Encouraged by experiments on 4He in aerogels, we confine planar spins in the pores of simulated aerogels (diffusion limited cluster-cluster aggregation) in order to study the effect of quenched disorder on the critical behavior of the three-dimensional XY model. Monte Carlo simulations and finite-size scaling are used to determine critical couplings K(c) and exponents. In agreement with experiments, clear evidence of change in the thermal critical exponents nu and alpha is found at nonzero volume fractions of impurities. These changes are explained in terms of hidden long-range correlations within disorder distributions.
Encouraged by experiments on 4He in aerogels, we confine planar spins in the pores of simulated aerogels (diffusion limited cluster-cluster aggregation) in order to study the effect of quenched disorder on the critical behavior of the three-dimensional XY model. Monte Carlo simulations and finite-size scaling are used to determine critical couplings K(c) and exponents. In agreement with experiments, clear evidence of change in the thermal critical exponents nu and alpha is found at nonzero volume fractions of impurities. These changes are explained in terms of hidden long-range correlations within disorder distributions.
We review recent results on the effect of a specific type of quenched disorder on well known O(m)-vector models in three dimensions: the XY model (3DXY, m = 2) and the Ising model (3DIS, m = 1).Evidence of changes of criticality in both systems, when confined in aerogel pores, is briefly referenced.The 3DXY model represents the universality class to which the λ-transition of bulk superfluid 4 He belongs.Experiments report interesting changes of critical exponents for this transition, when superfluid 4 He is confined in aerogels.Numerical evidence has also been presented that the 3DXY model, confined in aerogel-like structures, exhibits critical exponents different from those of bulk, in agreement with experiments.Both results seem to contradict Harris criterion: being the specific heat exponent negative for the pure system (α 3DXY -0.011 < 0), changes should be explained in terms of the extended criterion due to Weinrib and Halperin, which requires disorder to be long-range correlated (LRC) at all scales.In numerical works, aerogels are simulated by the diffusion limited cluster-cluster aggregation (DLCA) algorithm, known to mimic the geometric features of aerogels.These objects, real or simulated, are fractal through some decades only, and present crossovers to homogeneous regimes at finite scales, so the violation to Harris criterion persists.The apparent violation has been explained in terms of hidden LRC subsets within aerogels [Phys.Rev. Lett., 2003, 90, 170602].On the other hand, experiments on the liquid-vapor (LV) transition of 4 He and N 2 confined in aerogels, also showed changes in critical-point exponents.Being the LV critical-point in the O(1) universality class, criticality may be affected by both short-range correlated (SRC) and LRC subsets of disorder.Simulations of the 3DIS in DLCA aerogels can corroborate experimental results.Both experiments and simulations suggest a shift in critical exponents to values closer to the SRC instead of those of the LRC fixed point.
The influence of correlated impurities on the critical behavior of the three-dimensional (3D) Ising model is studied using Monte Carlo simulations. Spins are confined into the pores of simulated aerogels (diffusion-limited cluster-cluster aggregation) in order to study the effect of quenched disorder on the critical behavior of this magnetic system. Finite-size scaling is used to estimate critical couplings and exponents. Long-range correlated disorder does not affect the critical behavior. Asymptotic exponents differ from those of the pure 3D Ising model, but it is impossible, with our precision, to distinguish them from the randomly diluted Ising model.
The influence of quenched nonmagnetic impurities on phase transitions in the three-dimensional Potts model with the number of spin states q = 3 is investigated using the Wolff single-cluster algorithm of the Monte Carlo method. The systems with linear sizes L = 20–44 at the spin concentrations p = 1.0, 0.9, 0.8, and 0.7 are analyzed. It is demonstrated with the use of the method of fourth-order Binder cumulants that the second-order phase transition occurs in the model under consideration at the spin concentrations p = 0.9, 0.8, and 0.7 and that the first-order phase transition is observed in the pure model ( p = 1.0). The static critical exponents α (heat capacity), γ (susceptibility), β (magnetization), and ν (correlation length) are calculated in the framework of the finite-size scaling theory. The problem regarding the universality classes of the critical behavior of weakly diluted systems is discussed.
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.
The influence of coexisting correlated and noncorrelated impurities on the critical behavior of the three-dimensional Ising model is studied using Monte Carlo numerical simulations and finite-size scaling. The amount of correlated and noncorrelated vacancies is modified and controlled during the simulations. The long-range correlated (LRC) critical behavior is always found for any value of the concentration of correlated vacancies. The smaller the amount of correlated vacancies the larger the system size needed to detect the LRC universality class. This result explains why critical values measured in xerogel liquid-vapor experiments, where the concentration of correlated vacancies is marginal, seem to correspond to a short-range correlated disorder.
We improve the theoretical estimates of the critical exponents for the three-dimensional XY universality class. We fine alpha=-0.0146(8), gamma =1.3177(5), nu =0.671 55(27), eta =0.0380(4), beta =0.3485(2), and delta =4.780(2). We observe a discrepancy with the most recent experimental estimate of alpha; this discrepancy calls for further theoretical and experimental investigations. Our results are obtained by combining Monte Carlo simulations based on finite-size scaling methods, and high-temperature expansions. Two improved models (with suppressed leading scaling corrections) are selected by Monte Carlo computation. The critical exponents are computed from high-temperature expansions specialized to these improved models. By the same technique we determine the coefficients of the small-magnetization expansion of the equation of state. This expansion is extended analytically by means of approximate parametric representations, obtaining the equation of state in the whole critical region. We also determine the specific-heat amplitude ratio.