Showing papers on "Ising model published in 1968"

Dynamics of the Ising Model near the Critical Point. I

Abstract: A simple dynamical model of interacting Ising spins is discussed. Each spin is assumed to flip spontaneously with a transition probability which depends on the temperature and the configuration of surrounding spins, but its functional form is assumed to be the simplest. The frequency-wave number dependent susceptibility χ( q , ω) is given exactly in the one-dimensional case. In two-and three-dimensional cases the model is treated in the molecular field and the generalized approximations.

Theory of a Two-Dimensional Ising Model with Random Impurities. I. Thermodynamics

Abstract: Recent experiments demonstrate that at the Curie temperature the specific heat may be a smooth function of the temperature. We propose that this effect can be due to random impurities and substantiate our proposal by a study of an Ising model containing such impurities. We modify the usual rectangular lattice by allowing each row of vertical bonds to vary randomly from row to row with a prescribed probability function. In the case that this probability is a particular distribution with a narrow width, we find that the logarithmic singularity of Onsager's lattice is smoothed out into a function which at ${T}_{c}$ is infinitely differentiable but not analytic. This function is expressible in terms of an integral involving Bessel functions and is computed numerically.

General Griffiths' Inequalities on Correlations in Ising Ferromagnets

Abstract: Let N = (1, 2, ⋯, n). For each subset A of N, let JA ≥ 0. For eachi∈N, let σi ± 1. For each subset A of N, define σA=∏i∈A σi. Let the Hamiltonian be − ΣACN JA σA. Then for each A, B⊂N, 〈σA〉≥0 and 〈σAσB〉−〈σA〉〈σB〉≥0. This weakens the hypothesis and widens the conclusion of a result due to Griffiths.

Analytic and clustering properties of thermodynamic functions and distribution functions for classical lattice and continuum systems

Abstract: Our most complete results concern the Ising spin system with purely ferromagnetic interactions in a magnetic fieldH (or the corresponding lattice gas model with fugacityz=const. exp(−2mHβ) wherem is the magnetic moment of each spin). We show that, in the limit of an infinite lattice, (i) the free energy per site and the distribution functionsn s (x 1, ...,x s ; β,z) are analytic in the two variables β andH if the reciprocal temperature β>0 and the complex numberH is not a limit point of zeros of the grand partition function ξ, and (ii) the Ursell functionsu s (x 1, ...,x s ; β,z) tend to 0 as Δ s ≡Max i, j |x i −x j | → ∞ if β>0 and ReH≠0; in particular, if the interaction potential vanishes for separations exceeding some fixed cutoff value λ, then |u s |0 and the complex fugacityz is less than the radius of convergence of the Mayerz expansion; for the continuum gas, however,n s andu s must be replaced by their values integrated over small volumes surrounding each of the pointsx 2, ...,x s . It is shown that the pressurep is analytic in both β andz, if it is analytic inz at fixed β over a suitable range of values of β andz, and further that, except for continuum systems without hard cores,p,n s andu s have convergent Maclaurin expansions in β for small enoughz.

General theoretical approach to the thermodynamic and kinetic properties of cooperative intramolecular transformations of linear biopolymers.

Abstract: A general method to calculate experimentally accessible thermodynamic and kinetic quantities for any type of cooperative transitions is developed. Special attention has been directed to transition curves and mean relaxation times. The procedure is applied to the most general case of nearest-neighbor cooperativity by using the linear Ising model and the matrix method of evaluation. The various potential types of end effects arid the resulting chain length dependences are discussed in detail. The significance of the theory with respect to the helix-coil transition of polypeptides as well as to the polyproline I-II transition is indicated.

Random Impurities as the Cause of Smooth Specific Heats Near the Critical Temperature

Abstract: We present a modification of the two-dimensional Ising model which incorporates random impurities. The specific heat of this model is infinitely differentiable even at the critical temperature where it possesses an essential singularity. We find this specific heat to be in perfect quantitative agreement with the smooth peak recently observed by van der Hoeven, Teaney, and Moruzzi for $T\ensuremath{\gtrsim}{T}_{c}$ in the specific heat of EuS.

