Showing papers on "Ising model published in 1978"

Infinite-ranged models of spin-glasses

Abstract: A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and $\mathrm{XY}$ spins are evaluated using a "many-replica" procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for $T$ greater than ${T}_{c}$, the spin-glass transition temperature. At and below ${T}_{c}$, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.

The Statistical Mechanics of Phase Transitions

Abstract: This article traces the development and study of phase transitions from late last century to the present day. We begin with a brief historical sketch and a description of the statistical mechanics of phase transitions. Particular attention is given to the modern era which began in 1944 with Onsager's celebrated solution of the two-dimensional Ising model. Points of development since Onsager, which are highlighted in this article, include the study of critical exponents, the scaling hypothesis, realisation of the universality of critical exponents and the recent renormalization group approach to critical phenomena. The basic idea of the renormalization group method, rather than the detailed application of the recipe to particular cases, is stressed and is discussed critically in some detail.

Order and disorder in gauge systems and magnets

Abstract: We show how phase transitions in Abelian two-dimensional spin and four-dimensional gauge systems can be understood in terms of condensation of topological objects. In the spin systems these objects are kinks and in the gauge systems either magnetic monopoles or fluxoids (quantized lines of magnetic flux). Four models are studied: two-dimensional Ising and $\mathrm{XY}$ models and four-dimensional ${Z}_{2}$ and U(1) gauge systems.

Phase transitions and reflection positivity. I. General theory and long range lattice models

Abstract: We systematize the study of reflection positivity in statistical mechanical models, and thereby two techniques in the theory of phase transitions: the method of infrared bounds and the chessboard method of estimating contour probabilities in Peierls arguments. We illustrate the ideas by applying them to models with long range interactions in one and two dimensions. Additional applications are discussed in a second paper.

Two-dimensional Ising Field Theory in a Magnetic Field: Breakup of the Cut in the Two Point Function

Abstract: We demonstrate that the cut which is present as the leading singularity in the two-point function of the Ising field theory for $Tl{T}_{c}$ and $H=0$ breaks up into a sequence of poles for $H\ensuremath{ e}0$. Both the positions and the residues of the low-lying poles are calculated.

Renormalization-group treatment of a Potts lattice gas for krypton adsorbed onto graphite

Abstract: Krypton atoms adsorbed in submonolayer quantities onto the basal graphite surface may be represented by a triangular lattice gas with nearest-neighbor exclusion and further-neighbor attraction decreasing with separation. We view this as a three-state Potts model with thermodynamic vacancies which are controlled by a chemical potential. A position-space renormalization-group treatment is performed by adapting Migdal's approximate recursion to the triangular lattice, and results are compared with experimental data. Our temperature versus density phase diagram for krypton submonolayers has an in-registry solid phase separated from a liquid phase by a line of continuous (Potts tricritical) transitions at higher temperatures. At lower temperatures, the solid phase is separated from a gas phase by first-order transitions. The Potts tricritical line meets the coexistence region of the first-order transitions at an isolated fourth-order transition point. This point may be related to the transition of the triplet Ising model, solved exactly by Baxter and Wu. Our "Potts lattice gas" global phase diagram is in a three-parameter space of pair-interaction constants and chemical potential. It contains solid, liquid, and gas phases, variously separated by first-order, Ising critical, three- and four-state Potts, and fourth-order transitions. The Lennard-Jones potential between krypton adatoms determines the planar subspace applicable to krypton submonolayers. Other planes, similarly determined, are applicable to adsorbed nitrogen, methane, and ethane, for which we estimate the temperatures of the fourth-order points. Our treatment also predicts a tricritical end-point topology, instead of the fourth-order point topology, when second-neighbor adatom pair attraction is not much stronger than third- and fourth-neighbor attractions.

Classification of continuous order-disorder transitions in adsorbed monolayers. II

Abstract: We extend the classification of continuous order-disorder transitions in adsorbed systems to encompass additional situations of physical interest. Among these are transitions on substrate arrays which are not simple Bravais lattices. The honeycomb and kagom\'e lattices are treated in detail. Their transitions belong to the universality class of the Ising, three- or four-state Potts model, and the Heisenberg model with cubic anisotropy. A simple case of the transitions of diatomic molecules is also considered. Those of ${\mathrm{Br}}_{2}$ on graphite are predicted to be first order. Lastly, transitions between ordered states are analyzed and an example of experimental interest is discussed.

Variational approximations for square lattice models in statistical mechanics

Abstract: This paper concerns a square lattice, Ising-type model with interactions between the four spins at the corners of each face. These may include nearest and next-nearest-neighbor interactions, and interactions with a magnetic field. Provided the Hamiltonian is symmetric with respect to both row reversal and column reversal, a rapidly convergent sequence of variational approximations is obtained, giving the free energy and other thermodynamic properties. For the usual Ising model, the lowest such approximations are those of Bethe and of Kramers and Wannier. The method provides a new definition of corner transfer matrices.

