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We used a Monte Carlo simulation of the structurally disordered three dimensional Ising model. For the systems with spin concentrations p = 0.95, 0.8, 0.6 and 0.5 we calculated the correlation-length critical exponent ν by finite-size scaling. Extrapolations to the thermodynamic limit yield ν(0.95) = 0.705(5), ν(0.8) = 0.711(6), ν(0.6) = 0.736(6) and ν(0.5) = 0.744(6). The analysis of the results demonstrates the nonuniversality of the critical behavior in the disordered Ising model.

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We discuss universal and non-universal critical exponents of a three dimensional Ising system in the presence of weak quenched disorder. Both experimental, computational, and theoretical results are reviewed. Special attention is paid to the results obtained by the field theoretical renormalization group approach. Different renormalization schemes are considered putting emphasis on analysis of divergent series obtained.

The critical behavior of the disordered ferromagnetic Ising model is studied numerically by the Monte Carlo method in a wide range of variation of concentration of nonmagnetic impurity atoms. The temperature dependences of correlation length and magnetic susceptibility are determined for samples with various spin concentrations and various linear sizes. The finite-size scaling technique is used for obtaining scaling functions for these quantities, which exhibit a universal behavior in the critical region; the critical temperatures and static critical exponents are also determined using scaling corrections. On the basis of variation of the scaling functions and values of critical exponents upon a change in the concentration, the conclusion is drawn concerning the existence of two universal classes of the critical behavior of the diluted Ising model with different characteristics for weakly and strongly disordered systems.

The current state of Monte Carlo work on phase transitions and critical phenomena is reviewed. Both classical and quantum Monte Carlo results are discussed with emphasis on statistical lattice-model studies involving the highly accurate calculation of critical exponents. It is shown that finite-size scaling is an effective method whereby not only simple lattice models but also those of complex practical magnetic materials can be treated. For models allowing crossover phenomena, it is shown that both finite-size scaling theory and conventional power-law functions must be used in carrying out a Monte Carlo analysis.

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.

The author has investigated the critical behaviour of site-disordered Ising systems over a large range of dilution (0.6<or=p<or=1.0) investing about 1015 single spin flips with conventional local dynamics. Critical temperatures have been determined with high-precision using the cumulant method. The finite-size simulations lead to concentration-dependent exponents which seem to violate universality. The author shows that the concentration dependence is induced by a crossover phenomenon which governs a large interval of the critical region. The author finds that the scaling functions of systems with different disorder converge asymptotically to universal behaviour consistent with the fixed point of weak disorder.

A Monte Carlo method is applied to simulate the static critical behavior of a cubic-lattice 3D Ising model for systems with
quenched disorder. Numerical results are presented for the spin concentrations of p = 1.0, 0.95, 0.9, 0.8, 0.6 on L × L × L lattices with L = 20–60 under periodic boundary conditions. The critical temperature is determined by the Binder cumulant method. A finite-size
scaling technique is used to calculate the static critical exponents α, β, γ, and ν (for specific heat, susceptibility, magnetization,
and correlation length, respectively) in the range of p under study. Universality classes of critical behavior are discussed for three-dimensional diluted systems.

We use a high-precision Monte Carlo simulation to determine the universal
specific-heat amplitude ratio A+/A- in the three-dimensional Ising model via
the impact angle \phi of complex temperature zeros. We also measure the
correlation-length critical exponent \nu from finite-size scaling, and the
specific-heat exponent \alpha through hyperscaling. Extrapolations to the
thermodynamic limit yield \phi = 59.2(1.0) degrees, A+/A- = 0.56(3), \nu =
0.63048(32) and \alpha = 0.1086(10). These results are compatible with some
previous estimates from a variety of sources and rule out recently conjectured
exact values.

We present a new method for using the data from Monte Carlo simulations that can increase the efficiency by 2 or more orders of magnitude. A single Monte Carlo simulation is sufficient to obtain complete thermodynamic information over the entire scaling region near a phase transition. The accuracy of the method is demonstrated by comparison with exact results for the d=2 Ising model. New results for d=2 eight-state Potts model are also presented. The method is generally applicable to statistical models and lattice-gauge theories.

We perform a high-statistics simulation of the three-dimensional randomly dilute Ising model on cubic lattices L3 with L< or =256. We choose a particular value of the density, x=0.8, for which the leading scaling corrections are suppressed. We determine the critical exponents, obtaining nu=0.683(3), eta=0.035(2), beta=0.3535(17), and alpha=-0.049(9), in agreement with previous numerical simulations. We also estimate numerically the fixed-point values of the four-point zero-momentum couplings that are used in field-theoretical fixed-dimension studies. Although these results somewhat differ from those obtained using perturbative field theory, the field-theoretical estimates of the critical exponents do not change significantly if the Monte Carlo result for the fixed point is used. Finally, we determine the six-point zero-momentum couplings, relevant for the small-magnetization expansion of the equation of state, and the invariant amplitude ratio R(+)(xi) that expresses the universality of the free-energy density per correlation volume. We find R(+)(xi)=0.2885(15).

We study the phase diagram of the site-diluted Ising model in a wide dilution range, through Monte Carlo simulations and Finite-Size Scaling techniques. Our results for the critical exponents and universal cumulants turn out to be dilution-independent, but only after a proper infinite volume extrapolation, taking into account the leading corrections-to-scaling terms.