Nonperturbative functional renormalization group for random field models and related disordered systems. I. Effective average action formalism

We have developed a nonperturbative functional renormalization group approach for random field models and related disordered systems for which, due to the existence of many metastable states, conventional perturbation theory often fails. The approach combines an exact renormalization group equation for the effective average action with a nonperturbative approximation scheme based on a description of the probability distribution of the renormalized disorder through its cumulants. For the random field $O\left(N\right)$ model, the minimal truncation within this scheme is shown to reproduce the known perturbative results in the appropriate limits, near the upper and lower critical dimensions and at a large number $N$ of components, while providing a unified nonperturbative description of the full $\left(N,d\right)$ plane, where $d$ is the spatial dimension.

Figure 1

Schematic phase diagram of the RFIM in the disorder strength $\Delta$-temperature $T$ plane above the lower critical dimension $d_{lc}=2$ (temperature can be introduced at the bare level through the Boltzmann weight). At low disorder and low temperature, the system is ferromagnetic, and it is paramagnetic otherwise. The arrows describe how the renormalized parameters evolve under the RG flow at long distance, and $I$ and $RF$ denote the critical fixed points of the pure and random field Ising models, respectively.