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arXiv:cond-mat/9704088 · Europhysics Letters (EPL)

The 3d random field Ising model at zero temperature

J C Anglès d'Auriac, Nicolas Sourlas

Published Sep 1, 1997• 69 citations
arXivHTMLPDF

Abstract

We study numerically the zero-temperature random field Ising model on cubic lattices of various linear sizes L in three dimensions. For each random field configuration we vary the ferromagnetic coupling strength J . We find that in the infinite volume limit the magnetization is discontinuous in J . The energy and its first J derivative are continuous. The approach to the thermodynamic limit is slow, behaving like L − p with p ∼ 0.8 for the Gaussian distribution of the random field. We also study the bimodal distribution h i = ± h , and we find similar results for the magnetization but with a different value of the exponent p ∼ 0.6. This raises the question of the validity of universality for the random field problem.

DOI
10.1209/epl/i1997-00379-x

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We study numerically the zero-temperature random field Ising model on cubic lattices of various linear sizes L in three dimensions. For each random field configuration we vary the ferromagnetic coupling strength J . We find that in the infinite volume limit the magnetization is discontinuous in J . The energy and its first J derivative are continuous. The approach to the thermodynamic limit is slow, behaving like L − p with p ∼ 0.8 for the Gaussian distribution of the random field. We also study the bimodal distribution h i = ± h , and we find similar results for the magnetization but with a different value of the exponent p ∼ 0.6. This raises the question of the validity of universality for the random field problem.
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The random field Ising model in three dimensions with Gaussian random fields is studied at zero temperature for system sizes up to 603.{60}^{3}.603. For each realization of the normalized random fields, the strength of the random field, $\ensuremath{\Delta}$ and a uniform external, H is adjusted to find the finite-size critical point. The finite-size critical point is identified as the point in the $H\ensuremath{-}\ensuremath{\Delta}$ plane where three degenerate ground states have the largest discontinuities in the magnetization. The discontinuities in the magnetization and bond energy between these ground states are used to calculate the magnetization and specific heat critical exponents and both exponents are found to be near zero.
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.
The three-dimensional bimodal random-field Ising model is investigated using the N-fold version of the Wang-Landau algorithm. The essential energy subspaces are determined by the recently developed critical minimum energy subspace technique, and two implementations of this scheme are utilized. The random fields are obtained from a bimodal discrete (±Δ) distribution, and we study the model for various values of the disorder strength Δ, Δ=0.5,1,1.5 and 2, on cubic lattices with linear sizes L=4–24. We extract information for the probability distributions of the specific heat peaks over samples of random fields. This permits us to obtain the phase diagram and present the finite-size behavior of the specific heat. The question of saturation of the specific heat is re-examined and it is shown that the open problem of universality for the random-field Ising model is strongly influenced by the lack of self-averaging of the model. This property appears to be substantially depended on the disorder strength.
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