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

Europhysics Letters (EPL)

Monte Carlo Simulation of Disordered 2-Dimensional Ising Systems

Hans-O Heuer

Published Oct 1, 1991• 22 citations
Publisher

Abstract

We report extensive Monte Carlo simulations of the site-disordered 2-dimensional Ising system. For concentrations between p = 0.6 and p = 1.0 ( p c = 0.59) we have studied the magnetic and caloric properties as well as the cumulants of the magnetization distribution in the critical region. We have found that disorder induces a strong increase of γ eff (γ max = 2.16 for p = 0.7) in the usual experimental temperature range ( T - T c )/ T c = 10 -2 ÷ 10 0 . Asymptotically near T c , γ approaches the theoretical value γ = 1.75 independent of the concentration. We propose a model of weakly coupled compact clusters of spins to describe this crossover behaviour of disordered spin systems.

DOI
10.1209/0295-5075/16/5/016

Related papers

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

Critical dynamics of disordered two-dimensional Ising systems: a Monte Carlo study

Vladimir V. Prudnikov, O. N. Markov

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.

Journal of Physics A Mathematical and General•Mar 23, 1995•5 citations

Critical crossover phenomena in disordered Ising systems

Hans-O Heuer

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 10 15 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.

Journal of Physics A Mathematical and General•Mar 21, 1993•70 citations

Monte Carlo simulations in a disordered binary Ising model

Dorilson Silva Cambui, A.S. de Arruda, Maurício Godoy

In this work we study a disordered binary Ising model on the square lattice. The model system consists of two different particles with spin-1/2 and spin-1, which are randomly distributed on the lattice. It has been considered only spin nearest-neighbor exchange interactions with $J>0$. This system can represent a disordered magnetic binary alloy $A_{x}B_{1-x}$, obtained from the high temperature quenching of a liquid mixture. The results were obtained by the use of Monte Carlo simulations for several lattice sizes $L$, temperature $T$ and concentration $x$ of ions $A$ with spin-1/2. We found its critical temperature, through the reduced fourth-order Binder cumulant for the several values of the concentration $x$ of particles (spin-1/2, spin-1), and also the magnetization, the susceptibility and the specific heat as a function of temperature $T$.

arXiv:1109.3797•Sep 17, 2011•1 citation

The Two-Dimensional Random Bond Ising Model at Criticality—A Monte Carlo Study

Jian‐Sheng Wang, Walter Selke, Vl.S. Dotsenko +1 more

The two-dimensional random bond Ising model with two distinct ferromagnetic nearest-neighbour couplings, J and J ', distributed randomly over the lattice with concentration 1/2 is studied for various ratios J '/ J at criticality. The critical temperature is exactly known from self-duality. The size-dependent specific heat shows a crossover from the perfect Ising to the randomness-dominated critical behaviour of doubly logarithmic form. In the latter region, our estimates for the critical exponents of the size-dependent magnetization and susceptibility are in very good agreement with analytic results suggesting β/ν = 1/8 and γ/ν = 7/4.

Europhysics Letters (EPL)•Feb 15, 1990•74 citations
We report extensive Monte Carlo simulations of the site-disordered 2-dimensional Ising system. For concentrations between p = 0.6 and p = 1.0 ( p c = 0.59) we have studied the magnetic and caloric properties as well as the cumulants of the magnetization distribution in the critical region. We have found that disorder induces a strong increase of γ eff (γ max = 2.16 for p = 0.7) in the usual experimental temperature range ( T - T c )/ T c = 10 -2 ÷ 10 0 . Asymptotically near T c , γ approaches the theoretical value γ = 1.75 independent of the concentration. We propose a model of weakly coupled compact clusters of spins to describe this crossover behaviour of disordered spin systems.
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.
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 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 10 15 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.
In this work we study a disordered binary Ising model on the square lattice. The model system consists of two different particles with spin-1/2 and spin-1, which are randomly distributed on the lattice. It has been considered only spin nearest-neighbor exchange interactions with J>0J>0J>0. This system can represent a disordered magnetic binary alloy AxB1−xA_{x}B_{1-x}Ax​B1−x​, obtained from the high temperature quenching of a liquid mixture. The results were obtained by the use of Monte Carlo simulations for several lattice sizes LLL, temperature TTT and concentration xxx of ions AAA with spin-1/2. We found its critical temperature, through the reduced fourth-order Binder cumulant for the several values of the concentration xxx of particles (spin-1/2, spin-1), and also the magnetization, the susceptibility and the specific heat as a function of temperature TTT.
The two-dimensional random bond Ising model with two distinct ferromagnetic nearest-neighbour couplings, J and J ', distributed randomly over the lattice with concentration 1/2 is studied for various ratios J '/ J at criticality. The critical temperature is exactly known from self-duality. The size-dependent specific heat shows a crossover from the perfect Ising to the randomness-dominated critical behaviour of doubly logarithmic form. In the latter region, our estimates for the critical exponents of the size-dependent magnetization and susceptibility are in very good agreement with analytic results suggesting β/ν = 1/8 and γ/ν = 7/4.