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arXiv:cond-mat/9303045 · Journal of Physics A Mathematical and General

Critical exponents of the three-dimensional random field Ising model

Heiko Rieger, A. P. Young

Published Oct 21, 1993• 109 citations
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Abstract

The phase transition of the three-dimensional random field Ising model with a discrete (+or-h) field distribution is investigated by extensive Monte Carlo simulations. Values of the critical exponents for the correlation length, specific heat, susceptibility, disconnected susceptibility and magnetization are determined simultaneously via finite size scaling. While the magnetization appears to be discontinuous, the specific heat appears to saturate, indicating no latent heat. Sample-to-sample fluctuations of the susceptibility are consistent with the droplet picture for the transition.

DOI
10.1088/0305-4470/26/20/014

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The phase transition of the three-dimensional random field Ising model with a discrete (+or-h) field distribution is investigated by extensive Monte Carlo simulations. Values of the critical exponents for the correlation length, specific heat, susceptibility, disconnected susceptibility and magnetization are determined simultaneously via finite size scaling. While the magnetization appears to be discontinuous, the specific heat appears to saturate, indicating no latent heat. Sample-to-sample fluctuations of the susceptibility are consistent with the droplet picture for the transition.
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 ferromagnet-to-paramagnet transition of the four-dimensional random-field Ising model with Gaussian distribution of the random fields is studied. Exact ground states of systems with sizes up to 324{32}^{4}324 are obtained using graph theoretical algorithms. The magnetization, the disconnected susceptibility, the susceptibility, and a specific-heat-like quantity are calculated. Using finite-size scaling techniques, the corresponding critical exponents are obtained: $\ensuremath{\beta}=0.13(5),$ $\overline{\ensuremath{\gamma}}=2.83(50),$ $\ensuremath{\gamma}=1.42(20).$ For the specific heat both $\ensuremath{\alpha}=0$ (logarithmic divergence) and an algebraic divergence with $\ensuremath{\alpha}=0.26(5)$ are compatible with the data. Furthermore, values for the critical randomness hc=4.18(1){h}_{c}=4.18(1)hc​=4.18(1) and the correlation-length exponent $\ensuremath{\nu}=0.78(10)$ were found. These independently obtained exponents are compatible with all (hyper)scaling relations and support the two-exponent scenario $(\overline{\ensuremath{\gamma}}=2\ensuremath{\gamma}).$
We present new numerical simulations of the random field Ising model in three dimensions at zero temperature. The critical exponents are found to agree with previous results. We study the magnetic susceptibility by applying a small magnetic field perturbation. We find the critical amplitude ratio of the magnetic susceptibilities to be very large, equal to 233.1 ± 1.5. We find strong sample to sample fluctuations which obey finite size scaling. The probability distribution of the size of small energy excitations is maximally non-self-averaging, obeying a double peak distribution, and is finite size scaling invariant. We also study the approach to the thermodynamic limit of the ground state magnetization at the phase transition.
We investigate the critical properties of the d=3d=3d=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 hhh 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 hc{h}_{\mathrm{c}}hc​ 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.
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.
We study numerically the region above the critical temperature of the four-dimensional random field Ising model. Using a cluster dynamic we measure the connected and disconnected magnetic susceptibility and the connected and disconnected overlap susceptibility. We use a bimodal distribution of the field with for all temperatures and a lattice size L = 16. Through a least-square fit we determine the critical temperature at which the two susceptibilities diverge. We also determine the critical exponents and . We find that the magnetic susceptibility and the overlap susceptibility diverge at two different temperatures. This is coherent with the existence of a glassy phase above . Accordingly with other simulations we find . In this case we have a scaling theory with two independent critical exponents.