Nonperturbative functional renormalization group for random field models and related disordered systems. IV. Supersymmetry and its spontaneous breaking

We apply the nonperturbative functional renormalization group (NP-FRG) in the superfield formalism that we have developed in the preceding paper to study long-standing issues concerning the critical behavior of the random field Ising model. Through the introduction of an appropriate regulator and a supersymmetry-compatible nonperturbative approximation, we are able to follow the supersymmetry, more specifically, the superrotational invariance first unveiled by Parisi and Sourlas [Phys. Rev. Lett. 43, 744 (1979)] and its spontaneous breaking along the RG flow. Breaking occurs below a critical dimension d(DR) similar or equal to 5.1, and the supersymmetry-broken fixed point that controls the critical behavior then leads to a breakdown of the dimensional-reduction property. We solve the NP-FRG flow equations numerically and determine the critical exponents as a function of dimension down to d less than or similar to 3, with a good agreement in d = 3 and 4 with the existing numerical estimates.