First excitations in two- and three-dimensional random-field Ising systems

Abstract

Abstract. We present results on the first excited states for the random-field Ising model. These are based on an exact algorithm, with which we study the excitation energies and the excitation sizes for two-and three-dimensional random-field Ising systems with a Gaussian distribution of the random fields. Our algorithm is based on an approach of Frontera and Vives which, in some cases, does not yield the true first excited states. Using the corrected algorithm, we find that the order-disorder phase transition for three dimensions is visible via crossings of the excitations-energy curves for different system sizes, while in two-dimensions these crossings converge to zero disorder. Furthermore, we obtain in three dimensions a fractal dimension of the excitations cluster of ds = 2.42(2). We also provide analytical droplet arguments to understand the behavior of the excitation energies for small and large disorder as well as close to the critical point.

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

“…Hence the critical behavior is the same everywhere along the phase boundary and we can predict it simply by staying at T = 0 and crossing the phase boundary at the critical field point. This is a convenient approach because we can determine the ground states of the system exactly using efficient optimization algorithms [53][54][55][56][57][58][59][60][61][62][63][64][65][66]79,[88][89][90][91][92][93] through an existing mapping of the ground state to the maximum-flow optimization problem [94][95][96]. A clear advantage of this approach is the ability to simulate large system sizes and disorder ensembles in rather moderate computational times.…”

Random field Ising model: statistical properties of low-energy excitations and equilibrium avalanches

“…The various predictions are checked via numerics for the one-dimensional Dyson hierarchical version, where large systems can be studied, and where the value of the exponent θ can be varied as a function of the exponent σ of the long-range ferromagnetic interactions. Previous studies on these questions for the short-range model in dimension d = 3 can be found in [9,10] for the statistics of low-energy excitations, and in [11,12] for the statistics of equilibrium avalanches.…”

Low-energy excitations in the three-dimensional random-field Ising model

“…There are few results [25,37] which give evidence that the low-temperature behavior of the three-dimensional RFIM is well described by the droplet theory [38,39,40,41], which is one of the most important and most successful theories to describe finite-dimensional systems exhibiting quenched disorder. The droplet theory has already turned out to be useful to describe the behavior of twodimensional (2d) SGs [42,43,44,45].…”

“…For the 2d SG model, much evidence supporting the validity of the dropletmodel description has been accumulated over the years in particular by studying low-energy excitations. Nevertheless, for the three-dimensional (3d) RFIM, low-energy excitations have been investigated only in few cases [25,37] so far. In particular, since the 3d RFIM exhibits a phase transition at non-trivial disorder [46], in contrast to the 2d SG, it is of high interest to study the excitations as a function of the disorder strength.…”