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.     

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.     

Bounds for the anisotropic Ising model

Rigorous upper bounds are found for the magnetisation, susceptibility, critical temperature and crossover exponent in an anisotropic Ising system.     

Remarks on upper bounds for correlation functions of the Ising ferromagnet

We give a sufficient condition under which a general Ising ferromagnet correlation function 〈σ_R〉 is equal to tanhβJ_R. This lemma allows us to giveI. a new simple proof of the 3rd GKS inequality and to show that the r.h.s. of this inequality is the strongest bound for correlation functions from some class of functional bounds,II. the strongest form of the Thompson “mean field” bound and of the Krinsky and Emery bound,III. a generalization of a Krinsky inequality, what results in a good estimation of the pair correlation function in the Ising model.     

Correlations equalities and some upper bounds for the critical temperature for spin one systems

Starting from correlation identities for the Blume–Capel spin 1 systems and using correlation inequalities, we obtain rigorous upper bounds for the critical temperature. The obtained results improve over effective field type results.     

Universality in the partially anisotropic three-dimensional Ising lattice

Using transfer-matrix extended phenomenological renormalization-group methods, we study the critical properties of the spin-1/2 Ising model on a simple-cubic lattice with partly anisotropic coupling strengths \(\mathop J\limits^ \to = (J',J',J)\). The universality of both fundamental critical exponents y _t and y _h is confirmed. It is shown that the critical finite-size scaling amplitude ratios \(U = A_{\chi ^{(4)} } A_\kappa /A_\chi ^2 ,Y_1 = A_{\kappa ''} /A_\chi\), and \(Y_2 = A_{\kappa ^{(4)} } /A_{\chi ^{(4)} }\) are independent of the lattice anisotropy parameter Δ=J′/J. For the Y² invariant of the three-dimensional Ising universality class, we give the first quantitative estimate Y²≈2.013 (shape L×L×∞, periodic boundary conditions in both transverse directions).     

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.     

Ising correlations at the critical temperature

We demonstrate how the quadratic difference equations of Hirota's Toda lattice form, recently derived for the planar Ising model, provide a particularly easy way to obtain pair correlation functions at the critical temperature. The new results are also relevant for the dimer problem.     

Calculation of critical properties for the anisotropic two-layer Ising model on the Kagome lattice: Cellular automata approach

The critical point (K_c) of the two-layer Ising model on the Kagome lattice has been calculated with a high precision, using the probabilistic cellular automata with the Glauber algorithm. The critical point is calculated for different values of the inter- and intra-layer couplings (K₁≠K₂≠K₃≠K_z), where K₁, K₂ and K₃ are the nearest-neighbor interactions within each layer in the 1, 2 and 3 directions, respectively, and K_z is the intralayer coupling. A general ansatz equation for the critical point is given as a function of the inter- and intra-layer interactions, ξ=K₃/K₁,σ=K₂/K₁ and ω=K_z/K₁ for the one- and two-layer Ising models on the Kagome lattice.     

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.     

The anisotropic 3D Ising model

An expression for the free energy of an (001) oriented domain wall of the anisotropic 3D Ising model is derived. The order--disorder transition takes place when the domain wall free energy vanishes. In the anisotropic limit, where two of the three exchange energies (e.g. J_x and J_y ) are small compared to the third exchange energy (J_z ), the following asymptotically exact equation for the critical temperature is derived, sinh(2J_z /k _B T _c)sinh(2(J_x ?+?J_y )/k _B T _c))?=?1. This expression is in perfect agreement with a mathematically rigorous result (k _B T _c/J_z ?=?2[ln(J_z /(J_x ?+?J_y ))?ln(ln(J_z /(J_x ?+?J_y ))?+?O(1)]^?1) derived earlier by Weng, Griffiths and Fisher (Phys. Rev. 162, 475 (1967)) using an approach that relies on a refinement of the Peierls argument. The constant that was left undetermined in the Weng et al. result is estimated to vary from ～0.84 at ((H_x ?+?H_y )/H_z )?=?10^?2 to ～0.76 at ((H_x ?+?H_y )/H_z )?=?10^?20.     

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。     

Analytical properties of the anisotropic cubic Ising model

We combine an exact functional relation, the inversion relation, with conventional high-temperature expansions to explore the analytic properties of the anisotropic Ising model on both the square and simple cubic lattice. In particular, we investigate the nature of the singularities that occur in partially resummed expansions of the partition function and of the susceptibility.     

Interface delocalization in the three-dimensional Ising model

The interface delocalization in the three-dimensional Ising model is studied by real-space renormalization group methods. The first-order cumulant expansion approximation is used. Defect free energies for a boundary plane of defects and an internal plane of defects are calculated in the whole temperature region. The phase diagrams are also obtained. The method and the model analyzed may give a correct phase diagram only in the regime of continuous interface delocalization. The interface delocalization is obtained for the boundary defect and also for the internal defect if the systems on two sides of the internal defect plane have a different degree of order. The delocalization transition does not occur in the case of the internal defect plane between two equally ordered systems.