Showing papers on "Ising model published in 1975"

Solvable Model of a Spin-Glass

Abstract: We consider an Ising model in which the spins are coupled by infinite-ranged random interactions independently distributed with a Gaussian probability density. Both "spinglass" and ferromagnetic phases occur. The competition between the phases and the type of order present in each are studied.

Cooperative phenomena in systems far from thermal equilibrium and in nonphysical systems

Abstract: This article consists of two parts. The first part presents a tutorial approach to cooperative phenomena in systems far from thermal equilibrium and in nonphysical systems. Particular emphasis is placed on the question of how order and self-organization may arise. The following example is treated among others: the ordered phase of the laser giving rise to both coherently oscillating atomic dipole moments and coherent light emission. A complete analogy of the laser light distribution function to that of the Ginzburg-Landau theory of superconductivity is found mathematically which allows us to interpret the laser threshold as a quasi-second-order phase transition with soft modes, critical slowing down, etc. Similar phenomena, again closely resembling phase transitions, are found in tunnel diodes and in the nonlinear wave interaction which occurs, for example, in nonlinear optics. Remarkable analogies between the instability of the laser and those in hydro-dynamics are elaborated. While these phenomena show pronounced analogies to phase transitions in thermal equilibrium, there are further classes of instabilities and new phases which rather resemble hard excitations known in electrical engineering. Chemical oscillations are particularly important examples. In addition, the first part of this article contains the example of the cooperative behavior of neuron networks and shows the applicability of simple physical concepts, e.g., the Ising model, to the problem of the dynamics of social groups. All these above-mentioned examples demonstrate clearly that rather complex phenomena brought about by the cooperation of many subsystems can be understood and described by a few simple concepts. One of the main concepts is the order parameter; another is the adiabatic elimination of the variables of the subsystems, which is based upon a hierarchy of time constants present in most systems. The second part of this article gives a systematic account of the mathematical tools which allow us to deal with fluctuations. It contains the master equation, the Fokker-Planck equation, the generalized Fokker-Planck equation, and the Langevin equations, and gives several general methods for deriving the stationary and, in certain cases, the nonstationary solutions of master equations and the Fokker-Planck equations. Such general classes comprise those in which detailed balance is present or in which the coupling to the reservoirs is weak. In the quantum mechanical domain, the density matrix and the projection formalism for its reduction are presented. Finally, it is shown how the principle of quantum-classical correspondence allows us to translate quantum statistical problems completely into the classical domain.

New computational method in the theory of spinodal decomposition

Abstract: We present a new series of calculations in the theory of spinodal decomposition. The computational scheme is based on a simple ansatz for the two-point distribution function which leads to closure of the hierarchy of equations of motion for the high-order correlation functions. The resulting theory is accurate throughout the spinodal region of the phase diagram, including at the boundaries of this region where the spinodal mechanism is difficult to distinguish from nucleation and growth. The computational scheme is worked out in detail for parameters approximating those of the three-dimensional, kinetic, spin-exchange Ising model with nearest-neighbor interactions. Numerical agreement with recent Monte Carlo data appears to be satisfactory.

Generalized Ginzburg-Landau theory of pseudo-one-dimensional systems

Abstract: A generalized Ginzburg-Landau theory is suggested to describe the phase transition of an array of weakly coupled pseudo-one-dimensional chains. Using a mean-field approximation, the coupled-chain problem is reduced to that of a single chain in an effective field. The finite-range correlations which develop along the chain are treated using exact one-dimensional solutions. The results obtained are then used to construct a generalized Ginzburg-Landau theory. We argue that this approach provides a means of treating the remaining slowly varying long-range fluctuations. Results are given for a variety of arrays consisting of Ising, classical Heisenberg, real and complex ${\ensuremath{\psi}}^{4}$ chains.

Numerical evaluations of the critical properties of the two-dimensional Ising model

Abstract: Scaling transformations are used in numerical calculations of the properties of the two-dimensional Ising model near its critical point. When compared with the exact Onsager solution, the best approximation is seen to lead to 1 part in ${10}^{4}$ accuracy for the two largest scaling indices. This rather accurate calculation is obtained by utilizing a scaling transformation which depends upon a parameter. The parameter is set by demanding that two different evaluations of the magnetic eigenvalue agree with one another. One evaluation is found via the standard eigenvalue method; the other comes from a consistency condition for the spin-spin correlation function. This condition may also be used to distinguish scaling eigenvalues from other eigenvalues.

Interface sharpness in the Ising system

Abstract: A simple proof is given for the existence of a sharp interface in three-dimensional Ising systems, at least up to the critical temperature of the corresponding two-dimensional system.

Singularities in the Critical Surface and Universality for Ising-Like Spin Systems

Abstract: It is shown that singularities appear in the shape of the critical surface of Ising-like spin systems for special interactions such as occur in the symmetric eight-vertex model. The nature of the singularity and the connection with breakdown of universality are given.

