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

Physical review. B, Condensed matter

Critical behavior of Ising models with static site dilution

Joaquín Marro, A. Labarta, Javier Tejada

Published Jul 1, 1986• 37 citations
Publisher

Abstract

A Monte Carlo analysis of the ferromagnetic spin-1/2 Ising model with nearest-neighbor interactions and static (quenched) site dilution, where a fraction x, 0\ensuremath{\le}x\ensuremath{\le}0.2, of the spins in a simple-cubic lattice are randomly replaced by vacant sites, reveals effective critical exponents varying continuously with x.
DOI
10.1103/physrevb.34.347

Related papers

Critical behavior of the simple cubic Ising model with quenched site impurities

David Paul Landau

The thermal and magnetic properties of the simple cubic Ising model with random, quenched, site impurities have been determined by Monte Carlo calculations. The transition appears to remain sharp, and the critical temperature decreases monotonically towards zero as the impurity concentration is increased towards the percolation limit. No clear evidence is found for any change in the critical exponents away from the pure lattice values. These results are discussed in terms of Harris's predictions for the critical behavior of impure systems.

Physical review. B, Condensed matter•Sep 1, 1980•79 citations

Phase transition in the Ising ferromagnetic model with fixed spins

A. Labarta, Joaquín Marro, Benjamín Martínez +1 more

The ferromagnetic Ising spin-(1/2 model in a finite simple-cubic lattice \ensuremath{\Lambda} is studied by Monte Carlo methods when two subsets of the lattice sites in \ensuremath{\Lambda}, say ${\ensuremath{\Omega}}^{+}$ and ${\ensuremath{\Omega}}^{\mathrm{\ensuremath{-}}}$, contain (the same number of) spins fixed at \ifmmode\pm\else\textpm\fi{}1, respectively, the global defect concentration being x\ensuremath{\le}0.25. We study the thermodynamic properties of the model for different choices of \ensuremath{\Omega}=${\ensuremath{\Omega}}^{+}$\ensuremath{\cup}${\ensuremath{\Omega}}^{\mathrm{\ensuremath{-}}}$. A finite-size-scaling analysis reveals that the transition remains second order with pure critical exponents for regularly spaced defects, the critical temperature varying with the symmetry of \ensuremath{\Omega}. Any small randomness in \ensuremath{\Omega}, however, makes the transition weakly first order; the transition becomes more abrupt for defects located fully at random, and the long-range order is suppressed when the numbers of defects in ${\ensuremath{\Omega}}^{+}$ and ${\ensuremath{\Omega}}^{\mathrm{\ensuremath{-}}}$ differ from each other. We also discuss our findings in relation to the random-field and frustration problems.

Physical review. B, Condensed matter•Jul 1, 1988•3 citations

Finite-size-scaling analysis of subsystem data in the dilute Ising model

Michael Hennecke

Monte Carlo simulation results for the magnetization of subsystems of finite lattices are used to determine the critical temperature and a critical exponent of the simple-cubic Ising model with quenched site dilution, at a concentration of p=40%. Particular attention is paid to the effect of the finite size of the systems from which the subsystem results are obtained. This finiteness of the lattices involved is shown to be a source of large deviations of critical temperatures and exponents estimated from subsystem data from their values in the thermodynamic limit. By the use of different lattice sizes, the results ${\mathit{T}}_{\mathit{c}}$(40%)=1.209\ifmmode\pm\else\textpm\fi{}0.002 and \ensuremath{\nu}(40%)=0.78\ifmmode\pm\else\textpm\fi{}0.01 could be extrapolated.

Physical review. B, Condensed matter•Sep 1, 1993•20 citations

Critical behavior of a 3-D Ising model in a random field

David Paul Landau, H. H. Lee, W. P. Kao

A Monte Carlo method has been used to study a simple cubic Ising ferromagnet in a random quenched magnetic field. The Hamiltonian for this model is ℋ==JΣ(ij)σiσj−ΣiHiσi, where σi,σj=±1, J is the nearest-neighbor interaction constant, and the field Hi=tH is fixed at each site with ti=±1 at random and Σti=0. L×L×L lattices with periodic boundary conditions have been studied for a range of H and T. As expected we find a ferromagnetic ordered state which for small H undergoes a second order phase change to the paramagnetic state with increasing temperature. A finite size scaling analysis of the preliminary data suggests that the critical exponent β is substantially smaller (β∼0.2) than the usual 3-dim Ising value of 0.31. Results obtained for small lattices indicate that below kT/J-2 the transition becomes first order suggesting that a tricritical point appears on the critcal field curve.

Journal of Applied Physics•Mar 1, 1978•66 citations

Short-time dynamics and critical behavior of the three-dimensional site-diluted Ising model

Vladimir V. Prudnikov, Pavel Vladimirovich Prudnikov, Aleksandr Krinitsyn +3 more

Monte Carlo simulations of the short-time dynamic behavior are reported for three-dimensional weakly site-diluted Ising model with spin concentrations p=0.95 and 0.8 at criticality. In contrast to studies of the critical behavior of the pure systems by the short-time dynamics method, our investigations of site-diluted Ising model have revealed three stages of the dynamic evolution characterizing a crossover phenomenon from the critical behavior typical for the pure systems to behavior determined by the influence of disorder. The static and dynamic critical exponents are determined with the use of the corrections to scaling for systems starting separately from ordered and disordered initial states. The obtained values of the exponents demonstrate a universal behavior of weakly site-diluted Ising model in the critical region. The values of the exponents are compared to results of numerical simulations which have been obtained in various works and, also, with results of the renormalization-group description of this model.

arXiv:1005.5218•Physical Review E•Jan 26, 2010•44 citations

Monte Carlo renormalization-group study of the site-diluted simple-cubic Ising model

T. Holey, M. Fähnle

The Monte Carlo renormalization-group approach is applied to investigate the critical behavior of the simple-cubic Ising model diluted randomly with nonmagnetic atoms of concentration x. For x=0.2 a value of \ensuremath{\nu}=0.688(13) is found which is well above the result \ensuremath{\nu}=0.629(4) for the pure model and above the exact lower bound of \ensuremath{\nu}=2/3, and the relative width of the asymptotic regime is definitely larger than 3\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}4}$. For x=0.1 the obtained value is below the exact lower bound, possibly indicating that the asymptotic critical regime has not been penetrated, that this regime has a width smaller than 1.3\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}4}$, and that it can hardly be explored by experiments or computer simulations.

