Topological and energetic aspects of the random-field Ising model

We show that random-field Ising model spin states are organized in equivalence classes (basins), each class containing the states that can be mutually reached from one another by some field history. Of these basins, only one contains the field-reachable states which can be generated by applying a suitable field history to one of the saturation states. We show that the basins form an oriented graph of which the basin of field-reachable states represents the bottom. The graph exhibits a hierarchical structure which reflects the organization in real space of the spin blocks that can be reversed back and forth by the field in some appropriate field interval without affecting the state of surrounding spins.