Experimental data for critical exponents in some magnetic materials are compared with recent theoretical results on the three-dimensional (3D) Ising model, as derived by one of us (ZDZ) based on two conjectures [Z.D. Zhang, Conjectures on the exact solution of three-dimensional (3D) simple orthorhombic Ising lattices, Phil. Mag. 87 (2007), pp. 5309–5419]. It is found that critical exponents in some bulk magnetic materials indeed form a 3D Ising universality. Our attention is then focused on the critical indices at fluid–fluid phase transition. We suggest to use Zhang's exponent β = 3/8 to fit the experimental data over the wider asymptotic region near the critical point of a fluid–fluid phase transition. The 3D Ising universality should exist for critical indices in a certain class of magnets and at fluid–fluid phase transition.
Experimental data for critical exponents in some magnetic materials are compared with recent theoretical results on the three-dimensional (3D) Ising model, as derived by one of us (ZDZ) based on two conjectures [Z.D. Zhang, Conjectures on the exact solution of three-dimensional (3D) simple orthorhombic Ising lattices, Phil. Mag. 87 (2007), pp. 5309–5419]. It is found that critical exponents in some bulk magnetic materials indeed form a 3D Ising universality. Our attention is then focused on the critical indices at fluid–fluid phase transition. We suggest to use Zhang's exponent β = 3/8 to fit the experimental data over the wider asymptotic region near the critical point of a fluid–fluid phase transition. The 3D Ising universality should exist for critical indices in a certain class of magnets and at fluid–fluid phase transition.
With the aid of exact seriesexpansions the shape of the critical isotherm of an Ising ferromagnet (or fluid) is re-analysed and it is conjectured that δ = 5 for all three-dimensional lattices. (The analysis strengthens previous conjectures that δ = 15 for all two-dimensional lattices.) Estimates are given for the amplitude of the singularity for the various lattices. Evidence is presented to support the conjecture that α' = α = 1/8, β = 5/16 and γ' = γ = 1¼ for all three-dimensional lattices.
Usually, the impact of structural disorder on the magnetic phase transition in the 3D Ising model is analyzed within the framework of quenched dilution by a nonmagnetic component, where some lattice sites are occupied by Ising spins, while others are nonmagnetic. This kind of quenched dilution, according to the Harris criterion, leads to a change in the critical exponents that govern the asymptotics in the vicinity of the phase transition point. However, the inherent reason for the emergence of a new, random Ising model universality class is not the presence of a nonmagnetic component, but the disorder in structure of spin arrangement. To demonstrate this fact, in this paper we set up extensive Monte Carlo simulations of a random mixture of two Ising-like magnets that differ in spin length s and concentration c. In doing so, we analyze the effect of structural disorder per se without appealing to the presence of a nonmagnetic component. We support our numerical simulations with renormalization group calculations. Our results demonstrate the emergence of the 3D randomly diluted Ising model universality class in a random mixture of two Ising magnets. While the asymptotic critical exponents coincide with those known for the site-diluted 3D Ising model, the effective critical behavior is triggered by parameters s and c. The impact of their interplay is a subject of detailed analysis.
Liquid-gas phase transitions are usually assumed to belong to the same universality class as Ising magnets. However, in the liquid-gas case, the free-energy expansion contains odd powers of the order parameter. We show in this work that, even though the odd terms are irrelevant near four dimensions, they become relevant at dimensionality ${d}_{c}=\frac{(10\ensuremath{-}5\ensuremath{\eta})}{3}\ensuremath{\approx}3.33$. A new, stable, fixed point may appear. We approximately calculate, to order ${\ensuremath{\epsilon}}^{\ensuremath{'}}={d}_{c}\ensuremath{-}d$, the difference between the critical exponents for this new fixed point and the Ising critical exponents. This difference turns out to be numerically small. We conclude that, even though in three dimensions the liquid-gas and Ising-magnet phase transition do not belong to the same universality class, the effect on the critical exponents is probably too small to be observed.
