We analyze low field hysteresis close to the demagnetized state in disordered ferromagnets using the zero temperature random-field Ising model. We solve the demagnetization process exactly in one dimension and derive the Rayleigh law of hysteresis. The initial susceptibility a and the hysteretic coefficient b display a peak as a function of the disorder width. This behavior is confirmed by numerical simulations d = 2, 3 showing that in limit of weak disorder demagnetization is not possible and the Rayleigh law is not defined. These results are in agreement with experimental observations on nanocrystalline magnetic materials.

We compare the ground state of the random-field Ising model with Gaussian distributed random fields, with its nonequilibrium hysteretic counterpart, the demagnetized state. This is a low-energy state obtained by a sequence of slow magnetic-field oscillations with decreasing amplitude. The main concern is how optimized the demagnetized state is with respect to the best-possible ground state. Exact results for the energy in d =1 show that in a paramagnet, with finite spin-spin correlations, there is a significant difference in the energies if the disorder is not so strong that the states are trivially almost alike. We use numerical simulations to better characterize the difference between the ground state and the demagnetized state. For d ജ 3, the random-field Ising model displays a disorder induced phase transition between a paramagnetic and a ferromagnetic state. The locations of the critical points R c ͑DS͒ and R c ͑GS͒ differ for the demagnetized state and ground state. We argue based on the numerics that in d = 3 the scaling at the transition is the same in both states. This claim is corroborated by the exact solution of the model on the Bethe lattice, where the critical points are also different.

Demagnetization, commonly employed to study ferromagnets, has been proposed as the basis for an optimization tool, a method to find the ground state of a disordered system. Here we present a detailed comparison between the ground state and the demagnetized state in the random field Ising model, combing exact results in d = 1 and numerical solutions in d = 3. We show that there are important differences between the two states that persist in the thermodynamic limit and thus conclude that AC demagnetization is not an efficient optimization method. Disordered systems are widely studied both for their conceptual importance and because the presence of randomness provides prototypical examples of complex optimization problems [1]. A disordered system can be non-trivial even at zero temperature due to the presence of a complex energy landscape. The properties of the ground-state (GS) are often difficult to determine analytically, and exact numerical evaluation becomes computationally prohibitive for large systems. Thus one is lead to construct approximate schemes, typically based on a non-equilibrium dynamics to find low energy states. In this respect a recently proposed method is hysteretic optimization [2]. Its basis is an analogy to a ferromagnetic demagnetization procedure: an external oscillating field with decreasing amplitude and low frequency is applied to the system. In ferromagnetic materials, one obtains at zero field after this procedure the demagnetized state (DS), which is used as a reference state for material characterization.Here we analyze the differences between the GS and the DS in the ferromagnetic random field Ising model (RFIM), which has been extensively studied in literature as a paradigmatic example of disordered system [3]. In the RFIM, a spin s i = ±1 is assigned to each site i of a d−dimensional lattice. The spins are coupled to their nearestneighbors spins by a ferromagnetic interaction of strength J and to the external field H. In addition, to each site of the lattice it is associated a random field h i taken from a Gaussian probability ρ(h) = exp(−h 2 /2R 2 )/ √ 2πR), with variance R. The Hamiltonian thus reads