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arXiv:cond-mat/0604592 · Physical Review B

Three-dimensional Ising model confined in low-porosity aerogels: A Monte Carlo study

Ricardo Paredes V., Carlos Vásquez

Published Aug 2, 2006• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.74.054201

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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.
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.
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 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.
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). These results are compatible with some previous estimates from a variety of sources. The analysis of the results demonstrates the nonuniversality of the critical behavior in the disordered Ising model.
The critical behavior of the Ising model in three dimensions on a lattice with site disorder is studied by applying Monte Carlo simulation techniques. Two cases for the site disorder are considered: uncorrelated disorder and long-range correlated disorder with a spatial correlation function that decays according to a power law ${r}^{\ensuremath{-}a}$. The critical exponents $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ as well as updated results for the critical exponent $\ensuremath{\nu}$ and confluent correction exponent $\ensuremath{\omega}$ are provided for a variety of different correlation exponents aaa and disorder concentrations pd{p}_{d}pd​. The estimation is done by using finite-size scaling analyses and a global fit procedure which combines the results obtained for different concentrations of defects. From the estimated critical exponents, the validity of hyperscaling relations is studied and finally the critical temperatures are provided for different aaa and pd{p}_{d}pd​.