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Journal of Statistical Mechanics Theory and Experiment

Dynamical Monte Carlo studies of the three-dimensional bimodal random-field Ising model

Wanjie Xiong, Yafei Dai

Published May 16, 2012• 6 citations
Publisher

Abstract

The nature of the three-dimensional random-field Ising model with a bimodal probability distribution is investigated using finite-time scaling combined with the Monte Carlo renormalization group method, in the presence of a linearly varying temperature. Our results support the existence of a first-order phase transition for this model, obtained through reducing the influence of finite-size effects. The critical exponents are estimated to be ν = 0.74(3), β = 0.27(1), α = − 0.023(5), and γ = 1.48(3) with corrections to the scaling. The Rushbrooke scaling law is satisfied with these exponents, and in turn identifies the asymptotic critical behavior.

DOI
10.1088/1742-5468/2012/05/p05018

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The nature of the three-dimensional random-field Ising model with a bimodal probability distribution is investigated using finite-time scaling combined with the Monte Carlo renormalization group method, in the presence of a linearly varying temperature. Our results support the existence of a first-order phase transition for this model, obtained through reducing the influence of finite-size effects. The critical exponents are estimated to be ν = 0.74(3), β = 0.27(1), α = − 0.023(5), and γ = 1.48(3) with corrections to the scaling. The Rushbrooke scaling law is satisfied with these exponents, and in turn identifies the asymptotic critical behavior.
We investigate the low-temperature critical behavior of the three-dimensional random-field Ising ferromagnet. By a scaling analysis we find that in the limit of temperature T→0 the usual scaling relations have to be modified as far as the exponent α of the specific heat is concerned. At zero temperature, the Rushbrooke equation is modified to α+2β+γ=1, an equation which we expect to be valid also for other systems with similar critical behavior. We test the scaling theory numerically for the three-dimensional random-field Ising system with Gaussian probability distribution of the random fields by a combination of calculations of exact ground states with an integer optimization algorithm and Monte Carlo methods. By a finite-size scaling analysis we calculate the critical exponents ν≈1.0, β≈0.05, γ̄≈2.9, γ≈1.5 and α≈−0.55.
Abstract We present a dynamical numerical study on a three-dimensional (3D) Gaussian distribution random-field Ising model in a wide range of disorder strengths by means of the finite-time dynamics method. By validating the Rushbrooke scaling law and analyzing the variation of exponents with disorder amplitudes, the asymptotic critical regime controlled by a hysteresis fixed point is identified with exponents , , and . The phase transition is found to be continuous and there is not a tricritical point in this version of Gaussian distribution. Moreover, our results support that the hysteresis and zero-temperature fixed points belong to a different universality class.
We have investigated the critical behavior of a three-dimensional random-bond Ising model for a series of the disorder strength by a finite-time scaling combining with Monte Carlo renormalization-group method in the presence of a linearly varying temperature. The method enables us to estimate a lot of critical exponents of both static and dynamic nature independently as well as the critical temperatures. The static exponents obtained agree well with most existing results, verify both the hyperscaling and the Rushbrooke scaling laws and their combined scaling law, which in turn validate their asymptotic nature, and corroborate the universality of the relevant random fixed point with respect to the forms of disorder. The dynamic critical exponent z is estimated to 2.114(51), which is compatible with those obtained from experiments and renormalization-group analyses. The exponents at low and high disorder strengths do not satisfy all scaling laws and are argued to be crossover exponents that reflect crossover from the random fixed point to the pure and the percolation fixed point. They also indicate that the exponents that were previously suggested to be a distinct universality class for strong disorder strength in the literature may be just crossover. Our results demonstrate the effectiveness of the finite-time scaling method.
In extensive Monte Carlo simulations the phase transition of the random-field Ising model in three dimensions is investigated. The values of the critical exponents are determined via finite-size scaling. For a Gaussian distribution of the random fields it is found that the correlation length \ensuremath{\xi} diverges with an exponent \ensuremath{\nu}=1.1\ifmmode\pm\else\textpm\fi{}0.2 at the critical temperature and that \ensuremath{\chi}\ensuremath{\sim}${\ensuremath{\xi}}^{2\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$ with \ensuremath{\eta}=0.50\ifmmode\pm\else\textpm\fi{}0.05 for the connected susceptibility and ${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$\ensuremath{\sim}${\ensuremath{\xi}}^{4\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$ with \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=1.03\ifmmode\pm\else\textpm\fi{}0.05 for the disconnected susceptibility. Together with the amplitude ratio A=${\mathrm{lim}}_{\mathit{T}\ensuremath{\rightarrow}\mathit{T}\mathit{c}}$${\mathrm{\ensuremath{\chi}}}_{\mathrm{dis}}$/${\mathrm{\ensuremath{\chi}}}^{2}$(hr{\mathit{h}}_{\mathit{r}}hr​/T)2{)}^{2})2 being close to one this gives further support for a two-exponent scaling scenario implying \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\eta}. The magnetization behaves discontinuously at the transition, i.e., \ensuremath{\beta}=0. However, no divergence for the specific heat and in particular no latent heat is found. Also the probability distribution of the magnetization does not show a multipeak structure that would be characteristic for the phase-coexistence at first-order phase-transition points.
The critical behavior of the random-field Ising model with a bimodal field distribution is studied using standard and histogram Monte Carlo calculations. It is definitely found that the transition is second order for weak fields while it becomes first order for higher fields. The existence of this crossover, discovered here in Monte Carlo simulations, is in contradiction with earlier Monte Carlo works, but in agreement with mean-field predictions. Estimates of the critical exponents of the model at low field are given.
Using a cluster-flipping Monte Carlo algorithm combined with a generalization of the histogram reweighting scheme of Ferrenberg and Swendsen [Phys. Rev. Lett. 61, 2635 (1988); 63, 1195 (1989)], we have studied the equilibrium properties of the thermal random-field Ising model on a cubic lattice in three dimensions. We have equilibrated systems of L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L spins, with values of L up to 32, and for these systems the cluster-flipping method appears to a large extent to overcome the slow equilibration seen in single-spin-flip methods. From the results of our simulations we have extracted values for the critical exponents and the critical temperature and randomness of the model by finite size scaling. For the exponents we find \ensuremath{\nu}=1.02\ifmmode\pm\else\textpm\fi{}0.06, \ensuremath{\beta}=0.06\ifmmode\pm\else\textpm\fi{}0.07, \ensuremath{\gamma}=1.9\ifmmode\pm\else\textpm\fi{}0.2, and \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2.9\ifmmode\pm\else\textpm\fi{}0.2, where \ensuremath{\nu}, \ensuremath{\beta}, \ensuremath{\gamma}, and \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{} govern the critical singularities in the correlation length, magnetization, and connected and disconnected susceptibilities, respectively. \textcopyright{} 1996 The American Physical Society.