The bethe approximation to the three-state chiral clock model

The three-state chiral clock model is studied by means of the Bethe approximation. While the phase diagram obtained by this method resembles the mean-field phase diagram in the vicinity of the boundary to the paramagnetic phase, a significant improvement is achieved in the low and intermediate temperature regions: By a low-temperature expansion of the free energy, which is carried out to third order, we find that, up to this order, the Bethe approximation exactly reproduces the results of the low-temperature analysis of the full model by Yeomans and Fisher. This and the numerical evaluation of the free energy show that, as far as the longer wavelength phases are concerned, the Bethe approximation is in keeping with predictions of Yeomans and Fisher for low temperature, where mean-field theory is qualitatively misleading. At higher temperatures more complicated structures are found to evolve from the basic low-temperature phases by structure combination branching processes in the same fashion as in the phase diagram of the ANNNI model.     

Bethe-Peierls approximation for the disordered Ising model

We study the Ising model for an alloy with an arbitrary number of components. We develop an approximation which reduces to that of Bethe and Peierls when the concentration of one of the components is unity. We investigate within this approximation the dependence of the various thermodynamic quantities, in particularT _c, on the composition of the alloy and the magnetic properties of its constituents. Comparison with the only exact calculation available, that of F. T. Leeet al., for a linear chain, shows extremely satisfactory agreement.Research supported by ARO (D). It has also benefited from the general support of Materials Science at the University of Chicago by the NSF.     

The random field method for the Ising model

The random field method is used for investigation of the Ising model. The generalization of the variational principle and a new representation for the free energy of the Ising model are proposed.     

On the mean-field Ising model in a random external field

We use a method developed by van Hemmen to obtain the free energy of the mean-field Ising model in a random external magnetic field. Some results of previous mean-field calculations are confirmed and generalized. The tricritical point in the global phase diagram is discussed in detail. We also consider different probability distributions of the random fields and provide some proofs regarding the conditions for the existence of a tricritical point.     

Two-spin correlation function of spin 12 Ising model on a bethe lattice

The two-spin correlation function of the Ising model on a Bethe lattice as a function of temperature and magnetic field. It is found that the coherence length is finite at finite temperatures.     

A self-consistent Ornstein-Zernike approximation for the site-diluted Ising model

We propose a theory for the site-diluted Ising model which is an extension to disordered systems of the self-consistent Ornstein-Zernike approximation of Hoye and Stell. By using the replica method in the context of liquid-state theory, we treat the concentration of impurities as an ordinary thermodynamic variable. This approach is not limited to the weak-disorder regime or to the vicinity of the percolation point. A preliminary analysis using series expansion shows that it can predict accurately the dependence of the critical temperature on dilution and can reproduce the nonuniversal behavior of the effective exponents. The theory also gives a reasonable estimate of the percolation threshold.     

Variational principle for the distribution function of the effective field for the random Ising model in the Bethe approximation

We study the random Ising model in the pair approximation of the cluster variation method. We show that the distribution function of the effective field is determined either by a reducibility condition of the distribution function of two sites to that of one site or by a stationarity condition of the averaged free energy.     

Critical properties of a random transverse crystal field Ising model with bond dilution

Within effective field theory (EFT), the critical properties of a random transverse crystal field Ising model with bond dilution are studied on a square lattice. Under both weak and strong bond dilution conditions, we consider three cases (α = 0,±0.5) of a transverse crystal field ratio, obtaining global phase diagrams in T−D _x space for changes in the random transverse crystal field concentration. The phase diagrams obtained for a weak bond dilution are very similar in shape to those of pure bond but with decreases in corresponding ordered phases and critical values. However, the phase diagrams for a strong bond dilution exhibit varieties, including a change in reentrant phenomenon, the occurrence of transverse crystal field degeneration, and the opposite direction crossover of temperature peak value.     

The random field Ising model

We discuss the current status of random field systems, particularly those with Ising symmetry. Both theory and experiment agree that, in the equilibrium state, there is a transition to an ordered state in three dimensions and no such transition in two dimensions. The critical behavior in three dimensions is, however, not very well understood. More work remains to be done to understand the dynamics, both in the critical region and the low temperature phase.     

Ising model with variable bond strength

The partition function of the two-dimensional Ising model with variable bond strength along one dimension is computed exactly.     

