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arXiv:1409.1795 · Europhysics Letters (EPL)

Diluted antiferromagnetic 3D Ising model in a field

Marco Picco, Nicolas Sourlas

Published Feb 1, 2015• 12 citations
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Abstract

We present numerical simulations for the diluted antiferromagnetic 3D Ising model in an external magnetic field (DAFF) at zero temperature. Our results are compatible with the DAFF being in the same universality class as the random field Ising model, in agreement with the renormalization group prediction.
DOI
10.1209/0295-5075/109/37001

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The bond-diluted Ising antiferromagnet in a uniform field (DAFF) on a square lattice is studied and comparison with the random-field problem is made. Position-space renormalisation group calculations for the DAFF give rise to a staircase structure in the zero-temperature phase boundary H (critical field) versus p (bond concentration), separating ordered and disordered phases. Since there is no stable long-range order in the equivalent random-field system, the authors discuss ways in which a consistent picture can be obtained, incorporating both our results and the correspondence between the two problems.
We perform large-scale Monte Carlo simulations using the Machta-Newman-Chayes algorithms to study the critical behavior of both the diluted antiferromagnet in a field with 30% dilution and the random-field Ising model with Gaussian random fields for different field strengths. Analytical calculations by Cardy [Phys. Rev. B 29, 505 (1984)] predict that both models map onto each other and share the same universality class in the limit of vanishing fields. However, a detailed finite-size scaling analysis of the Binder cumulant, the two-point finite-size correlation length, and the susceptibility suggests that even in the limit of small fields, where the mapping is expected to work, both models are not in the same universality class. Based on our numerical data, we present analytical expressions for the phase boundaries of both models.
It is shown that the dipolar interaction breaks the universality of the low-temperature and critical behaviour between the random-field Ising model (RFIM) and the diluted antiferromagnet in an external field (DAFF). For the RFIM, the dipolar interaction changes the roughness and the critical exponents; in particular, the upper critical dimensionalities are diminished by one. The dipolar interaction exerts no influence on these exponents in the DAFF, nor on the lower critical dimensionality d l =2 of both systems.
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.
We study the purely relaxational dynamics (model A) at criticality in three-dimensional disordered Ising systems whose static critical behaviour belongs to the randomly diluted Ising universality class. We consider the site-diluted and bond-diluted Ising models, and the ± J Ising model along the paramagnetic–ferromagnetic transition line. We perform Monte Carlo simulations at the critical point using the Metropolis algorithm and study the dynamic behaviour in equilibrium at various values of the disorder parameter. The results provide a robust evidence of the existence of a unique model-A dynamic universality class which describes the relaxational critical dynamics in all considered models. In particular, the analysis of the size dependence of suitably defined autocorrelation times at the critical point provides the estimate z = 2.35(2) for the universal dynamic critical exponent. We also study the off-equilibrium relaxational dynamics following a quench from to T c . In agreement with the field-theory scenario, the analysis of the off-equilibrium dynamic critical behaviour gives an estimate of z that is perfectly consistent with the equilibrium estimate z = 2.35(2).
We have studied the equilibrium properties of the random-field (H) ferromagnetic Ising model (RFIM) and the dilute antiferromagnetic Ising model (DAF) in a uniform field. For RFIM we have used the transfer-matrix technique, thus bypassing serious equilibration problems which would arise in a Monte Carlo calculation for random systems. We have calculated the structure factor S versus H. Our results are consistent with S\ensuremath{\sim}exp(c/H2{H}^{2}H2) as predicted by theories which yield a lower critical dimension dc{d}_{c}dc​=2, whereas the expected behavior for dc{d}_{c}dc​=3, S\ensuremath{\sim}${H}^{\mathrm{\ensuremath{-}}4}$ is inconsistent with our results. We have carried out analogous calculations for DAF. Our results for DAF suffer seriously from finite-size effects. To circumvent this difficulty, we have also performed calculations on interface models. Together, these results indicate that DAF crosses over to RFIM behavior at H\ensuremath{\simeq}J.