# Disorder driven non-equilibrium phase transition: the Random field Ising model

In hard magnetic materials, the domain walls movement or even creation is suppressed, and other mechanisms, like domains nucleation and coherent spin rotation enter in the process of magnetization reversal. For these kind of materials a description in terms of spin models is more appropriate. We focused on the non-equilibrium properties of the random field Ising model (RFIM), to describe the competition between quenched disorder and exchange interactions and their effect on the hysteretic behavior.

Equilibrium versus non-equilibrium: Similarities and differences between equilibrium and non-equilibrium states in disordered systems have been widely studied both for their conceptual importance and because of the implications in the study of complex optimization problems. The central issue is to understand whether the equilibrium properties of disordered systems provide a faithful representation of the non-equilibrium states in which the system is likely to be found in practice. In optimization terms, the question is what is the relation between an approximate solution and the optimal one. We analyzed this issue in the case of the RFIM. We studied the ground state (GS) properties in order to compare the ground state with its non-equilibrium hysteretic counterpart: the demagnetized state (DS)- a low energy state obtained by a sequence of slow magnetic field oscillations with decreasing amplitude. We solved the model exactly in one dimension and on the Bethe lattice, both for the GS and the DS. The ground state properties are obtained by solving recursively the integral equation for the effective field distribution. The results show that there is a significant difference in the energies of the two states for all values of the disorder variance. We resorted to numerical simulations in d>1 to better characterize the origin of this difference.

Disorder-induced non-equilibrium phase transition: In three and higher dimensions, the RFIM shows a phase transition between a strong disorder phase, where the hysteresis loop is continuous and a weak disorder phase, where the hysteresis loop shows a macroscopic jump. We identified in the remanent magnetization of the DS the natural non-equilibrium order parameter at zero external field, to compare with the GS magnetization. Our results suggest that the two transitions are characterized by the same scaling exponents and scaling function, while the location of the critical point differs. This analysis is corroborated by the exact solution of the model on the Bethe lattice. Recently, hysteretic optimization was proposed as a method, alternative to simulated annealing, to find quasi-optimal solutions of complex optimization problems. Our comparison of equilibrium versus non-equilibrium phase transition has interesting implications on the use of demagnetization as an optimization tool: the DS ferromagnetic phase is the first to disappear as the disorder is increased. Therefore there is a region of the disorder strength where the DS (paramagnetic) is drastically different from the GS (ferromagnetic), suggesting that hysteretic optimization is likely to fail. However, our results indicate that this optimization scheme can be improved by demagnetizing the system at a different value of the disorder strength, or as well two different values of the coupling constant in the Hamiltonian.

Minor loops, low field hysteresis and the Rayleigh law: The hysteresis loops at low fields, starting from the demagnetized state, are usually described by the phenomenological Rayleigh law. We analyzed the demagnetization properties and the small field behavior of the RFIM on the Bethe lattice focusing on the region near the disorder induced phase transition. We derived an exact recursion relation for the magnetization as a function of external field for any given field history. Integrating numerically this recursive equations we can access all nested minor loops eventually up to the demagnetized state. Our analysis shows that demagnetization is possible only in the continuous high disorder phase, where at low field the loops are described by the Rayleigh law. In the low disorder phase, the saturation loop displays a discontinuity which is reflected by a non vanishing magnetization after a series of nested loops. In this case, at low fields the loops are not symmetric and the Rayleigh law does not hold.