Dynamics of the random field Ising model

Dynamics of the kinetic Ising model in the presence of static random fields is investigated using a self-consistent method. It is shown that if the interface fluctuations of the low temperature phase are small the system at low temperatures stays in a state without long range order. For this state the spin correlation function 〈S_q(t)S_?q(O)> averaged over all configurations of random fields decays exponentially in time with a single wavevector dependent relaxation time which is finite at the transition temperature T₀ and remains very long below T₀. In the mean field approximation the correlation time at the magnetic Bragg peak and at T₀ scales with the magnitude of the random field as τ∞h^?z_h with z_h = 1 for d = 2 and $\text{z}{}_{\mathrm{h}}\text{=}\text{4}\text{3}$ for d = 3, respectively.     

Pressure and variational principle for random Ising model

An Ising model traditionally is a model for a repartition of spins on a lattice. Griffiths and Lebowitz ([3. 5]) have considered distributions of spins which can occur only on some randomly prescribed sites—Edwards and Anderson have introduced models where the interaction was random ([6, 7]). In both cases, the formalism of statistical mechanics reduces mainly to a relativised variational principle, which has been proved recently by Walters and the author [1]. In this note, we show how that reduction works and formulate the corresponding results on an example of either model.     

The spin-one Ising model in the mean-spherical approximation

The spin-one Ising ferromagnet on a simple cubic lattice is treated in the mean-spherical approximation (MSA) for an exchange potentialJ(r) parametrized by a Kac-Baker inverse-range parameter γ. The mean-field result is recovered when γ→ 0; in this limit the result is exact. For γ≠ 0, a detailed analysis is given of the phase separation associated with the tricritical point that occurs. The analysis is made through the relation that gives the internal energy viaJ(r). It shows that the MSA result satisfactorily captures the important thermodynamic features of the tricritical point as long as γ is not too large. The case of CoulombicJ(r) is also considered; hereJ(r) is antiferromagnetic. An argument is given in support of the expectation that on the simple cubic and body-centered cubic lattices the CoulombicJ(r) will give rise to a tricritical point at which a λ-line of Néel points meets a paramagnetic-antiferromagnetic coexistence boundary.     

The hierarchical random field Ising model

We introduce and solve explicitly a hierarchical approximation to the random field Ising model. This approximation is defined in terms of Peierls' contours. It exhibits a spontaneous magnetization ind>2 and illustrates some of the ideas used in the proof of that result for the real RFIM. Ind=2, there is no spontaneous magnetization, but a very slow decay of correlations. However, we argue that this latter property is an artifact of the approximation. For the real RFIM, we expect exponential decay of correlations for any value of the disorder.     

Zero-temperature dynamics for the ferromagnetic Ising model on random graphs

We consider Glauber dynamics at zero temperature for the ferromagnetic Ising model on the usual random graph model on N vertices, with on average γ edges incident to each vertex, in the limit as N→∞. Based on numerical simulations, Svenson (Phys. Rev. E 64 (2001) 036122) reported that the dynamics fails to reach a global energy minimum for a range of values of γ. The present paper provides a mathematically rigorous proof that this failure to find the global minimum in fact happens for all γ>0. A lower bound on the residual energy is also given.     

Exact results on random bond Ising Chains in magnetic fields

In this contribution we present exact results on the random bond Ising Chain in a magnetic field. The original problem is reduced to the solution of a functional equation for a certain probability distribution, which can be used to evaluate thermodynamics and correlation functions. We give a sufficiently accurate solution for low temperatures, which yields the complete low-temperature behaviour. Comparison is made with different Monte-Carlo-calculations performed on this system.Work supported in part by the Deutsche Forschungs-gemeinschaft     

Critical behavior of the two-dimensional thermalized bond Ising model

A suitably modified Wolff single-cluster Monte Carlo simulation has been performed to investigate the critical behavior of a two-dimensional Ising model with temperature-dependent annealed bond dilution, also known as the thermalized bond Ising model, which is intended to simulate the thermal excitations of electronic bond degrees of freedom as in covalently bonded network liquids. A finite-size scaling analysis of the susceptibility and the fourth-order cumulant, results in a reliable estimation of the critical exponents in the thermodynamic limit. The exponents are found to be consistent with those predicted by the Fisher renormalization relations, despite the well known violations of the renormalization relations when approximate methods such as real space renormalization group are employed to investigate two-dimensional Ising model with annealed bond dilution, and the temperature variation of the bond concentration in thermalized bond model system.