Spin-3/2 Ising model for tricritical points in ternary fluid mixtures

Abstract: We present a lattice-gas model of ternary fluid mixtures. Within the mean-field approximation, we study a nonsymmetric tricritical point in this model. We compare our results to the experimental observations on the system ethanol-water-carbon-dioxide. In the course of our work, we have studied a Landau theory describing the neighborhood of a fourth-order critical point. Also, we have noted that mean-field theory indicates the existence of a fourth-order critical point in the spin-1 model of Blume, Emery, and Griffiths, corresponding to $Kl0$, $J+Kg0$.

Inequalities for Ising models and field theories which obey the Lee-Yang Theorem

Abstract: A series of inequalities for partition, correlation, and Ursell functions are derived as consequences of the Lee-Yang Theorem. In particular, then-point Schwinger functions ofeven φ4 models are bounded in terms of the 2-point function as strongly as is the case for Gaussian fields; this strengthens recent results of Glimm and Jaffe and shows that renormalizability of the 2-point function by fourth degree counter-terms implies existence of a φ4 field theory with a moment generating function which is entire of exponential order at most two. It is also noted that ifany (even) truncated Schwinger function vanishes identically, the resulting field theory is a generalized free field.

Order–disorder phenomena in adsorbed layers described by a lattice gas model

Abstract: Order–disorder transitions in adsorbed phases on single crystal surfaces manifest themselves by variations of the low energy electron diffraction (LEED) patterns. The present paper contains a theoretical treatment of the statistical properties of the simplest structure within this framework, namely a layer which may be described by a square lattice gas model with repulsive interactions between nearest neighbors and giving rise to a c2×2‐LEED pattern on the (100) surface of a fcc or bcc crystal. At a coverage ϑ=1/2 the relative intensities of the half‐order LEED spots are, within the kinematic approximation, shown to be identical to the expectation value of the spin‐correlation function of the two‐dimensional Ising model, averaged over an area corresponding to the coherence width of the electron beam. For ϑ<1/2 no analytic solutions are available, but the problem may be treated by means of the Monte Carlo technique, the results of which for ϑ=1/2 agree quite well with those from the analytic solution. The ...

Propagative spin relaxation in the Ising-like antiferromagnetic linear chain

Abstract: It is shown that at low temperatures the spin dynamics of an antiferromagnetic linear chain can be governed by propagation of boundaries between antiferromagnetic one-dimensional domains. A double maximum results in the neutron-scattering function S(q, ω) if the one-dimensional momentum transfer q is not too close to a one-dimensional superlattice point. In contrast with ordinary magnons of an anisotropic system, the dispersion law corresponding to the maximum exhibits no gap. If q = π, S(q, ω) is centered at ω = 0 but has a nonlorentzian shape. At moderately low temperatures the neutron linewidth at q = π amd the spin-lattice relaxation time are both proportional to the inverse correlation length K , whereas at very low temperature T, they are proportional to K T 1 2 . Much smaller values of T1 and of the neutron linewidth are expected if the spin dynamics is governed by thermally activated processes, as occurs if boundaries are trapped by lattice imperfection and cannot propagate. In the ferromagnetic Ising-like chain, spin dynamics is always governed by thermally activated processes.

’’Exactly soluble’’ two‐component lattice solution with upper and lower critical solution temperatures

Abstract: A decorated lattice model of a two−component liquid solution is presented which has closed−loop coexistence curves with both upper and lower critical solution temperatures analogous to the behavior found in the nicotine + water and m−toluidine + glycerol systems. The model can be transformed exactly into the spin−1/2 Ising model for which exact results are known in two dimensions and reliable estimates are available in three dimensions. The model exhibits nonclassical critical exponents at both upper and lower critical solution temperatures and has coexistence curves in qualitative agreement with those for real systems. The coexistence curves exhibit characteristic features found in most systems with closed−loop coexistence curves.

Quantum effects in the renormalization group approach to phase transitions

Abstract: The renormalization group approach to critical phenomena is developed for quantum mechanical problems with non-commuting operators. Applying the theory to the spin 1/2 Ising model with a transverse field Gamma it is found that the critical exponents are those of the Ising model with Gamma =0 if the transition occurs at T>0. However, for transitions at T=0 the critical behaviour of the d-dimensional transverse system corresponds to that of the (d+1)-dimensional Ising model with Gamma =0, in agreement with series expansion predictions. At T=0 the dynamic, as well as static, critical behaviour is given by mean field theory for d>3.

Violation of Dynamical Scaling for Randomly Dilute Ising Ferromagnets near Percolation Threshold

Abstract: Strongly anisotropic magnets at low temperatures with many (quenched) nonmagnetic impurities are shown to deviate from usual dynamical scaling assumptions near the critical concentration; instead of the time, the logarithm of the time is the basic variable. Analysis of previous Monte Carlo data confirms the static droplet picture used here, which fulfills for the first time the homogeneity, analyticity, and symmetry requirements of static scaling.

