Similarities and contrasts between critical point behavior of heavy fluid alkalis and d-dimensional Ising model

A brief summary is first given of recent progress in establishing the near-critical point behavior of the fluid alkalis Rb and Cs. Departure from the law of Rectilinear Diameters is emphasized, along with its consequences for theories emphasizing homogeneity and scaling. The behavior as the critical points of Rb and Cs are approached is compared and contrasted with the d-dimensional Ising model.     

Spontaneous magnetization of the Ising model on a layered square lattice

We have studied the Ising model on a layered square lattice with four different coupling constants and two different magnetic moments. The partition function at zero magnetic field is derived exactly. We propose a formula for the spontaneous magnetization which agrees with the exact low-temperature series expansion up to the 16th order and reduces to the exact result of Au-Yang and McCoy in a special case.     

Nonconcavity of the magnetization in Ising ferromagnets

Some Ising ferromagnets having nonconcave magnetization are presented as counterexamples to the often assumed case of concave magnetization.     

Tree graph inequalities and critical behavior in percolation models

Various inequalities are derived and used for the study of the critical behavior in independent percolation models. In particular, we consider the critical exponent associated with the expected cluster sizex and the structure of then-site connection probabilities =_n(x₁,..., x_n). It is shown that quite generally 1. The upper critical dimension, above which attains the Bethe lattice value 1, is characterized both in terms of the geometry of incipient clusters and a diagramatic convergence condition. For homogeneousd-dimensional lattices with (x, y)=O(¦x -y¦^–(d–2+), atp=p _c, our criterion shows that =1 if > (6-d)/3. The connectivity functions _n are generally bounded by tree diagrams which involve the two-point function. We conjecture that above the critical dimension the asymptotic behavior of _n, in the critical regime, is actually given by such tree diagrams modified by a nonsingular vertex factor. Other results deal with the exponential decay of the cluster-size distribution and the function ₂ (x, y). A. P. Sloan Foundation Research Fellow. Research supported in part by the National Science Foundation Grant No. PHY-8301493.Research supported in part by the National Science Foundation Grant No. MCS80-19384.     

Corner spontaneous magnetization

Details are given of a new method allowing an exact calculation of the spontaneous magnetization in the corner as well as along the edge at an arbitrary distance of the corner for a rectangular planar Ising ferromagnet.     

Spontaneous magnetization of the Ising model on a 4–8 lattice

The spontaneous magnetization of the Ising model on a 4–8 lattice with six different coupling constants and two different magnetic moments is studied. A formula for the spontaneous magnetization is proposed. The result agrees with the exact low-temperature series expansions up to the 12th order.     

The stochastic models method applied to the critical behavior of Ising lattices

The stochastic models (SM) computer simulation method for treating manybody systems in thermodynamic equilibrium is investigated. The SM method, unlike the commonly used Metropolis Monte Carlo method, is not of a relaxation type. Thus an equilibrium configuration is constructed at once by adding particles to an initiallyempty volume with the help of a model stochastic process. The probability of the equilibrium configurations is known and this permits one to estimate the entropy directly. In the present work we greatly improve the accuracy of the SM method for the two and three-dimensional Ising lattices and extend its scope to calculate fluctuations, and hence specific heat and magnetic susceptibility, in addition to average thermodynamic quantities like energy, entropy, and magnetization. The method is found to be advantageous near the critical temperature. Of special interest are the results at the critical temperature itself, where the Metropolis method seems to be impractical. At this temperature, the average thermodynamic quantities agree well with theoretical values, for both the two and three-dimensional lattices. For the two-dimensional lattice the specific heat exhibits the expected logarithmic dependence on lattice size; the dependence of the susceptibility on lattice size is also satisfactory, leading to a ratio of critical exponents/=1.85 ±0.08. For the three-dimensional lattice the dependence of the specific heat, long-range order, and susceptibility on lattice size leads to similarly satisfactory exponents:=0.12 ±0.03,=0.30 ±0.03, and=1.32 ±0.05 (assuming =2/3).     

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.     

On the continuity of the magnetization and energy in Ising ferromagnets

We investigate the phase diagram of ferromagnetic Ising spin systems satisfying the G.H.S. inequality. We show that they cannot have a normal first-order phase transition as the temperatureT is varied, i.e., one where the magnetization is discontinuous and there are three coexisting phases. Furthermore, for n.n. interactions, discontinuity in the magnetization at 0 <T ₀ T _c implies an uncountable infinity of pure phases atT ₀.     

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。     

On scaling properties of cluster distributions in Ising models

Scaling relations of cluster distributions for the Wolff algorithm are derived. We found them to be well satisfied for the Ising model ind=3 dimensions. Using scaling and a parametrization of the cluster distribution, we determine the critical exponent/=0.516(6) with moderate effort in computing time.     

On the equivalence of different order parameters for Ising ferromagnets

In a recent note Barber showed, for a spin-1/2 Ising system with ferromagnetic pair interactions, that some critical exponents of the triplet order parameter _{i j k} are the same as those of the magnetization _i . Here we prove such results for all odd correlations and dispense with the requirement of pair interactions. We also prove that the critical temperatureT _c , defined as the temperature below which there is a spontaneous magnetization, is for fixed even spin interactionsJ _e independent of the way in which the odd interactionsJ _o approach zero from above. This is achieved by using only the simplest, Griffiths-Kelley-Sherman (GKS), inequalities, which apply to the most general many-spin, ferromagnetic interactions.Research supported in part by NSF Grant #MPS 75-20638.     

Corner transfer matrices of the triangular Ising model

Recently, a new technique for investigating the zero-field, eight-vertex model on the square lattice using corner transfer matrices was suggested by Baxter. In this paper these ideas are applied to the anisotropic, ferromagnetic, triangular Ising lattice in zero field below its critical temperature. The diagonal form of the corner transfer matrix for the triangular lattice shows essentially the same structure as that for the square Ising lattice. The spontaneous magnetization can be obtained easily and agrees with that previously derived.     

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.     

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.     

Monte Carlo study of the critical behavior of pure and site-diluted Ising ferro-and ferrimagnets

Monte Carlo simulations are performed for pure and site-diluted Ising ferro- and ferrimagnets on a simple cubic lattice with up to 40³ sites and with impurity concentrationx. For the diluted ferromagnet (x=0.2) the exponent= 0.392±0.03 is definitely larger than the pure model value of=0.304±0.03. In contrast, for ferrimagnetic systems (x=0, 0.1, 0.2) the values appear to be independent ofx and within the error limits consistent with the value for the pure ferromagnet, possibly because the width of the asymptotic random critical regime (or of the crossover regime) is even smaller than in the case of ferromagnets.     

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.