Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Journal of Modern Physics B

PHASE TRANSITION IN THREE-DIMENSIONAL ISING SYSTEMS WITH QUENCHED NON-MAGNETIC IMPURITIES: A REVIEW OF MONTE CARLO STUDIES

Peter R. Braun, U. Staaden, T. Holey, M. Fähnle

Published Sep 1, 1989• 13 citations
Publisher

Abstract

A review is given of Monte Carlo simulations for the phase transition in simple cubic Ising ferro-, ferri-, and antiferromagnets with quenched non-magnetic impurities of concentration x. Special emphasis is given on the description of a fully vectorized multispin coding program and on the problems related to the analysis of the Monte Carlo data. The following three subjects are discussed: 1) For ferro- and antiferromagnets, the critical behaviour of the order parameter is different from the one of the corresponding pure system, in agreement with the prediction of the Harris criterion. The possibly effective exponent β depends on x, whereas for ferrimagnets no such concentration dependence is revealed by the present study for x=0, 0.1, and 0.2.2) For strongly diluted systems, there is a maximum for the Kouvel-Fisher exponent γ(T) of the order-parameter susceptibility outside the critical regime, in accordance with the predictions of the correlated molecular field theory. 3) For the diluted antiferromagnet (x=0.5), a cusp emerges at T c for the uniform susceptibility, in agreement with the theory of Fishman and Aharony.

DOI
10.1142/s0217979289000889

Related papers

Ising ferromagnet with quenched impurities

E. Stoll, T. Schneider

Using the Monte Carlo technique the authors have studied a two-dimensional Ising model with quenched non-magnetic impurities. In particular they calculated the temperature dependence of the order parameter, of the zero-field susceptibility and of the specific heat. These results indicate that a second-order phase transition is preserved for p>or=0.9 where 1-p denotes the concentration of randomly distributed non-magnetic sites. Moreover, they yield beta approximately=1/8 and gamma approximately=7/4, indicating that these exponents equal the corresponding ones for the pure Ising model.

Journal of Physics A Mathematical and General•Jul 1, 1976•26 citations

Phase transition properties of three-dimensional systems described by diluted potts model

Akai Kurbanovich Murtazaev, Albert Babaevich Babaev, G. Ya. Aznaurova

Monte Carlo simulations are performed to analyze phase transitions in three-dimensional systems described by the 3-state Potts model with nonmagnetic impurities. Numerical results are presented for systems with spin concentrations p = 1.00, 0.95, 0.90, 0.80, 0.70, and 0.65 on lattices of size L varying between 20 and 44. Binder’s cumulant analysis shows that the introduction of quenched disorder in the form of non-magnetic impurities induces a crossover from first-order to second-order phase transition. The finite-size scaling method is used to calculate the static critical exponents α, γ, β, and ν for specific heat, susceptibility, magnetization, and correlation length, respectively.

Journal of Experimental and Theoretical Physics•Sep 1, 2009•14 citations

Influence of quenched nonmagnetic impurities on phase transitions in a three-dimensional diluted Potts model with spin state number q = 4

Akai Kurbanovich Murtazaev, Albert Babaevich Babaev, G. Ya. Aznaurova

The influence of quenched nonmagnetic impurities on phase transitions and critical phenomena in the 3D Potts model with the spin state number q = 4 is studied using the Monte Carlo method. Systems with the linear size L = 20–32 and spin concentrations p = 1.00, 0.90, 0.65 are considered. The fourth order Binder cumulant method is used to demonstrate that in the strongly diluted regime, a phase transition of the second kind is observed in this model for the spin concentration p = 0.65, and a phase transition of the first kind is observed for the pure ( p = 1.00) and weakly diluted ( p = 0.90) models. The theory of finite-dimensional scaling is used to calculate the static critical parameters of heat capacity α, susceptibility γ, magnetization β, and correlation radius ν.

Bulletin of the Russian Academy of Sciences Physics•May 1, 2011•2 citations

Investigation of the influence of quenched nonmagnetic impurities on phase transitions in the three-dimensional Potts model

Akai Kurbanovich Murtazaev, Albert Babaevich Babaev, G. Ya. Aznaurova

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.

