions in certain one-dimensional long-range random systems

An Ising chain is considered with a potential of the formJ(i, j)/|i–j|, where theJ(i, j) are independent random variables with mean zero. The chain contains both randomness and frustration, and serves to model a spin glass. A simple argument is provided to show that the system does not exhibit a phase transition at a positive temperature if>1. This is to be contrasted with a ferromagnetic interaction which requires>2. The basic idea is to prove that the surfacefree energy between two half-lines is finite, although the surface energy may be unbounded. Ford-dimensional systems, it is shown that the free energy does not depend on the specific boundary conditions if>(1/2)d.     

二维伊辛模型相变临界点温度的模拟计算

用计算模拟方法计算了二维伊辛模型的相变临界点温度,其结果接近严格解,明显布喇格－威廉斯近似和贝特近似的结果。     

A Markov-property Monte Carlo method: One-dimensional Ising model

In this paper we introduce a new Monte Carlo procedure based on the Markov property. This procedure is particularly well suited to massively parallel computation. We illustrate the method on the critical phenomena of the well known one-dimensional Ising model. In the course of this work we found that the autocorrelation time for the Metropolis Monte Carlo algorithm is closely given by the square of the correlation length. We find speedup factors of the order of 1 million for the method as implemented on the CM2 relative to a serial machine. Our procedure gives error estimates which are quite consistent with the observed deviations from the analytically known exact results.