Essential Singularity in Percolation Problems and Asymptotic Behavior of Cluster Size Distribution

Abstract: It is rigorously proved that the analog of the free energy for the bond and site percolation problem on\(\mathbb{Z}^v \) in arbitrary dimensionΝ (Ν> 1) has a singularity at zero external field as soon as percolation appears, whereas it is analytic for small concentrations. For large concentrations at least, it remains, however, infinitely differentiable and Borel-summable. Results on the asymptotic behavior of the cluster size distribution and its moments, and on the average surface-to-size ratio, are also obtained. Analogous results hold for the cluster generating function of any equilibrium state of a lattice model, including, for example, the Ising model, but infinite-range andn-body interactions are also allowed.

Critical and thermodynamic properties of the randomly dilute Ising model

Abstract: The randomly bond-dilute two-dimensional nearest-neighbor Ising model on the square lattice is studied by renormalization-group methods based on the Migdal-Kadanoff approximate recursion relations. Calculations give both thermal and magnetic exponents associated with the percolative fixed point. Differential recursion relations yield a phase diagram which is in quantitative agreement with all known results. Curves for the specific heat, percolation probability, and magnetization are displayed. The critical region of the specific heat becomes unobservably narrow well above the percolation threshold ${p}_{c}$. This provides a possible explanation for the apparent specific-heat rounding in certain experiments.

On the theory of spinodal decomposition in solid and liquid binary mixtures

Abstract: The kinetics of phase separation is discussed with emphasis on the transition between spinodal decomposition and nucleation. A reanalysis of the theory of Langer, Baron and Miller shows that it exhibits a spinodal line somewhat closer to the coexistence curve than the meanfield spinodal. There the same (as we think unphysical) critical singularities occur as in Cahn-Hilliard theory. The precise location of this spinodal line depends on the cell size of the coarse graining. For concentrations less than the spinodal one the structure factorS(k, t) converges then towards the structure factor of the metastable onephase state, implying an infinite lifetime of the latter. In order to include the effects of nucleation and growth we hence present an alternative treatment, extending our previous work on cluster dynamics. From a simple approximation for the radial concentration distribution function of clustersS(k, t) is computed numerically. Even at rather low concentrations the time evolution ofS(k, t) is then similar to what Langer et al. find at high concentrations, implying a very gradual transition from nucleation and growth to spinodal decomposition, at least for parameter values appropriate to the Ising model. This treatment, which is consistent with Lifshitz-Slyozov's coarsening law at late times, is extended to the early stages of phase separation in liquid mixtures.

Spin-flop multicritical points in systems with random fields and in spin glasses

Abstract: Mean-field theory and renormalization-group arguments are used to study the phase diagram of an anisotropic $n$-component $d$-dimensional magnetic system with a uniaxially random magnetic field. The resulting phase diagram is shown to be very similar to that of anisotropic antiferromagnets in a uniform field: For small random fields, the system orders along the direction of uniaxial anisotropy, with exponents which are related to those of nonrandom Ising systems in $d\ensuremath{-}2$ dimensions. For larger random fields, parallel to the direction of uniaxial anisotropy, the transverse $n\ensuremath{-}1$ spin components order, with exponents which are unaffected by the random field. The two regions are separated by a spin-flop first-order line, by an intermediate "mixed" phase, and by a tetracritical (or bicritical) point. The exponents at this multicritical point are shown to coincide, near $d=6$, with those of the random-field Ising model. This phase diagram is shown to describe the behavior of random-site spin glasses in a uniform magnetic field. Other types of anisotropic random fields, related experimental realizations and other generalizations are also mentioned. Although some of the quantitative results are found only near $d=6$, qualitative results are believed to apply at $d=3$ as well.

Lattice Coulomb gas representations of two-dimensional problems

Abstract: Many of the standard two-dimensional problems of statistical physics can be transformed into 'Coulomb gas' problems in which there are two kinds of 'charges' represented by integers n and m. Such a transformation works for the Ising model, the three- and four-state Potts models, the Ashkin-Teller model, any many others. In general the n-n and m-m interactions have the Coulomb character in which the interaction is, for large separations, proportional to the logarithm of the distance. On the other hand, the n(r)-m(R) interaction is for large distances proportional to i times the angle Phi (r-R) which measures the angular position of R relative to r. This latter interaction is akin to that between a magnetic monopole and an electric charge.

399th solution of the Ising model

Abstract: The authors show that the nearest-neighbour correlations of the honeycomb, triangular, and square Ising models can be obtained by using only the star-triangle relations and simple assumptions concerning the thermodynamic limit and differentiability. This gives the internal energy, and hence the free energy and specific heat.

