  We study a quasi-statically driven random field Ising model (RFIM) at zero
temperature with interactions mediated by the long-range anisotropic Eshelby
kernel. Analogously to amorphous solids at their yielding transition, and
differently from ferromagnetic and dipolar RFIMs, the model shows a
discontinuous magnetization jump associated with the appearance of a band-like
structure for weak disorder and a continuous magnetization growth, yet
punctuated by avalanches, for strong disorder. Through a finite-size scaling
analysis in 2 and 3 dimensions we find that the two regimes are separated by a
finite-disorder critical point which we characterize. We discuss similarities
and differences between the present model and models of sheared amorphous
solids.
