Standard Ising Model Research Articles

Dynamics of phase separation in alloys with internal elastic energy: a Monte Carlo approach

A two-dimensional Monte Carlo approach of phase separation in binary alloys with internal elastic energy has been developed. Based on the standard Ising model, this approach considers both the binding potential and the internal elastic interaction between two species, which are assumed to decay respectively with the distance via power functions. It has been found that the phase separation exhibits quite different dynamics from the current mode which only takes into account the binding potential interaction. This dynamics is characterized by strong lattice anisotropy of the microstructure and structure function, periodically modulating correlation functions and an exponent of the late stage coarsening kinetics much smaller than 0.33 as well, demonstrating the crucial role played by the elastic interaction in dynamics of phase separation.

Boundary effects in the three-dimensional Ising model

Simulations of the standard three-dimensional ising model with a variety of boundary conditions, for medium and large systems, are compared with field-theoretical predictions. In particular, setting boundary spins equal to zero is not the best approximation for the Dirichtet boundary condition of zero magnetization at the boundary. The simulations for the specific heat agree semiquantitatively with the field theory for periodic and Dirichtet boundary conditions.

Percolation and the hard-core lattice gas model

Recently a uniqueness condition for Gibbs measures in terms of disagreement percolation (a type of dependent percolation involving two realizations) has been obtained. In general this condition is sufficient but not necessary for uniqueness. In the present paper we study the hard-core lattice gas model which we abbreviate as hard-core model. This model is not only relevant in Statistical Physics, but was recently rediscovered in Operations Research in the context of certain communication networks. First we show that the uniqueness result mentioned above implies that the critical activity for the hard-core model on a graph is at least P c (1 − P c) , where P c is the critical probability for site percolation on that graph. Then, for the hard-core model on bi-partite graphs, we study the probability that a given vertex v is occupied under the two extreme boundary conditions, and show that the difference can be written in terms of the probability of having a ‘path of disagreement’ from v to the boundary. This is the key to a proof that, for this case, the uniqueness condition mentioned above is also necessary, i.e. roughly speaking, phase transition is equivalent with disagreement percolation in the product space. Finally, we discuss the hard-core model on Z d with two different values of the activity, one for the even, and one for the odd vertices. It appears that the question whether this model has a unique Gibbs measure, can, in analogy with the standard ferromagnetic Ising model, be reduced to the question whether the third central moment of the surplus of odd occupied vertices for a certain class of finite boxes is negative.

EXACT PHASE DIAGRAMS FOR THE CONDENSATION OF A KAGOMÉ LATTICE GAS WITH THREE-PARTICLE INTERACTIONS

The condensation of a two-dimensional kagomé lattice gas having purely three-particle interactions is first theoretically investigated. The Hamiltonian Hℓg=−∈3Σ<i, j, k> ninjnk, where ∈3>0 is the strength. parameter of the short-range attractive triplet interaction, the sum is taken over all elementary triangles of the kagomé lattice, and nℓ=0, 1 is an idempotent site-occupation number. The method initially involves transforming the lattice-gas model into a generalized kagomé Ising model having both pair and triplet interactions as well as a magnetic field. Since the canonical partition function of a generalized kagomé Ising model is equivalent (aside from known prefactors) to the canonical partition function of a standard honeycomb Ising model in a magnetic field, one can deduce the exact liquid-vapor phase diagrams of the triplet-interaction kagomé lattice gas from its grand canonical partition function. As results, the liquid-vapor phase boundary (reduced chemical potential μ/∈3 vs reduced temperature T/Tc) is found to be curvilinear with a positive slope, originating at zero temperature with μ/∈3=−2/3 and analytic at its terminating critical point whose coordinates are T/Tc=1, μc/∈3=−0.64469…, where ∈3/kBTc=3.96992…. The companion coexistence curve (particle number density ρ vs. reduced temperature T/Tc) exhibits an asymmetric rounded shape with a positive-slope curvilinear diameter, and the value of the critical density ρc=0.58931…. At criticality, the expression for the coexistence curve superposes a pair of branch point singularities resulting in an infinite (vertical) slope at the critical point (T/Tc=ρ/ρc=1). The case of a kagomé lattice gas having mixed attractive pair interactions and very weak repulsive triplet interactions (Axilrod-Teller) is next considered. Perturbation analyses upon exact expressions relating to the phase diagrams reveal, over chosen ranges of reduced temperatures, that the phase boundary and the diameter of the two-phase coexistence region each have a negative slope due to the repulsive three-particle interactions.

