Showing papers on "Ising model published in 1974"

The critical properties of the two-dimensional xy model

Abstract: The critical properties of the xy model with nearest-neighbour interactions on a two-dimensional square lattice are studied by a renormalization group technique. The mean magnetization is zero for all temperatures, and the transition is from a state of finite to one of infinite susceptibility. The correlation length is found to diverge faster than any power of the deviation from the critical temperature. Analogues of the strong scaling laws are derived and the critical exponents, eta , and delta , are the same as for the two-dimensional Ising model.

Effect of random defects on the critical behaviour of Ising models

Abstract: A cumulant expansion is used to calculate the transition temperature of Ising models with random-bond defects. For a concentration, x, of missing interactions in the simple-square Ising model the author finds -Tc-1 dTc/dx mod x=0=1.329 compared with the mean-field value of one. If the interactions are independent random variable with a width delta J/J identical to epsilon , the result is -Tc-1 dTc/d epsilon 2 mod epsilon =0=0.312 compared with the mean-field results of zero. An approximation yields the specific heat in the critical regime as C approximately C0/(1+x gamma 2C0), where gamma is a constant and C0 is the unperturbed specific heat at a renormalized temperature. Thus, the specific heat divergence is broadened over a temperature interval Delta T, with Delta T/Tc approximately x(1 alpha )/, where alpha is the critical exponent for the specific heat, and a maximum value of order x-1 is attained. Heuristic arguments show that this smoothing effect occurs if alpha >0.

On the ground state properties of the anisotropic triangular antiferromagnet

Abstract: Our aim is to present further evidence supporting a recent suggestion by Anderson (1973) that the ground state of the triangular antiferromagnet is different from the conventional three-sublattice Neel state. The anisotropic Heisenberg model is investigated. Near the Ising limit a peculiar, possibly liquid-like state is found to be energetically more favourable than the Neel-state. It seems to be probable that this type of ground state prevails in the anisotropy region between the Ising model and the isotropic Heisenberg model. The implications for the applicability of the resonating valence bond picture to the S = ½ antiferromagnets are also discussed.

Superposition approximation and natural iteration calculation in cluster‐variation method

Abstract: In the cluster‐variation method of cooperative phenomena and also in the quasichemical method, the bottleneck step has been to solve simultaneous equations. This paper proposes a new iterative procedure for solving the equations. This iteration does not use differentiation nor matrix inversion and may be called the natural iteration method. The free energy always decreases as the iteration proceeds, with a consequence that the iteration always converges to a stable solution (a local minimum of free energy) as long as the initial state is a physically acceptable one. The method derives in its introductory step a superposition approximation which writes the distribution variables of the basic cluster as a product of those of subclusters. The method is first explained with the pair approximation of the Ising ferromagnet, and then is applied to the fcc binary alloys to derive a phase diagram which is compared with the one reported recently by van Baal.

Ising model for tricritical points in ternary mixtures

Abstract: An Ising model with spin $S=1$ at each lattice point which exhibits multiple tricritical points is presented. This is a generalization of the Blume-Emery-Griffiths model which was used for describing the tricritical point in ${\mathrm{He}}^{3}$-${\mathrm{He}}^{4}$ mixtures. The model is solved in the molecular-field approximation. It is found that its thermodynamic behavior near the tricritical point is in qualitative agreement with the thermodynamic behavior of ternary fluids near their tricritical points.

Renormalization-Group Approach to the Critical Behavior of Random-Spin Models

Abstract: A renormalization-group technique is used to study the critical behavior of spin models in which each interaction has a small independent random width about its average value. The cluster approximation of Niemeyer and Van Leeuwen indicates that the two-dimensional Ising model has the same critical behavior as the homogeneous system. The e expansion for n-component continuous spins shows that this behavior holds to first order in e for n>4. For n<4, there is a new stable fixed point with 2ν=1+[3n/16(n−1)]e.

