Equimagnetization lines in the hybrid‐order phase diagram of the d=3 random‐field Ising model (invited)

The three‐dimensional random‐field Ising model has been approximately solved by pursuing the global renormalization‐group trajectories of the full, coupled probability distributions of local fields and bonds. This study, which in effect is the global analysis of the renormalization‐group flows of 220 physical quantities (based on a generalization of the Migdal–Kadanoff approximation), yields several results that were previously unsuspected. The boundary between the ferromagnetic and paramagnetic phases is hybrid order, in the sense that the magnetization is discontinuous (as in a first‐order transition) and the specific heat has a power‐law singularity (as in a second‐order transition). The magnetic properties of the d=3 random‐field Ising model are presented in the form of equimagnetization lines. The study has been repeated for a variety of dimensions d. The lower critical dimension d_l=2 is correctly obtained. Moreover, this study indicates the existence of an intermediate‐critical dimension d_t (between the lower‐ and upper‐critical dimensions) where a singularity occurs in the critical properties as a function of dimension.