Crossover finite-size scaling at first-order transitions

In a recent paper we developed a method which allows one to control rigorously the finite-size behavior in long cylinders near first-order phase transitions at low temperature. Here we apply this method to asymmetric transitions with two competing phases, and to theq-state Potts model as a typical model of a temperature-driven transition, whereq low-temperature phases compete with one high-temperature phase. We obtain the finite-size scaling of the firstN eigenvalues (whereN is the number of competing phases) of the transfer matrix in a periodic box of volumeL × ... ×L ×t, and, as a corollary, the finite-size scaling of the shape of the order parameter in a hypercubic box (t=L), the infinite cylinder (t=∞), and the crossover regime from hypercubic to cylindrical scaling. For the two-phase case (N=2 we find that the crossover lengthξL is given by O(Lw)exp(ΒσLv), whereΒ is the inverse temperature, σ is the surface tension, and w=1/2 if v+1=2 whilew=0 if v+1 >2. For the standard Ising model we also consider free boundary conditions, showing that ξL=exp[ΒσLv+O(Lv− 1)] for any dimension v+1⩾2. For v+1=2 we finally discuss a class of boundary conditions which interpolate between free (corresponding to the interpolating parameter g=0) and periodic boundary conditions (corresponding to g=1), finding thatξL=O(Lw)exp(ΒσLv) withw=0 forg=0 andw=1/2 for 0<g⩽1.

ISING DROPLETS, NUCLEATION, AND STRETCHED EXPONENTIAL RELAXATION

The standard Ising model in two and three dimensions was simulated with Metropolis-Glauber kinetics on the Intel Hypercube with 32 MIMD processors of i860 type, each with 16 megabytes of distributed memory, and on 8 such processors of the Alliant FX with shared memory. The nucleation time in reverse magnetic fields h. was found to increase as 1/h2 in three dimensions, for times up to millions of iterations. The relaxation towards the equilibrium spontaneous magnetization was a simple exponential in three dimensions and a stretched exponential in two below the critical temperature; its power-law decay at the critical point lead to z=2.18(d=2) and =2.09 (d=3). The Becker-Döring equation for the growth of microdroplets leads at the critical temperature to a growth rate vanishing with some power of the time.

Kinetics of domain growth in finite Ising strips

Monte Carlo simulations are presented for the kinetics of ordering of the two-dimensional nearest-neighbor Ising models in an L x M geometry with two free boundaries of length M ⪢ L. This geometry models a “terrace” of width L on regularly stepped surfaces, adatoms adsorbed on neighboring terraces being assumed to be noninteracting. Starting out with an initially random configuration of the atoms in the lattice gas at coverage θ = 1 2 in the square lattice, quenching experiments to temperatures in the range 0.85⩽ T/ T c⩽1 are considered, assuming a dynamics of the Glauber model type (no conservation laws being operative). At T c the ordering behavior can be described in terms of a time-dependent correlation length ξ( t), which grows with the time t after the quench as ξ(t)∼t 1 z with the dynamic exponent z≈2.1, until the correlation length settles down at its equilibrium value 2 L/π (for correlations in the direction of the steps). Below T c a two-stage growth is observed: in the first stage, the scattering intensity 〈 m 2( t)〉 grows linearly with time, as in the standard kinetic Ising model, until the domain size is of the same size as the terrace width. The further growth of 〈 m 2( t)〉 in the second stage is consistent with a logarithmic law.

