  We study the energy landscape of the soft-spin random field model in the
mean-field limit and compute analytically the quenched complexity of the
metastable states as a function of their magnetization and energy at a given
external magnetic field. The shape of the domain within which the complexity is
positive (and the number of typical metastable states grows exponentially with
system size) changes with the amount of disorder and becomes non-convex and
disconnected at low disorder. As a consequence, phase transitions occur both at
equilibrium and out of equilibrium along the saturation hysteresis loop. We
focus on the zero complexity curve in the field-magnetization plane and its
relationship with the hysteresis loop. We also study the response of the system
when the magnetization is externally controlled instead of the magnetic field.
The main features of the model that should survive in finite dimensions are
discussed.