Random Spin Systems: Some Rigorous Results

Abstract: Several general results are obtained for a system of spins on a lattice in which the various lattice sites are occupied at random, and the spins, if present, interact via a general Heisenberg or Ising interaction decreasing sufficiently rapidly with distance. It is shown that the free energy per site exists in the limit of an infinite system, is a continuous function of concentration, and has the usual convexity (stability) properties. For Ising systems with interactions of finite range, the free energy is an analytic function of concentration and magnetic field for a suitable range of these variables. The random Ising ferromagnet on a square lattice (or simple cubic lattice) with nearest‐neighbor interactions is shown to exhibit a spontaneous magnetization at sufficiently high concentrations and low temperatures.

Weak‐Graph Method for Obtaining Formal Series Expansions for Lattice Statistical Problems

Abstract: A unified exposition of the weak‐graph method for obtaining formal series expansions for lattice statistical problems is presented. The prototype of this method is the derivation of the hyperbolictangent high‐temperature expansion for the spin‐½ Ising model. Also, recent expansions of the monomer‐dimer problem and various hydrogen‐bonded problems have been treated by essentially the same method. In this paper the method is further illustrated by obtaining series expansions for the low‐temperature spin‐½ Ising problem, the low‐density hard‐core lattice‐gas problem, the high‐temperature spin‐1 Ising problem, the k‐color problem, and two new model problems, the ramrod model and a special ternary model. The weak‐graph method enables one to obtain especially useful series expansions for a certain class of problems, including the spin‐½ Ising problem and the monomer‐dimer problem, which have essentially a binary nature.

One-Dimensional Ising Model with General Spin

Abstract: The one-dimensional Ising model is investigated by generalizing the Bethe approximation, which, in this case, gives exact solutions. The energy, specific heat and the zero field susceptibility for S =1, 3/2 and 2 except the susceptibility for S =2 are calculated exactly and compared with the results of Suzuki et al. . As a direct application of this method, the special lattice called the Bethe lattice is treated, and the Curie temperature of the lattice for S =1 is obtained. This method is also applied to solve the one-dimensional Ising model with the second neighbor interactions as well as the first neighbor ones, and the energy and specific heat for S =1/2 are calculated.

Critical behaviour of the Ising model above and below the critical temperature

Abstract: Previous series expansion studies of the magnetization and susceptibility of the Ising model are extended to the higher derivatives of the free energy with respect to the magnetic field. It is found that just above the critical temperature the 2nth derivative is given by where in two dimensions ? = 1?, ? = 1fraction seven-eighths and in three dimensions ? = 1?, ? = 1fraction nine-sixteens. In two dimensions ? is exact and ? is correct to 0.6%; the three-dimensional indices are correct to 0.2%. Just below the critical temperature where in three dimensions 0.303

Statistical Thermodynamics of Finite Ising Model. II

Abstract: The partition functions of the two- and three-dimensional finite Ising models in the presence of a magnetic field have been calculated with use of a high-speed computer. Periodic boundary conditions were imposed. All the numbers of configurations for the 3×3×3, 5×5, and 4×6 lattices are tabulated. The distribution of zeros of the partition functions in the complex magnetic-field plane has been obtained both for the ferro- and antiferromagnetic cases. The zeros in the ferromagnetic cases are distributed on the unit circle according to the Yang-Lee theorem, and the dependence on temperature of the distribution functions has been studied. In the antiferromagnetic case most of zeros are on the negative real axis, but a certain number of complex roots appear in the left half of the plane. The magnetization and the magnetic susceptibility have been calculated as functions of temperature.