Investigation of metastable states and nucleation in the kinetic Ising model

Abstract: The relaxation of a two-dimensional Ising ferromagnet after a sudden reversal of the applied magnetic field is studied from various points of view, including nucleation theories, computer experiments, and a scaling theory, to provide a description for the metastable states and the kinetics of the magnetization reversal. Metastable states are characterized by a "flatness" property of the relaxation function. The Monte Carlo method is used to simulate the relaxation process for finite $L\ifmmode\times\else\texttimes\fi{}L$ square lattices ($L=55, 110, 220 \mathrm{and} 440, \mathrm{respectively}$); no dependence on $L$ is found for these systems in the range of magnetic fields calculated. The metastable states found for small enough fields terminate at a rather well-defined "coercive field," where no singular behavior of the susceptibility can be detected, within the accuracy of the numerical calculation. In order to explain these results an approximate theory of cluster dynamics is derived from the master equation, and Fisher's static-cluster model, generalizing the more conventional nucleation theories. It is shown that the properties of the metastable states (including their lifetimes) derived from this treatment are quite consistent with the numerical data, although the details of the dynamics of cluster distributions are somewhat different. This treatment contradicts the mean-field theory and other extrapolations, predicting the existence of a spinodal curve. In order to elucidate the possible analytic behavior of the coercive field we discuss a generalization of the scaling equation of state, which includes the metastable states in agreement with our data.

Ising Model on a Triangular Lattice with Three-spin Interactions. I. The Eigenvalue Equation

Abstract: It is shown that the Ising model with three-spin interactions on a triangular lattice is equivalent to a site-colouring problem on a hexagonal lattice. The transfer matrix method is then used to solve the colouring problem. The colouring of two neighbouri

Polycritical Points and Floplike Displacive Transitions in Perovskites

Abstract: The critical behavior occurring at displacive phase transitions in anisotropically stressed perovskite crystals is examined. It is shown that, under various conditions, the asymptotic critical behavior may be of Ising, $\mathrm{XY}$-model, or Heisenberg type. The existence of a "spin-flop"-like transition (at zero stress) between two differently ordered phases is predicted. An explanation of the discrepancy between the measured exponents and those predicted theoretically is proposed, and several new experiments are suggested.

Kinetic ising model study of phase separation in binary alloys

Abstract: The decomposition of binary alloys after a sudden quench below the critical temperature of unmixing is studied on a microscopic basis, using a stochastic model analog to the Kawasaki spin exchange Ising model. For interactions of long range the model reduces to the standard phenomenological equation. We relate this nonlinear equation to our recent treatment of the time-dependent Ginzburg-Landau theory, and calculate the nonequilibrium relaxation functions. For interactions of short range we present some computer simulations. As example, we treat a 55×55 square lattice, with a gradient in the chemical potential. Details are given for the relaxation process, which is a coarsening at the initial stage of the decomposition, while later the gradient produces a more macroscopic phase separation.

Phase Separation in the Two-Dimensional Ising Ferromagnet

Abstract: The interface density profile of the two-dimensional Ising ferromagnet is investigated for all temperatures below the critical point. The width of the interface diverges in the thermodynamic limit.

Griffiths singularities in the randomly dilute one-dimensional Ising model

Abstract: Exact solutions for the thermodynamic functions of the randomly dilute $s=\frac{1}{2}$ nearest-neighbor Ising chain in a magnetic field are examined. Both site and bond impurities are treated. Behavior is nonanalytic at $T=h=0$. The divergences of the pure-chain thermodynamics are replaced at nonzero dilution by essential singularities of the Griffiths type at which all functions are finite and infinitely differentiable. The simplicity of the solution allows the origin and form of the Griffiths singularities to be traced in detail.

Monte Carlo calculation of phase separation in a two-dimensional Ising system

Abstract: Monte Carlo calculations were carried out for a two-dimensional Ising model of a binary alloy with nearest-neighbor attractive interactions between like atoms. The pair correlation observed had the form of an exponentially damped cosine function with parameters varying as the one-sixth power of the time.

Zeroes of the partition function for the Ising model in the complex temperature plane

Abstract: For special boundary conditions, the zeroes of the partition function of the square Ising model are shown to lie on Fisher's two circles in the complex exp(− 2βJ) plane. For some more general boundary conditions, the zeroes distribute asymptotically on these circles.

Upper bounds for the transition temperatures of generalized Ising models

Abstract: Fisher's rigorous bound chi < chi saw, where chi saw is the susceptibility calculated from the self-avoiding walk (SAW) generating function, is applied to (i) the ferromagnetic Ising alloy, (ii) the dilute Ising ferromagnet, (iii) the layer susceptibility of a free surface, (iv) the Ising ferromagnet in a transverse field, and (v) the dilute Ising ferromagnet in a transverse field. The author obtains an upper bound TUB for the temperature T0 below which a spontaneous magnetization occurs. The use of a microscopically varying molecule field is analysed. A heuristic argument is given to show that, for the dilute Ising ferromagnet in a transverse field, the critical transverse field at zero temperature jumps from zero to a value at least equal to that of the one-dimensional ground state as x passes through the critical percolation concentration.

