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

Abstract

With respect to the usual thermal ferromagnetic transitions, thezero-temperature finite-disorder critical point of the random field Isingmodel (RFIM) has the peculiarity of involving some ‘droplet’ exponentθ that enters the generalized hyperscaling relation2 − α = ν(d − θ). In the present paper, to better understand the meaning of this droplet exponentθ beyond its role in the thermodynamics, we discuss the statistics of low-energyexcitations generated by an imposed single spin-flip with respect to the groundstate, as well as the statistics of equilibrium avalanches, i.e. the magnetizationjumps that occur in the sequence of ground states as a function of the externalmagnetic field. The droplet scaling theory predicts that the distributiondl/l1 + θ of thelinear size l of low-energy excitations transforms into the distributionds/s1 + θ/df for thesize s (number of spins) of excitations of fractal dimensiondf (s ∼ ldf). In the non-mean-field regiond < dc, droplets are compactdf = d, whereas in the mean-fieldregion d > dc, droplets havea fractal dimension df = 2θ leading to the well-known mean-field resultds/s3/2. Zero-field equilibrium avalanches are expected to display the same distributionds/s1 + θ/df. We also discuss the statistics of equilibrium avalanches integrated over the externalfield and finite-size behaviors. These expectations are checked numericallyfor the Dyson hierarchical version of the RFIM, where the droplet exponentθ(σ) can be varied as a function of the effective long-range interactionJ(r) ∼ 1/rd + σ ind = 1.