Physics of the Solid State•Apr 1, 2008•25 citations

Monte Carlo Simulation of Strongly Disordered Ising Ferromagnets

Hans-O Heuer

We report about extensive Monte Carlo simulations of disordered Ising systems with concentrations of magnetic sites between p = 1.0 and p = 0.5. We have measured the magnetization, the susceptibility and the caloric properties in the critical region. In addition the magnetization-energy correlation function has been investigated and the critical exponent ζ = 1 - β = (α + γ)/2 has been determined. We find that power law behaviour and scaling is fulfilled in the whole concentration range. The effective critical exponents γ, β and ζ are shown to be concentration dependent. In the range of strong disorder γ and β increase until they reach a kind of asymptotic behaviour at β = 0.335 ± 0.01 and γ = 1.49 ± 0.02 for p = 0.5. In accord with the exponent relation ζ = 1 - β the critical exponent ζ decreases with dilution to ζ = 0.675 ± 0.01.

Europhysics Letters (EPL)•Jul 15, 1990•27 citations

Monte Carlo simulation of strongly disordered Ising ferromagnets

Hans-Otto Heuer

We report extensive Monte Carlo simulations of disordered Ising systems in the ferromagnetic region with concentrations of magnetic sites between p=1.0 and 0.5. The magnetization, the susceptibility, and the caloric properties have been studied in the critical region. The critical exponents \ensuremath{\beta} and \ensuremath{\gamma} as well as the universal amplitude A of the magnetization and the ratio ${\mathit{C}}_{+}$/${\mathit{C}}_{\mathrm{\ensuremath{-}}}$ of the susceptibility amplitudes have been determined with high precision. In addition to the cusplike specific heat, we have also measured the magnetization-energy correlation function \ensuremath{\Gamma}, which is a divergent thermal quantity. The corresponding critical exponent \ensuremath{\zeta}, which is related to the other exponents by \ensuremath{\zeta}=1-\ensuremath{\beta}=(\ensuremath{\alpha}+\ensuremath{\gamma})/2, has been determined. We have found that all quantities show power-law behavior within the temperature range of our simulation. All critical exponents change continuously with dilution. Even in the range of weak dilution (p\ensuremath{\ge}0.8), the effective critical exponents are concentration dependent and are clearly different from their pure system values. In the strongly diluted regime the critical exponents gradually reach new asymptotic behavior at p=0.5--0.6 with values of \ensuremath{\beta}=0.335\ifmmode\pm\else\textpm\fi{}0.01 and \ensuremath{\gamma}=1.49\ifmmode\pm\else\textpm\fi{}0.02. The exponent \ensuremath{\alpha} of the specific heat becomes -0.17\ifmmode\pm\else\textpm\fi{}0.04, which corresponds to a cusplike singularity. We conclude that disorder profoundly changes the critical behavior for weakly as well as strongly disordered spin systems.

