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Physica A Statistical Mechanics and its Applications

SCALING ANALYSIS OF THE d = 3 RANDOM FIELD ISING MODEL

Laurence J. Jacobs, Michael Nauenberg

Published Jan 1, 1983• 0 citations
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Abstract

We present a Monte Carlo simulation of the d = 3 Ising model in a random magnetic field with values ±H on each lattice site. Calculations have been performed on cubic lattices with linear dimension L = 6, 8, 10 and 12, and several values of H. A finite-size scaling analysis of the data for H = 0.5, 0.75 and 1.0 is consistent with hyperscaling, and a correlation exponent v = 0.7 ± 0.1. For H = 1.50, 1.75 and 2.0 some of the L = 12 data show a discontinuity as a function of the inverse temperature β. The implications of these results are discussed.

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We present a Monte Carlo simulation of the d = 3 Ising model in a random magnetic field with values ±H on each lattice site. Calculations have been performed on cubic lattices with linear dimension L = 6, 8, 10 and 12, and several values of H. A finite-size scaling analysis of the data for H = 0.5, 0.75 and 1.0 is consistent with hyperscaling, and a correlation exponent v = 0.7 ± 0.1. For H = 1.50, 1.75 and 2.0 some of the L = 12 data show a discontinuity as a function of the inverse temperature β. The implications of these results are discussed.
We report extensive Monte Carlo simulations and finite-size scaling analysis of the correlation length (${\ensuremath{\xi}}_{\mathrm{L}}$) for the spin-spin correlation function and the renormalized coupling (gL{\mathrm{g}}_{\mathrm{L}}gL​) of the simple-cubic Ising model with a wide range of lattice sizes (L\ensuremath{\le}96). New methods of analysis which avoid assumptions related to the correlation length were introduced. The exponents \ensuremath{\nu} and 2-\ensuremath{\alpha} were estimated from separate analysis of ${\ensuremath{\xi}}_{\mathrm{L}}$ and gL{\mathrm{g}}_{\mathrm{L}}gL​, respectively. A value of ${\mathrm{\ensuremath{\omega}}}^{\mathrm{*}}$\ensuremath{\approxeq}0.001\ifmmode\pm\else\textpm\fi{}0.07 for the Fisher exponent is obtained. This is consistent with a very small or no violations of hyperscalings.
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.
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.
We report single-cluster Monte Carlo simulations of the Ising model on three-dimensional Poissonian random lattices with up to $128000\ensuremath{\approx}{50}^{3}$ sites which are linked together according to the Voronoi-Delaunay prescription. For each lattice size quenched averages are performed over 96 realizations. By using reweighting techniques and finite-size scaling analyses we investigate the critical properties of the model in the close vicinity of the phase transition point. Our random lattice data provide strong evidence that, for the available system sizes, the resulting effective critical exponents are indistinguishable from recent high-precision estimates obtained in Monte Carlo studies of the Ising model and ${\ensuremath{\varphi}}^{4}$ field theory on three-dimensional regular cubic lattices.
The three-state Potts model with nearest ferromagnetic interaction J on the simple cubic lattice, exposed to a field which randomly favours one of the states, is studied by finite size scaling analysis of Monte Carlo simulations. The consequences of the hyperscaling violations for the finite size scaling description are worked out, and it is shown that at criticality the order parameter distribution develops a trivial character (superposition of three delta functions), unlike the nontrivial distributions in models where hyperscaling holds. The numerical data are compatible with this interpretation, and are also compatible with a description in terms of only two independent static critical exponents, as proposed by Schwartz et al . Slow relaxation and the need of averaging over a large number of random field samples restricts the range of linear dimensions L to , and in view of crossover problems only rough estimates of critical exponents are obtained.
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.