The Random Field Ising Model (RFIM) represents one of the simplest models of cooperative behaviour with quenched disorder, and it is, in a way, complementary to the Ising Spin Glass which will be extensively treated later in this book. It accounts for the presence of a random external magnetic field which antagonizes the ordering induced by the ferromagnetic spin–spin interactions. From an experimental point of view, on the other hand, as shown by Fishman and Aharony (1979) and Cardy (1984), it is equivalent to a dilute anti-ferromagnet in a uniform field (see Belanger, 1998 for a recent review on experimental results).

Despite twenty-five years of active and continuous research the RFIM is not yet completely understood. The problem seems related to the presence of bound states in the ferromagnetic phase, which make the standard theoretical approaches not adequate to analyze the critical behaviour. Here we discuss the RFIM in the context of perturbative field theory. The chapter is organized as follows: in Section 2.1 we define the model and outline the main expectations for its qualitative behaviour. In Section 2.2 we introduce an effective replicated ϕ4 field model where the disorder has been integrated out. Then we perform a perturbative analysis on this model (Section 2.3) and illustrate how the so-called dimensional reduction arises (Section 2.4). Finally, in Section 2.5 we introduce some generalized couplings which need to be taken into account to properly describe the system; we perform a perturbative Renormalization Group (RG) close to the upper critical dimension (Sections 2.6 and 2.7) and discuss the occurrence of a vitrous transition (Section 2.8).