Infinite‐Spin Ising Model in One Dimension

Abstract: The partition function z, the pair correlation function ρ, and the zero‐field susceptibility χ for the one‐dimensional Ising model with infinite spin, are expressed in terms of the eigenvalues and eigenfunctions of an integral equation. The eigenfunctions of the integral equation are shown to be the oblate spheroidal wavefunctions, and, from known asymptotic expansions, high‐ and low‐temperature expansions are given for z, ρ, and χ. It is shown that the low‐temperature behavior of z, ρ, and χ differs qualitatively from the corresponding behavior for all finite spin.

Theorems on the Ising Model with General Spin and Phase Transition

Abstract: The theorem of Lee and Yang has been extended to the ferromagnetic Ising model with arbitrarily mixed spin values of Sj = ½, 1, and 32, including the case of equal spin values as a special one. Namely, it has been proved that the zeros of the partition function for the above Ising model with higher spin values lie on the unit circle in the fugacity plane (or complex magnetic‐field plane). Expressions for general correlation functions in Ising ferromagnets with higher spin values have been derived in terms of the above generalized theorem. By the use of these expressions, the relations among the critical indices are discussed and the same results are obtained as those predicted by the scaling‐law approach.

The specific heat of the three-dimensional Ising model below Tc

Abstract: A representation for the specific heat below Tc of a three-dimensional Ising ferromagnet (or fluid) is constructed. On expansion, it yields the first twelve terms of the exact low-temperature series for the tetrahedral lattice, and behaves asymptotically as where ?prime = fraction one-eighth, A- = 0?20 and B- = 0?13. However, such behaviour is only becoming evident at t identical with 1 - T/Tc similar, equals 10-4. Indeed, from t similar, equals 4 ? 10-2 to 4 ? 10-4, CH appears to be increasing roughly logarithmically. The specific heat of the Ising model is compared directly with experimental results for four `classical' fluids and three Ising-like antiferromagnets. Difficulties in establishing experimental values for the critical exponent ?prime are discussed. A simple explanation is proposed of why the `rounding-off', observed in most magnetic transitions, is often more pronounced below Tc.

Magnetic Anisotropies in Antiferromagnetic Rare Earth Antimonide Single Crystals

Abstract: The magnetic properties of the rare earth antimonides are anisotropic below ordering temperatures. In the compounds of Ce, Nd, Dy, and Ho the crystalline field confines the spin to the [100] axis. The observed uniaxial metamagnetism is explained in terms of the Ising model. The induced magnetic moments depend strongly on the direction of the applied field. This anisotropy is less pronounced in the cases of TbSb and ErSb, in good agreement with theoretical considerations. Metastable spin structures are observed in DySb, HoSb, and CeSb.

The Effect of Lattice Vibration on the Phase Transition of the Ising Model

Abstract: The effect of lattice vibration on the second-order phase transition for the Ising model is investigated. The total Hamiltonian of the system is separated into the phonon part and the spin part, if the displacement of the atoms from their equilibrium position is assumed to be small. The spin part of the free energy is then calculated (i) in the molecular-field approximation for the square and the simple cubic lattices, and (ii) by the method due to Kadanoff, a modification of the Onsager's exact solution, for the square lattice. The results show that (i) the phase transition remains to be of the second order, (ii) the critical temperature increases, and (iii) the magnetic part of the heat capacity decreases, both for ferro- and antiferromagnetic Ising models.

Critical Properties of Isotropically Interacting Classical Spins Constrained to a Plane

Abstract: High-temperature expansions of the susceptibility and internal energy (specific heat) are presented for general lattice structure for a system of isotropically interacting unit vectors (or "classical spins") which are constrained to lie in a plane. A phase transition (${T}_{c}g0$) is indicated for two-dimensional lattices; the expected result ${T}_{c}=0$ is found in one dimension, but only upon choosing a more suitable expansion parameter than $\frac{J}{\mathrm{kT}}$. Similarities with the corresponding expansions of the $S=\frac{1}{2}$ Ising and classical Heisenberg models are pointed out; in particular, it is found that certain critical properties of this planar model appear to be bounded on one side by the Ising model and on the other side by the Heisenberg model.

