Exactly solvable model of avalanches dynamics for Barkhausen crackling noise

We review the present state of understanding of the Barkhausen effect in soft ferromagnetic materials. Barkhausen noise (BN) is generated by the discontinuous motion of magnetic domains as they interact with impurities and defects. BN is one of the many examples of crackling noise, arising in a variety of contexts with remarkably similar features, and occurring when a system responds in a jerky manner to a smooth external forcing. Among all crackling system, we focus on BN, where a complete and consistent picture emerges thanks to an exactly solvable model of avalanche dynamics, known as the ABBM model, which ultimately describes the system in terms of a Langevin equation for the velocity of the avalanche front. Despite its simplicity, the ABBM model is able to accurately reproduce the phenomenology observed in the experiments on a large class of magnetic materials, as long as universal properties are involved. To complete the picture and to understand the long-standing discrepancy between the ABBM theory and the experiments, which otherwise agree exceptionally well, consisting of the puzzling asymmetric shape of the noise pulses, microscopic details must be taken into account, namely the effects of eddy current retardation. These effects can be incorporated in the model, and result, to a first-order approximation, in a negative effective mass associated with the wall. The progress made in understanding BN is potentially relevant for other crackling systems: on the one hand, the ABBM model turns out to be a paradigmatic model for the universal behaviour of avalanche dynamics; on the other hand, the microscopic explanation of the asymmetry in the noise pulses suggests that inertial effects may also be at the origin of pulses asymmetry observed in other crackling systems.