Combinatorial applications of an inequality from statistical mechanics

Abstract: The main part of this paper shows how an inequality of statistical mechanics has several applications in combinatorial theory.The inequality (known as the FKG inequality) was derived by Fortuin, Kasteleyn and Ginibre(4) in their work on the statistical mechanics of Ising ferromagnets. We first show how it leads to new properties of log supermodular functions, Bernstein polynomials, and log convex sequences.

Correlation inequalities for Ising spin lattices

Abstract: A change of spin representation is used to present expectation inequalities on Ising lattices directly as sums of terms of like sign. The technique is extended to correlation inequalities by introducing replica variables which convert correlations into expectations on a larger space. Second order correlations are analyzed in full from this viewpoint, recovering the FKG set, among others. Third order correlations are examined in some detail, and the sign of the multi-site Ursell correlationsF3,F4,F6 established under appropriate restrictions.

Crossover scaling, correlation function and connectivity in dilute low temperature ferromagnets

Abstract: For quenched dilute ferromagnets with a fractionp of spins (nearest neighbor exchange energyJ) and a fraction 1 —p of randomly distributed nonmagnetic atoms, a crossover assumption similar to tricritical scaling theory relates the critical exponents of zero temperature percolation theory to the low temperature critical amplitudes and exponents near the critical lineT c (p)>0. For example, the specific heat amplitude nearT c (p) is found to vanish, the susceptibility amplitude is found to diverge forT c (p →p c ) → 0. (Typically,p c =20%.) AtT=0 the spin-spin correlation function is argued from a droplet picture to obey scaling homogeneity but (at fixed distance) not to vary like the energy; instead it varies as const + (p c —p)2β +⋯ for fixed small distances. A generalization of the correlation function to finite temperatures nearT c (p) allows to estimate the number of effective percolation channels connecting two sites in the infinite (percolating) network forp>p c ; this in turn gives, via a dynamical scaling argument, a good approximation for theT=0 percolation exponent 1.6 in the conductivity of random three-dimensional resistor networks. This channel approximation also givesΦ=2 for the crossover exponent; i.e. exp(−2J/kT c (p)) is an analytic function ofp nearp=p c . An appendix shows that cluster-cluster correlations atT=0 (excluded volume effects) are responsible for the difference between percolation exponents and the (pure) Ising exponents atT c (p=1).

Equilibrium states of the two-dimensional Ising model in the two-phase region

Abstract: We prove that at zero external field and for any temperature below the critical temperature, all translationally invariant equilibrium states for the two-dimensional Ising ferromagnet, are a convex combination of only two extremal states.

Magnetization of the three-spin triangular Ising model

Abstract: Below its critical temperature, the triangular Ising model with pure three-spin interactions has a spontaneous magnetization M. The first 13 terms of a series expansion for M are known. Here these are re-arranged into a form which can be extrapolated to all terms in a natural way. Thus a conjecture is obtained for M. It fits the scaling prediction beta =1/12. Another order parameter is similarly discussed, together with the difficulties encountered in applying the technique to the two-spin Ising model susceptibility.

Staggered eight-vertex model

Abstract: An eight-vertex model with staggered (site-dependent) vertex weights is considered. The model is an extension of the usual one with translationally invariant weights and contains sixteen independent vertex weights. From its Ising representation it is seen that there are actually only eleven independent parameters. After discussing some general symmetry properties of this model, we consider in detail the soluble case of a free-fermion model. We find that the staggered free-fermion model may exhibit up to three phase transitions. Generally the specific heat has logarithmic singularities, expect in some special cases it has an exponent $\ensuremath{\alpha}=\frac{1}{2}$ and the system is frozen below a unique transition point. Conditions for these special cases are given.

Scaling theory for the pair-connectedness in percolation models

Abstract: Percolation theory is considered as the limit J>>kBT of a dilute Ising model. The thermodynamic and correlation scaling arguments for the Ising model are shown to give corresponding scaling laws for the cluster size distribution and the pair-connectivity respectively.

Self‐interacting, boson, quantum, field theory and the thermodynamic limit in d dimensions

Abstract: By use of a finite volume, lattice approximation, we set up an approximation to the analytic continuation of a polynomial, self‐interacting boson quantum field theory from Minkowski space to Euclidean space. The infinite volume limit for various boundary conditions is shown to exist and to be asymptotic to the perturbation expansion in the coupling constant g at g=0. For g:φ4:d theory we prove mass renormalizability and show how, by use of Nelson’s reconstruction theorem, the corresponding Minkowski space quantum field theory can be obtained. We discuss, at least for d?4, how statistical mechanical techniques, used to analyze the Ising model in the critical region just above the critical temperature, can be used to compute the properties of quantum field theory.

Cluster shapes in lattice gases and percolation

Abstract: An analysis is undertaken of the mean surface s of clusters of size n from Monte Carlo data simulating a two-dimensional Ising model. At sufficiently high temperatures the data represent a percolation process and it is found that the clusters are completely ramified (tree- or sponge-like). At temperatures just below Tc the data do not correspond to circular droplets as assumed in standard nucleation theory and typical samples show a great deal of ramification. It is concluded that the parameters in Fisher's droplet model are empirical and should not be given a direct physical interpretation.