Impurities and defects in Ising lattices

Abstract: The changes in the properties of an Ising lattice caused by various types of defect are related to the spin correlations of the perfect lattice. The critical indices associated with these changes obey the scaling laws with the same gap index Δ as the bulk properties and a susceptibility index which exceeds the bulk index by unity. For planar lattices in zero field an alternative approach leads to a confirmation of the isotropy of the boundary tension near Tc and, incidentally, to an unlimited number of relations between the spin correlations of the normal lattice and its dual.

Theorems on Extended Ising Model with Applications to Dilute Ferromagnetism

Abstract: It is proved that the zeros of the partition function of an extended Ising model lie on a unit circle in the fugacity piane under certain conditions. Each spin assumes a general value. The key point of the proof is to derive a simple condition sufficient for zeros of polynomials to lie on a unit circle. Conjectured theorems on the Heisenberg model are also discussed.

Combinatorial Theorem for Graphs on a Lattice

Abstract: Several problems in lattice statistical mechanics, such as the spin‐½ Ising problem and the monomer‐dimer problem, can be formulated in terms of the p‐generating function for the weak subgraphs of a regular lattice. This paper presents an algebraic transformation theorem which allows the p‐generating function for the weak subgraphs of a lattice to be determined from the p‐generating function for the far less numerous subset consisting of the closed weak subgraphs. This result will be especially useful in reducing the labor required to obtain exact finite series for various problems. The theorem also enables one to give a straightforward proof of the Ising susceptibility graph theorem due to Sykes.

Phase Transition and Surface Tension in the Quasichemical Approximation

Abstract: The statistical formulation of phase transition is derived for a three‐dimensional Ising model using Guggenheim's approximation. As the temperature approaches the critical temperature, the difference equations governing the transition can be replaced by differential equations, easily integrated. Comparisons are made with the Cahn‐Hilliard continuum model, and criticisms of that model by Widom are also examined.

On the Ising Model with Long‐Range Interaction

Abstract: We have obtained an expansion of the free energy per spin of an Ising model with long‐range interaction in the absence of an external field, for temperatures above the Curie temperature of the Weiss‐Bragg‐Williams approximation (BWCP). We use as expansion parameter the reciprocal (γ) of an effective number of neighbors. Terms through order γ2 are obtained by extracting factors from a representation of the partition function as an average over random fields. For terms of higher order, we give a diagrammatic series in which all terms through order γn are contained in the diagrams with not more than 2(n − 1) bonds. The terms of order γ3 are given explicitly. For temperatures below the BWCP we have calculated terms through order γ. Since after a few finite terms the coefficients in this γ expansion become infinite at the BWCP, we exhibit a modification of the random field representation which avoids this difficulty. We have compared our results with those of previous authors wherever available—that is, throug...

Generalized Lee-Yang's Theorem

Abstract: Lee-Yang's theorem which states that all the roots of the partition function of the ferromagnetic Ising model of spin 1/2 lie on the unit circle in the complex `fugacity' plane is generalized to the case of S =1 and 3/2, where S is the magnitude of the spin.

The frequency-dependent initial perpendicular susceptibility of the Ising model

Abstract: The Kubo linear response theory is applied to the quantum mechanical Ising model to obtain a general expression which is valid for all regular lattices for the perpendicular response function in terms of spin correlations. For the honeycomb and the square lattices, exact expressions are found for the frequency-dependent initial perpendicular susceptibility, valid for all temperatures. For the honeycomb lattice, the present result for zero frequency agrees with Fisher's static calculation, whereas there is disagreement in the case of the square lattice. The discrepancy is resolved by using an extended calculation which explicitly illustrates the difference between isothermal and adiabatic susceptibilities.