Physical review. B, Condensed matter•Oct 1, 1990•55 citations
A review is given of Monte Carlo simulations for the phase transition in simple cubic Ising ferro-, ferri-, and antiferromagnets with quenched non-magnetic impurities of concentration x. Special emphasis is given on the description of a fully vectorized multispin coding program and on the problems related to the analysis of the Monte Carlo data. The following three subjects are discussed: 1) For ferro- and antiferromagnets, the critical behaviour of the order parameter is different from the one of the corresponding pure system, in agreement with the prediction of the Harris criterion. The possibly effective exponent β depends on x, whereas for ferrimagnets no such concentration dependence is revealed by the present study for x=0, 0.1, and 0.2.2) For strongly diluted systems, there is a maximum for the Kouvel-Fisher exponent γ(T) of the order-parameter susceptibility outside the critical regime, in accordance with the predictions of the correlated molecular field theory. 3) For the diluted antiferromagnet (x=0.5), a cusp emerges at T c for the uniform susceptibility, in agreement with the theory of Fishman and Aharony.
Using the Monte Carlo technique the authors have studied a two-dimensional Ising model with quenched non-magnetic impurities. In particular they calculated the temperature dependence of the order parameter, of the zero-field susceptibility and of the specific heat. These results indicate that a second-order phase transition is preserved for p>or=0.9 where 1-p denotes the concentration of randomly distributed non-magnetic sites. Moreover, they yield beta approximately=1/8 and gamma approximately=7/4, indicating that these exponents equal the corresponding ones for the pure Ising model.
Monte Carlo simulations are performed to analyze phase transitions in three-dimensional systems described by the 3-state Potts model with nonmagnetic impurities. Numerical results are presented for systems with spin concentrations p = 1.00, 0.95, 0.90, 0.80, 0.70, and 0.65 on lattices of size L varying between 20 and 44. Binder’s cumulant analysis shows that the introduction of quenched disorder in the form of non-magnetic impurities induces a crossover from first-order to second-order phase transition. The finite-size scaling method is used to calculate the static critical exponents α, γ, β, and ν for specific heat, susceptibility, magnetization, and correlation length, respectively.
The influence of quenched nonmagnetic impurities on phase transitions and critical phenomena in the 3D Potts model with the spin state number q = 4 is studied using the Monte Carlo method. Systems with the linear size L = 20–32 and spin concentrations p = 1.00, 0.90, 0.65 are considered. The fourth order Binder cumulant method is used to demonstrate that in the strongly diluted regime, a phase transition of the second kind is observed in this model for the spin concentration p = 0.65, and a phase transition of the first kind is observed for the pure ( p = 1.00) and weakly diluted ( p = 0.90) models. The theory of finite-dimensional scaling is used to calculate the static critical parameters of heat capacity α, susceptibility γ, magnetization β, and correlation radius ν.
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.
We report about extensive Monte Carlo simulations of disordered Ising systems with concentrations of magnetic sites between p = 1.0 and p = 0.5. We have measured the magnetization, the susceptibility and the caloric properties in the critical region. In addition the magnetization-energy correlation function has been investigated and the critical exponent ζ = 1 - β = (α + γ)/2 has been determined. We find that power law behaviour and scaling is fulfilled in the whole concentration range. The effective critical exponents γ, β and ζ are shown to be concentration dependent. In the range of strong disorder γ and β increase until they reach a kind of asymptotic behaviour at β = 0.335 ± 0.01 and γ = 1.49 ± 0.02 for p = 0.5. In accord with the exponent relation ζ = 1 - β the critical exponent ζ decreases with dilution to ζ = 0.675 ± 0.01.
We report extensive Monte Carlo simulations of disordered Ising systems in the ferromagnetic region with concentrations of magnetic sites between p=1.0 and 0.5. The magnetization, the susceptibility, and the caloric properties have been studied in the critical region. The critical exponents \ensuremath{\beta} and \ensuremath{\gamma} as well as the universal amplitude A of the magnetization and the ratio C+{\mathit{C}}_{+}C+​/${\mathit{C}}_{\mathrm{\ensuremath{-}}}$ of the susceptibility amplitudes have been determined with high precision. In addition to the cusplike specific heat, we have also measured the magnetization-energy correlation function \ensuremath{\Gamma}, which is a divergent thermal quantity. The corresponding critical exponent \ensuremath{\zeta}, which is related to the other exponents by \ensuremath{\zeta}=1-\ensuremath{\beta}=(\ensuremath{\alpha}+\ensuremath{\gamma})/2, has been determined. We have found that all quantities show power-law behavior within the temperature range of our simulation. All critical exponents change continuously with dilution. Even in the range of weak dilution (p\ensuremath{\ge}0.8), the effective critical exponents are concentration dependent and are clearly different from their pure system values. In the strongly diluted regime the critical exponents gradually reach new asymptotic behavior at p=0.5--0.6 with values of \ensuremath{\beta}=0.335\ifmmode\pm\else\textpm\fi{}0.01 and \ensuremath{\gamma}=1.49\ifmmode\pm\else\textpm\fi{}0.02. The exponent \ensuremath{\alpha} of the specific heat becomes -0.17\ifmmode\pm\else\textpm\fi{}0.04, which corresponds to a cusplike singularity. We conclude that disorder profoundly changes the critical behavior for weakly as well as strongly disordered spin systems.