Physical review. B, Condensed matter•Jun 1, 1990•28 citations
The thermal and magnetic properties of the simple cubic Ising model with random, quenched, site impurities have been determined by Monte Carlo calculations. The transition appears to remain sharp, and the critical temperature decreases monotonically towards zero as the impurity concentration is increased towards the percolation limit. No clear evidence is found for any change in the critical exponents away from the pure lattice values. These results are discussed in terms of Harris's predictions for the critical behavior of impure systems.
The ferromagnetic Ising spin-(1/2 model in a finite simple-cubic lattice \ensuremath{\Lambda} is studied by Monte Carlo methods when two subsets of the lattice sites in \ensuremath{\Lambda}, say ${\ensuremath{\Omega}}^{+}$ and ${\ensuremath{\Omega}}^{\mathrm{\ensuremath{-}}}$, contain (the same number of) spins fixed at \ifmmode\pm\else\textpm\fi{}1, respectively, the global defect concentration being x\ensuremath{\le}0.25. We study the thermodynamic properties of the model for different choices of \ensuremath{\Omega}=${\ensuremath{\Omega}}^{+}$\ensuremath{\cup}${\ensuremath{\Omega}}^{\mathrm{\ensuremath{-}}}$. A finite-size-scaling analysis reveals that the transition remains second order with pure critical exponents for regularly spaced defects, the critical temperature varying with the symmetry of \ensuremath{\Omega}. Any small randomness in \ensuremath{\Omega}, however, makes the transition weakly first order; the transition becomes more abrupt for defects located fully at random, and the long-range order is suppressed when the numbers of defects in ${\ensuremath{\Omega}}^{+}$ and ${\ensuremath{\Omega}}^{\mathrm{\ensuremath{-}}}$ differ from each other. We also discuss our findings in relation to the random-field and frustration problems.
Monte Carlo simulation results for the magnetization of subsystems of finite lattices are used to determine the critical temperature and a critical exponent of the simple-cubic Ising model with quenched site dilution, at a concentration of p=40%. Particular attention is paid to the effect of the finite size of the systems from which the subsystem results are obtained. This finiteness of the lattices involved is shown to be a source of large deviations of critical temperatures and exponents estimated from subsystem data from their values in the thermodynamic limit. By the use of different lattice sizes, the results Tc{\mathit{T}}_{\mathit{c}}Tc​(40%)=1.209\ifmmode\pm\else\textpm\fi{}0.002 and \ensuremath{\nu}(40%)=0.78\ifmmode\pm\else\textpm\fi{}0.01 could be extrapolated.
A Monte Carlo method has been used to study a simple cubic Ising ferromagnet in a random quenched magnetic field. The Hamiltonian for this model is ℋ==JΣ(ij)σiσj−ΣiHiσi, where σi,σj=±1, J is the nearest-neighbor interaction constant, and the field Hi=tH is fixed at each site with ti=±1 at random and Σti=0. L×L×L lattices with periodic boundary conditions have been studied for a range of H and T. As expected we find a ferromagnetic ordered state which for small H undergoes a second order phase change to the paramagnetic state with increasing temperature. A finite size scaling analysis of the preliminary data suggests that the critical exponent β is substantially smaller (β∼0.2) than the usual 3-dim Ising value of 0.31. Results obtained for small lattices indicate that below kT/J-2 the transition becomes first order suggesting that a tricritical point appears on the critcal field curve.
Monte Carlo simulations of the short-time dynamic behavior are reported for three-dimensional weakly site-diluted Ising model with spin concentrations p=0.95 and 0.8 at criticality. In contrast to studies of the critical behavior of the pure systems by the short-time dynamics method, our investigations of site-diluted Ising model have revealed three stages of the dynamic evolution characterizing a crossover phenomenon from the critical behavior typical for the pure systems to behavior determined by the influence of disorder. The static and dynamic critical exponents are determined with the use of the corrections to scaling for systems starting separately from ordered and disordered initial states. The obtained values of the exponents demonstrate a universal behavior of weakly site-diluted Ising model in the critical region. The values of the exponents are compared to results of numerical simulations which have been obtained in various works and, also, with results of the renormalization-group description of this model.
The Monte Carlo renormalization-group approach is applied to investigate the critical behavior of the simple-cubic Ising model diluted randomly with nonmagnetic atoms of concentration x. For x=0.2 a value of \ensuremath{\nu}=0.688(13) is found which is well above the result \ensuremath{\nu}=0.629(4) for the pure model and above the exact lower bound of \ensuremath{\nu}=2/3, and the relative width of the asymptotic regime is definitely larger than 3\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}4}$. For x=0.1 the obtained value is below the exact lower bound, possibly indicating that the asymptotic critical regime has not been penetrated, that this regime has a width smaller than 1.3\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}4}$, and that it can hardly be explored by experiments or computer simulations.