Supersymmetry and Its Spontaneous Breaking in the Random Field Ising Model

We provide a resolution of one of the long-standing puzzles in the theory of disordered systems. By reformulating the functional renormalization group for the critical behavior of the random field Ising model in a superfield formalism, we are able to follow the associated supersymmetry and its spontaneous breaking along the functional renormalization group flow. Breaking is shown to occur below a critical dimension d(DR) similar or equal to 5.1 and leads to a breakdown of the dimensional reduction'' property. We compute the critical exponents as a function of dimension and give evidence that scaling is described by three independent exponents.