Critical properties of the three-dimensional Ising model with quenched disorder

The static critical properties of the three-dimensional Ising model with quenched disorder are studied by the Monte-Carlo (MC) method on a simple cubic lattice, in which the quenched disorder is distributed as nonmagnetic impurities by the canonical manner. The calculations are carried out for systems with periodic boundary conditions and spin concentrations p=1.0; 0.95; 0.9; 0.8; 0.7; 0.6. The systems of non-linear sizes L×L×L, L=20-60 are researched. On the basis of the finite-size scaling (FSS) theory, the static critical exponents of specific heat α, susceptibility γ, magnetization β, and an exponent of the correlation radius in a studied interval of concentrations p are calculated. It is shown that the three-dimensional Ising model with quenched disorder has two regimes of the critical behavior universality in a dependence on nonmagnetic impurities.     

Critical behavior of the 3D Ising model on a poissonian random lattice

The single-cluster Monte Carlo algorithm and the reweighting technique are used to simulate the 3D ferromagnetic Ising model on 3D Voronoi-Delauney lattices. It is assumed that the coupling factor J varies with the distance r between the first neighbors as J(r)∝e^−ar, with a≥0. The critical exponents γ/ν, β/ν, and ν are calculated, and according to the present estimates for the critical exponents, we argue that this random system belongs to the same universality class of the pure 3D ferromagnetic Ising model.     

Wang-Landau study of the 3D Ising model with bond disorder

We implement a two-stage approach of the Wang-Landau algorithm to investigate the critical properties of the 3D Ising model with quenched bond randomness. In particular, we consider the case where disorder couples to the nearest-neighbor ferromagnetic interaction, in terms of a bimodal distribution of strong versus weak bonds. Our simulations are carried out for large ensembles of disorder realizations and lattices with linear sizes L in the range L=8-64L=8{-}64. We apply well-established finite-size scaling techniques and concepts from the scaling theory of disordered systems to describe the nature of the phase transition of the disordered model, departing gradually from the fixed point of the pure system. Our analysis (based on the determination of the critical exponents) shows that the 3D random-bond Ising model belongs to the same universality class with the site- and bond-dilution models, providing a single universality class for the 3D Ising model with these three types of quenched uncorrelated disorder.     

Critical energy relaxation in 2D Ising model

The lack of a logarithmic factor in the critical energy relaxation at the Curie point is explained by a cluster approximation.     

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

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 T_c 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(/dp) exp(–2J ₁/kT_c¦_p=p_c + = 1, in agreement with Bergstresser's rigorous result for the diluted ferromagnet near the percolation threshold.Work supported in part by National Science Foundation Grant No. DMR 76-21703, Office of Naval Research Grant No. N00014-76-C-0106, and National Science Foundation MRL program Grant No. DMR 76-00678.Paper presented at the 37th Yeshiva University Statistical Mechanics Meeting, May 10, 1977.     

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.     

Ferromagnetic phase breakdown of a quenched bond-mixed Ising model

Within the framework of an effective field theory beyond Mean Field Approximation, we discuss the ferromagnetic phase stability limit in the temperature-concentration space of a quenched bond-mixed spin-$\text{1}\text{2}$ Ising model in square lattice for both competing and noncompeting interactions J₁ and J₂. Quite reasonable results are obtained in both situations. In particular for the case of competing interactions, numerical estimates of the vanishing temperature critical bond concentrations are predicted for particular values of the ratio $\text{J}{}_1\text{J}{}_2$.     

Excess noise in a hopping model for a resistor with quenched disorder

We consider a model for independent charged particles, hopping on a lattice with static disorder in the waiting times. The excess current noise is calculated and shown to be related to resistance noise and arising from mobility fluctuations. It is also related to the four point super-Burnett-function. The strength of the noise is calculated at small frequencies for weak disorder (classical long time tails) and for strong disorder, when it may behave like I/f. In that case the Hooge factor equals the fraction of deep trapping centers.     

New method for the 3D Ising model

A simple, general and practically exact method is developed for the equilibrium properties of the macroscopic physical systems with translational symmetry. Applied to the Ising model in two and three dimension, a modest calculation gives the spontaneous magnetization and the specific heat to less than 1% error.     

The anisotropic 3D Ising model

An expression for the free energy of an (001) oriented domain wall of the anisotropic 3D Ising model is derived. The order--disorder transition takes place when the domain wall free energy vanishes. In the anisotropic limit, where two of the three exchange energies (e.g. J_x and J_y ) are small compared to the third exchange energy (J_z ), the following asymptotically exact equation for the critical temperature is derived, sinh(2J_z /k _B T _c)sinh(2(J_x ?+?J_y )/k _B T _c))?=?1. This expression is in perfect agreement with a mathematically rigorous result (k _B T _c/J_z ?=?2[ln(J_z /(J_x ?+?J_y ))?ln(ln(J_z /(J_x ?+?J_y ))?+?O(1)]^?1) derived earlier by Weng, Griffiths and Fisher (Phys. Rev. 162, 475 (1967)) using an approach that relies on a refinement of the Peierls argument. The constant that was left undetermined in the Weng et al. result is estimated to vary from ～0.84 at ((H_x ?+?H_y )/H_z )?=?10^?2 to ～0.76 at ((H_x ?+?H_y )/H_z )?=?10^?20.     

Critical Casimir effects in 2D Ising model with curved defect lines

This work is aimed at studying the influence of critical Casimir effects on energetic properties of curved defect lines in the frame of 2D Ising model. Two types of defect curves were investigated. We start with a simple task of globule formation from four-defect line. It was proved that an exothermic reaction of collapse occurs and the dependence of energy release on temperature was observed. Critical Casimir energy of extensive line of constant curvature was also examined. It was shown that its critical Casimir energy is proportional to curvature that leads to the tendency to radius decreasing under Casimir forces. The results obtained can be applied to proteins folding problem in polarized liquid.     

Critical behaviour of a compressible Ising model

Renormalization-group methods are applied to the critical behaviour of Ising-like systems on an elastic solid. The lattice part has been treated quantum mechanically. This makes the renormalized Ising fixed point realizable.     

Critical behavior of the specific heat for pure and site-diluted simple cubic Ising systems

The exponent of the specific heatC is determined for the pure and the site-diluted simple cubic Ising model (concentrationx=0, 0.2, 0.4 of nonmagnetic sites) by a finite-size scaling analysis of the peak value C_max(L) for systems of linear dimensionsL=8, 16, 32, and 64. The C_max values are obtained by the Ferrenberg-Swendsen algorithm, using Monte Carlo data from a fully-vectorized multi-spin coding program. We obtain =0.11 for x=0 and a crossover to a negative value upon dilution, with =–0.029(4) both forx=0.2 andx=0.4.