Exact solutions for even-number correlations of the square Ising model.

To obtain an unusually large set of exact solutions for even-number localized correlations of the standard square Ising model, a method is developed that combines traditional Pfaffian techniques with linear-algebraic systems of correlation identities having interaction-dependent coefficients. Two eight-site clusters (subclusters of the ``Greek cross'' cluster) are studied, and altogether sixty different even-number correlations are determined exactly at all temperatures. Besides demonstrating the existence of degeneracies within the set of exact solutions, the solution curve for an eight-site correlation is evidently the first of such large order to be displayed for an Ising model on any regular planar lattice. Significant time as well as labor efficiency of the method is clearly demonstrated since relatively few correlations need to be actually calculated by Pfaffian techniques in order to obtain large additional numbers of multisite correlation solutions using much simpler linear-algebraic procedures.

Monte Carlo simulations of surface morphologies during mineral dissolution

The interdependence of surface morphology and the dissolution kinetics of minerals is studied by means of Monte Carlo simulations. A square lattice without dislocations is used. The standard kinetic Ising model is compared critically to facts and concepts from the coordination chemistry of oxide dissolution. Activated complex theory provides a link between the reversible Ising model and irreversible weathering reactions. Two sets of parameters, the surface roughness and the mol fractions of different sites (like steps and kinks), are used to quantify surface morphologies. These parameters are monitored in the simulations for a wide range of dissolution rates. The results indicate that steady-state surface morphologies are formed after a few lattice layers have been dissolved. The Monte Carlo data follow simple equations which link the kinetics of the model with the resulting surface morphologies. A partition function governs the mol fractions of sites in the steady state. On the basis of this result rigorous expressions for the Monte Carlo rate law are presented.

Monte Carlo simulation of very large dilute random-field Ising models using multi-spin coding

We have developed an efficient algorithm for simulating with computers very large dilute Ising models in random external magnetic fields using logical bit-by-bit operations with a multi-spin coding technique. The algorithm is written in such a way that the same algorithm can also be used for simulating the standard Ising model, the dilute and the random-field Ising models. We present results for domain growth in the two-dimensional dilute random-field Ising model as an illustration of the algorithm.

Continuum limit on a 2-dimensional random lattice

We investigate the continuum limit of the Ising model on a 2-dimensional random lattice using Monte Carlo and strong coupling techniques. No evidence is found for deviation from the behaviour of the standard Ising model on a regular lattice; we do not see the modified critical properties found in the bond-diluted Ising model. Our results suggest that the continuum limit is the usual non-interacting Majorana fermion theory.

From self-avoiding random loops to Ising systems an interpolating spin model and its Monte Carlo study

We develop a new lattice model involving Ising spins on links with next-neighbor couplings which contains a parameter ξ that interpolates between the standard Ising model ( ξ=1) and an ensemble of self-avoiding random loops ( ξ=0). This model is studied by Monte Carlo techniques. We calculate the average length and its variance as a function of temperature. The reliability of our results is checked at several steps by comparison with the exactly known Ising case on finite lattices.

The exact solution of a nonplanar Ising model

A generalisation of the star-triangle transformation is used to establish a correspondence between a nonplanar Ising lattice with two interaction constants and the standard triangular Ising model. The critical point of the nonplanar lattice is obtained exactly for various values of the ratio of the interaction constants. The singularity in the zero-field free energy of the nonplanar lattice is investigated. It is shown to be of the same form as that displayed by the corresponding standard Ising model.

Series expansions for a spin-glass model

Series expansions are investigated for a spin-glass Ising model with nearest-neighbour interactions J which can be randomly positive or negative. For the high-temperature phase the following conclusions result: (i) the free energy has a singularity at about w = µ-1/2 (w = tanh β J) (where µ is the self-avoiding walk limit), (ii) the magnetic susceptibility corresponds to uncoupled spins, (iii) the second derivative of susceptibility is simply related to the susceptibility of the standard Ising model and has a singularity at w = wc1/2 (where wc refers to the standard model).