Fluctuations and criticality in the random-field Ising model

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 V = L × L × L, with L 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 c and the correlation length exponent ν. 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 α and the violation of the hyperscaling exponent θ. Finally, a data collapse analysis of the order parameter and disconnected susceptibility provides accurate estimates for the critical exponent ratios β/ν andγ /ν, respectively.

Efficient numerical methods for the random-field Ising model: Finite-size scaling, reweighting extrapolation, and computation of response functions

“…Obviously, as we have no command over this value, what is usually done is to use some candidate values of the critical field around the best known estimate and then repeat the simulations for all those candidate values. However, even in this case the results are ambiguous, as a change in the value of σ (c) by a factor of δσ (c) = 10 −3 results, on a average, in a change of the order of δα ≈ 0.04 in the value of α [112]. Following the discussion above, our numerical studies of disordered systems are carried out near their critical points using finite samples; each sample is a particular random realization of the quenched disorder.…”

Random-field Ising model: Insight from zero-temperature simulations

“…Hence, the critical behavior is the same everywhere along the phase boundary of figure 1, and we can predict it simply by staying at T = 0 and crossing the phase boundary at h = h c . This is a convenient approach, because we can determine the ground states of the system exactly using efficient optimization algorithms [20,21,25,65,66,[71][72][73][74][75][76] through an existing mapping of the ground state to the maximum-flow optimization problem [77]. A clear advantage of this approach is the ability to simulate large system sizes and disorder ensembles in rather moderate computational times.…”

“…Periodic global updates are often crucial to the practical speed of the algorithm [79,80]. Following the suggestions of references [21,79,80], we have also applied global updates here every V relabels, a practice found to be computationally optimum [25,76,79,80]. Using this scheme we performed large-scale simulations of the RFIM with both type of distributions discussed above in section 1.…”

“…Currently, even the well-known correspondence among the RFIM and its experimental analogue, the diluted antifferomagnet in a field (DAFF), has been severely questioned by extensive simulations performed on both models at positive-and zero-temperature [27]. In any case, the whole issue of the model's critical behavior is still under intense investigation [20][21][22][23][24][25][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43].…”

“…The criteria for determining the order of the low-temperature phase transition and its dependence on the form of the field distribution have been discussed throughout the years [15][16][17][18][19]. Although the view that the phase transition of the RFIM is nowadays considered to be of second order [20][21][22][23][24][25], the extremely small value of the exponent β continues to cast some doubts. Moreover, a rather strong debate with regard to the role of disorder, i.e., the dependence, or not, of the critical exponents on the particular choice of the distribution for the random fields and the value of the disorder strength, analogously to the mean-field theory predictions [15][16][17], was only recently put on a different basis [26].…”