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Journal of Physics Conference Series

Monte Carlo simulation of the critical behavior near the percolation threshold with invaded cluster algorithm

Maria Shlyakhtich, Pavel Vladimirovich Prudnikov

Published Jan 1, 2021• 1 citation
Publisher

Abstract

Abstract We investigated the critical behavior of the disordered three-dimensional Ising model with an impurity concentration close to the threshold of impurity percolation p = pc = 0.688 313 500 on high-performance computing systems. The system was updated using Metropolis algorithm and invaded cluster algorithm from initial configurations. The values of dynamic critical exponent z and static critical exponents β and v were obtained in this paper.

DOI
10.1088/1742-6596/1740/1/012009

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arXiv:cond-mat/9806127•International Journal of Modern Physics C•Feb 1, 1999•16 citations
Abstract We investigated the critical behavior of the disordered three-dimensional Ising model with an impurity concentration close to the threshold of impurity percolation p = pc = 0.688 313 500 on high-performance computing systems. The system was updated using Metropolis algorithm and invaded cluster algorithm from initial configurations. The values of dynamic critical exponent z and static critical exponents β and v were obtained in this paper.
The critical relaxation of the magnetization in a two-dimensional Ising model with quenched random non-magnetic impurities has been studied by numerical simulation. A squared lattice of size 400 2 with spin concentrations p=1.0; 0.95, 0,9, 0.85, 0.8, 0.75 and 0.7 was considered. The dynamical critical exponent z was determined by the Monte Carlo method combined with the dynamical renormalization group method. The following values were received for z(p): z(1)=2.24+or-0.07, z(0.95)=2.24+or-0.06, z(0.9)=2.24+or-0.06, z(0.85)=2.38+or-0.05, z(0.8)=2.51+or-0.06, z(0.75)=2.66+or-0.07 and z(0.7)=2.88+or-0.06. A singular dynamic scaling behaviour with z=A' mod ln(p-p c ) mod +B' was found for systems with spin concentrations p<or=0.85 near the percolation threshold p c , where A'=0.56+or-0.07 and B'=1.62+or-0.07 are temperature-independent constants.
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 critical relaxation of the magnetization in a 2D Ising model with quenched random non-magnetic impurities is studied by numerical simulation. A square lattice of size 400 2 and spin concentrations p = 1.0, 0.95, 0.9, 0.85, 0.8, 0.75 and 0.7 are considered. The dynamical critical exponent z is determined by the dynamical renormalization group Monte Carlo method. The values for z(p) are: z (1) = 2.24 ± 0.07, z (0.95) = 2.24 ± 0.06, z (0.9) = 2.24 ± 0.06, z (0.85) = 2.38 ± 0.05, z (0.8) = 2.51 ± 0.06, z (0.75) = 2.66 ± 0.07 and z (0.7) = 2.88 ± 0.06. A singular dynamic-scaling behaviour with z(p) = A '|ln( p − p c )| + B ' is found for systems with spin concentrations near the percolation threshold p c , where A ' = 0.56 ± 0.07 and B ' = 1.62 ± 0.07 are temperature-independent constants.
Phase transitions and critical phenomena in several spin models are studied. These models include a two and three dimensional XY model and Ising models in structured and random field. The methods of study are cluster Monte Carlo simulations and network flow optimization algorithms. The XY model and a related soft spin model belonging to the O(2) universality class are studied using a modified version of the Invaded Cluster algorithm. The static critical properties of the models and the dynamic characteristics of the algorithm are studied. The critical exponent η for the O(2) universality class in two and three dimensions is measured. The Invaded Cluster algorithm for the XY model does not show any critical slowing for global variables. Benzene adsorption in a nanoporous solid is modeled with an Ising model with both internal and external magnetic fields. The first order phase transition and the critical properties of the system are studied. The system is simulated using the two replica cluster algorithm. The critical temperature of the system is found to decrease with increasing difference of the values of the internal fields. It is found that the system undergoes a phase transition for a relatively wide, experimentally obtainable, region of values of its parameters. The random field Ising model in three dimensions is studied at zero temperature. The nature of randomness induced phase transition is studied using a numerical network flow optimization method. It is found that the latent heat at the phase transition vanishes in the limit of infinite system size which can be taken as a proof of a second order phase transition.