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Physical Review E

Fluctuations and criticality in the random-field Ising model

Panagiotis E. Theodorakis, Ioannis Georgiou, Nikolaos G. Fytas

Published Mar 8, 2013• 11 citations
Publisher

Abstract

We investigate the critical properties of the $d=3$ random-field Ising model with a Gaussian field distribution at zero temperature. By implementing suitable graph-theoretical algorithms, we perform a large-scale numerical simulation of the model for a vast range of values of the disorder strength $h$ and system sizes $\mathcal{V}=L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$, with $L\ensuremath{\le}156$. Using the sample-to-sample fluctuations of various quantities and proper finite-size scaling techniques we estimate with high accuracy the critical disorder strength ${h}_{\mathrm{c}}$ and the correlation length exponent $\ensuremath{\nu}$. Additional simulations in the area of the estimated critical-field strength and relevant scaling analysis of the bond energy suggest bounds for the specific heat critical exponent $\ensuremath{\alpha}$ and the violation of the hyperscaling exponent $\ensuremath{\theta}$. Finally, a data collapse analysis of the order parameter and disconnected susceptibility provides accurate estimates for the critical exponent ratios $\ensuremath{\beta}/\ensuremath{\nu}$ and $\overline{\ensuremath{\gamma}}/\ensuremath{\nu}$, respectively.

DOI
10.1103/physreve.87.032119

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We enlighten some critical aspects of the three-dimensional (d = 3) random-field Ising model from simulations performed at zero temperature.We consider two different, in terms of the field distribution, versions of model, namely a Gaussian random-field Ising model and an equal-weight trimodal random-field Ising model.By implementing a computational approach that maps the ground-state of the system to the maximum-flow optimization problem of a network, we employ the most up-to-date version of the push-relabel algorithm and simulate large ensembles of disorder realizations of both models for a broad range of random-field values and systems sizes V = L × L × L, where L denotes linear lattice size and L max = 156.Using as finite-size measures the sampleto-sample fluctuations of various quantities of physical and technical origin, and the primitive operations of the push-relabel algorithm, we propose, for both types of distributions, estimates of the critical field h c and the critical exponent ν of the correlation length, the latter clearly suggesting that both models share the same universality class.Additional simulations of the Gaussian random-field Ising model at the best-known value of the critical field provide the magnetic exponent ratio β/ν with high accuracy and clear out the controversial issue of the critical exponent α of the specific heat.Finally, we discuss the infinite-limit size extrapolation of energyand order-parameter-based noise to signal ratios related to the self-averaging properties of the model, as well as the critical slowing down aspects of the algorithm.
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We investigate the one-dimensional long-range random-field Ising magnet with Gaussian distribution of the random fields. In this model, a ferromagnetic bond between two spins is placed with a probability $p\ensuremath{\sim}{r}^{\ensuremath{-}1\ensuremath{-}\ensuremath{\sigma}}$, where rrr is the distance between these spins and $\ensuremath{\sigma}$ is a parameter to control the effective dimension of the model. Exact ground states at zero temperature are calculated for system sizes up to L=219L={2}^{19}L=219 via graph theoretical algorithms for four different values of $\ensuremath{\sigma}\ensuremath{\in}{0.25,0.4,0.5,1.0}$ while varying the strength hhh of the random fields. For each of these values several independent physical observables are calculated, i.e., magnetization, Binder parameter, susceptibility, and a specific-heat-like quantity. The ferromagnet-paramagnet transitions at critical values ${h}_{c}(\ensuremath{\sigma})$ as well as the corresponding critical exponents are obtained. The results agree well with theory, and interestingly we find for $\ensuremath{\sigma}=1/2$ the data is compatible with a critical random-field strength hc>0{h}_{c}>0hc​>0.
The disorder-driven phase transition of the random-field Ising model is observed using exact ground-state computer simulations for hypercubic lattices in d=5,6,7d=5,6,7d=5,6,7 dimensions. Finite-size scaling analyses are used to calculate the critical point and the critical exponents of the specific heat, magnetization, susceptibility, and of the correlation length. For dimensions d=6,7d=6,7d=6,7 which are larger or equal to the assumed upper critical dimension, du=6{d}_{u}=6du​=6, mean-field behavior is found, i.e., $\ensuremath{\alpha}=0, \ensuremath{\beta}=1/2, \ensuremath{\gamma}=1, \ensuremath{\nu}=1/2$. For the analysis of the numerical data, it appears to be necessary to include recently proposed corrections to scaling at and beyond the upper critical dimension.