We report conjectures on the three-dimensional (3D) Ising model of simple orthorhombic lattices, together with details of calculations for a putative exact solution. Two conjectures, an additional rotation in the fourth curled-up dimension and weight factors on the eigenvectors, are proposed to serve as a boundary condition to deal with the topologic problem of the 3D Ising model. The partition function of the 3D simple orthorhombic Ising model is evaluated by spinor analysis, employing these conjectures. Based on the validity of the conjectures, the critical temperature of the simple orthorhombic Ising lattices could be determined by the relation of KK* = KK′ + KK″ + K′K″ or sinh 2K · sinh 2(K′ + K″ + (K′K″/K)) = 1. For a simple cubic Ising lattice, the critical point is putatively determined to locate exactly at the golden ratio , as derived from K* = 3K or sinh 2K · sinh 6K = 1. If the conjectures would be true, the specific heat of the simple orthorhombic Ising system would show a logarithmic singularity at the critical point of the phase transition. The spontaneous magnetization of the simple orthorhombic Ising ferromagnet is derived explicitly by the perturbation procedure, following the conjectures. The spin correlation functions are discussed on the terms of the Pfaffians, by defining the effective skew-symmetric matrix Aeff. The true range κx of the correlation and the susceptibility of the simple orthorhombic Ising system are determined by procedures similar to those used for the two-dimensional Ising system. The putative critical exponents derived explicitly for the simple orthorhombic Ising lattices are α = 0, β = 3/8, γ = 5/4, δ = 13/3, η = 1/8 and ν = 2/3, showing the universality behaviour and satisfying the scaling laws. The cooperative phenomena near the critical point are studied and the results based on the conjectures are compared with those of approximation methods and experimental findings. The putative solutions have been judged by several criteria. The deviations of the approximation results and the experimental data from the solutions are interpreted. Based on the solution, it is found that the 3D-to-2D crossover phenomenon differs with the 2D-to-1D crossover phenomenon and there is a gradual crossover of the exponents from the 3D to the 2D values. Special attention is also paid to the extra energy caused by the introduction of the fourth curled-up dimension and the states at/near infinite temperature revealed by the weight factors of the eigenvectors. The physics beyond the conjectures and the existence of the extra dimension are discussed. The present work is not only significant for statistical and condensed matter physics, but also fill the gap between the quantum field theory, cosmology theory, high-energy particle physics, graph theory and computer science.
We study the ±J three-dimensional (3D) Ising model with a spatially uniaxial anisotropic bond randomness on the simple cubic lattice. The ±J random exchange is applied on the xy planes, whereas, in the z direction, only a ferromagnetic exchange is used. After sketching the phase diagram and comparing it with the corresponding isotropic case, the system is studied at the ferromagnetic-paramagnetic transition line using parallel tempering and a convenient concentration of antiferromagnetic bonds (p(z)=0;p(xy)=0.176). The numerical data clearly point out a second-order ferromagnetic-paramagnetic phase transition belonging in the same universality class with the 3D random Ising model. The smooth finite-size behavior of the effective exponents, describing the peaks of the logarithmic derivatives of the order parameter, provides an accurate estimate of the critical exponent 1/ν=1.463(3), and a collapse analysis of magnetization data gives an estimate of β/ν=0.516(7). These results are in agreement with previous papers and, in particular, with those of the isotropic ±J three-dimensional Ising model at the ferromagnetic-paramagnetic transition line, indicating the irrelevance of the introduced anisotropy.
Four combinations of critical exponents pertaining to the Ising model are constructed which yield simply the dimensionality d. One of these, connecting γ and η, is shown to lead to an immediately useful estimates of the sum α + η in three dimensions, about which exponents controversy still exists. This sum in three dimensions is shown to be very near or equal to 1/8, a condition which Zhang’s predictions, for the exact solution of the 3D Ising model, satisfy with α = 0 and η = 1/8. But the pioneering work of Wilson based on renormalisation group theory gave the estimate η = 0.037, while the Perk advocates α = 0.1, without however, proceeding a proof. We therefore finally stress the current importance of further experimental measurements. The critical exponent δ may be the most favourable focus, to distinguish between Zhang’s prediction of 13/3 and a value around 4.8 following for d = 3 from Wilson’s value of η quoted above.