Theory of lower critical solution points in aqueous mixtures

Abstract: Two decorated lattice models of hydrogen bonded mixtures are presented that exhibit lower critical solution temperatures and closed‐loop coexistence curves and that account for the strongly asymmetric closed‐loop coexistence curves found in many aqueous mixtures. The models are extensions of an earlier decorated lattice model that produces only symmetric closed‐loop curves. They incorporate asymmetries in both directional and nondirectional energies as well as the possibility of multiple hydrogen bonding in water. The models are exactly soluble in terms of the spin‐1/2 Ising model, and exhibit nonclassical critical behavior at both upper and lower critical solution temperatures. The hydrogen bond energies in the models are in good agreement with those in real systems, and one of the models gives coexistence curves that are in reasonable quantitative agreement with most of the mixtures considered.

Essential Singularity in the Percolation Model

Abstract: It is rigorously proved that the analog of free energy for the percolation models has a singularity at zero external field as soon as percolation appears. The singularity is an essential one at least for large concentrations. Results on the asymptotic behavior of the cluster-size distribution are also obtained for percolation models and for the Ising model at low temperatures.

Critical temperature for two-dimensional ising ferromagnets with quenched bond disorder

Abstract: The configuration-averaged free energy of a quenched, random bond Ising model on a square lattice which contains an equal mixture of two types of ferromagnetic bonds J1 and J2 is shown to obey the same duality relation as the ordered rectangular model with the same two bond strengths. If the random.system has a single, sharp critical point, the critical temperature Tc must be identical to that of the ordered system, i.e., sinh(2J 1/kT c) sinh(2J 2/kT c) = 1. Since c (B) = 1/2, we can takeJ 2 → 0 and use Bergstresser-type inequalities to obtain(ρ/ρdp) exp(−2J 1/kTc¦p=pc + = 1, in agreement with Bergstresser's rigorous result for the diluted ferromagnet near the percolation threshold.

Band theory for order-disorder phase transformations: CuAu

Abstract: We present a band theory for the internal energy of binary alloys which undergo order-disorder transformations. The general method involves the self-consistent solution of a tight-binding Hartree Hamiltonian in the presence of both long- and short-range-order correlations through the use of an extended cluster-Bethe-lattice method. A complete scheme for the derivation of the alloy thermodynamics is obtained by coupling the electronic theory to the cluster-variation combinatorial theory for the alloy configurational entropy. The theory is applied to the study of the CuAu order-disorder transition and comparisons are made to existing data and Ising models.

The particle structure of v-dimensional Ising models at low temperatures

Abstract: In this work we study thev-dimensional Ising model at low temperatures and establish the existence of an upper gap in the energy-momentum spectrum of the two-point function forv≧3. Forv=2, it is known that this gap is absent.

Critical behavior of a 3‐D Ising model in a random field

Abstract: A Monte Carlo method has been used to study a simple cubic Ising ferromagnet in a random quenched magnetic field. The Hamiltonian for this model is H==JΣ(ij)σiσj−ΣiHiσi, where σi,σj=±1, J is the nearest‐neighbor interaction constant, and the field Hi=tH is fixed at each site with ti=±1 at random and Σti=0. L×L×L lattices with periodic boundary conditions have been studied for a range of H and T. As expected we find a ferromagnetic ordered state which for small H undergoes a second order phase change to the paramagnetic state with increasing temperature. A finite size scaling analysis of the preliminary data suggests that the critical exponent β is substantially smaller (β∼0.2) than the usual 3‐dim Ising value of 0.31. Results obtained for small lattices indicate that below kT/J‐2 the transition becomes first order suggesting that a tricritical point appears on the critcal field curve.

Criticality and crossover in the bond-diluted random Ising model

Abstract: By using the replica trick and a duality transformation, the bond-diluted random Ising model is mapped onto a new Hamiltonian. This demonstrates the higher order critical nature of the percolation point and identifies the appropriate crossover scaling variables. Taking the n to 0 limit of the replica method near the percolation point is shown to be equivalent to the q to 1 limit of the Potts model. The critical line near pc is calculated, yielding for a square lattice exp(-2Kc)=2ln2(p-1/2)+0(p-1/2)2.

Necessary and sufficient conditions for the ghs inequality with applications to analysis and probability

Abstract: AssrRAcr. The GHS inequality is an important tool in the study of the Ising model of ferromagnetism (a model in equilibrium statistical mechanics) and in Ellc-lidean quantum field theoer. This paper derives necessary and sufficient conditions on an Ising spin system for the GHS nnequality to be valid. Applications to convexity-preserving properties of certad differential equations and diffusion processes are given.