Ground state optimization and hysteretic demagnetization: the random-field Ising model

We compare the ground state of the random-field Ising model with Gaussian distributed random fields, with its nonequilibrium hysteretic counterpart, the demagnetized state. This is a low-energy state obtained by a sequence of slow magnetic-field oscillations with decreasing amplitude. The main concern is how optimized the demagnetized state is with respect to the best-possible ground state. Exact results for the energy in d=1 show that in a paramagnet, with finite spin-spin correlations, there is a significant difference in the energies if the disorder is not so strong that the states are trivially almost alike. We use numerical simulations to better characterize the difference between the ground state and the demagnetized state. For d⩾3, the random-field Ising model displays a disorder induced phase transition between a paramagnetic and a ferromagnetic state. The locations of the critical points R(DS)c and R(GS)c differ for the demagnetized state and ground state. We argue based on the numerics that in d=3 the scaling at the transition is the same in both states. This claim is corroborated by the exact solution of the model on the Bethe lattice, where the critical points are also different.