On the exponential decay of correlation functions

Abstract: We use the properties of subharmonic functions to prove the following results, First, for any lattice system with finite-range forces there is a gap in the spectrum of the transfer matrix, which persists in the thermodynamic limit, if the fugacityz lies in a regionE of the complex plane that contains the origin and is free of zeros of the grand partition function (with periodic boundary conditions) as the thermodynamic limit is approached. Secondly, if the transfer matrix is symmetric (for example, with nearest and next-nearest neighbor interactions in two dimensions), and if infinite-volume Ursell functions exist that are independent of the order in which the various sides of the periodicity box tend to infinity, then these Ursell functions decay exponentially with distance for all positivez inE. (For the Ising ferromagnet with two-body interactions, exponential decay holds forz inE even if the range of interaction is not restricted to one lattice spacing). Thirdly, if the interaction potential decays moreslowly than any decaying exponential, then so do all the infinite-volume Ursell functions, for almost all sufficiently small fugacities in the case of general lattice systems, and for all real magnetic fields in the case of Ising ferromagnets.

A study of metastability in the Ising model

Abstract: We consider a two-dimensional Ising ferromagnet with (+) boundary conditions and negative external field, where a Markovian time evolution is assumed.

The pseudo-spin model and its application to KDP-type materials

Abstract: The applicability of the pseudo-spin model is discussed in relation to the existence of constituents with a restricted number of low-lying energy states. The situation certainly occurs in materials with low-lying electronic levels which undergo co-operative Jahn-Teller phase transitions. The Slater-Takagi model of KDP is analysed to show that under certain approximations it leads to a double well in the collective co-ordinates with two levels in each unit cell and hence to a pseudo-spin model of one S = 1/2 per cell. The spin hamiltonian is the Ising model in a transverse field together with four spin interactions. The thermodynamic properties of this model are discussed and best values of the parameters obtained. The elastic constant anomalies are interpreted in terms of cs', in the adiabatic regime. The spin dynamics is discussed in terms of random phase approximation for spin waves, and more complete treatments are reviewed. Raman scattering data is analysed in terms of a coupled spin-phonon m...

Ising Model on a Triangular Lattice with Three-spin Interactions. I. Free Energy and Correlation Length

Abstract: Following the demonstration in Part I that the Ising mogel with three-spin interactions on a triangular lattice is equivalent to a colouring problem on a hexagonal lattice, and that a generalized Bethe ansatz can be used to obtain equations for the eigenv

Monte Carlo computer experiments on critical phenomena and metastable states

Abstract: Ising and Heisenberg models are studied by the Monte Carlo method. Several hundred up to 60 000 spins located at two- and three-dimensional lattices are treated and various boundary conditions used to elucidate various aspects of phase transitions. Using free boundaries the finite size scaling theory is tested and surface properties are derived, while the periodic boundary condition or the effective field-like ‘self-consistent’ boundary condition are used to derive bulk critical properties. Since Monte Carlo averages can be interpreted as time averages of a stochastic model, ‘critical slowing down of convergence’ occurs. The critical dynamics is investigated in the case of the single spin-flip kinetic Ising model. Also non-equilibrium relaxation processes are treated, e.g. switching on small negative fields the magnetization reversal and nucleation processes are studied. The metastable states found can be understood in terms of a scaling theory and the droplet model. Using a spin exchange model t...

Eight‐vertex model on the honeycomb lattice

Abstract: The most general vertex model defined on a honeycomb lattice is the eight‐vertex model. In this paper it is shown that the symmetric eight‐vertex model reduces to an Ising model with a nonzero real or pure imaginary magnetic field H. The equivalent Ising model is either ferromagnetic with e2H/kT real or antiferromagnetic with e2H/kT unimodular. The exact transition temperature and the order of phase transition in the former case are determined. As an application of the result we verify the absence of a phase transition in the monomer‐dimer system on the honeycomb lattice.

High-temperature series for the susceptibility of the spin-S Ising model

Abstract: We have extended the series for the zero-field susceptibility of the spin-$S$ Ising model to eighth order in the reduced temperature $K$, on the triangular, simple cubic, body-centered-cubic, and face-centered-cubic lattices. The coefficients of these series ${h}_{n}(S)$ are expressed as simple polynomials in $X=S(S+1)$. For the face-centered-cubic lattice, an accurate polynomial fit to the critical point ${K}_{c}(S)$ is presented; and the apparent spin dependence of the critical exponent $\ensuremath{\gamma}$ is briefly discussed. The series are quite well behaved for all $S$. However, the large-$S$ series seems to exhibit more rapid apparent convergence.