Dimensional reduction and its breakdown in the 3-dimensional long-range random field Ising model

Understand

We investigate dimensional reduction, the property that the critical behavior of a system in the presence of quenched disorder in dimension d is the same as that of its pure counterpart in d-2, and its breakdown in the case of the random-field Ising model in which both the interactions and the correlations of the disorder are long-ranged, i.e.

• power-law decaying.
• To some extent the power-law exponents play the role of spatial dimension in a short-range model, which allows us to probe the theoretically predicted existence of a nontrivial critical value separating a region where dimensional reduction holds from one where it is broken, while still considering the physical dimension d=3.
• By extending our recently developed approach based on a nonperturbative functional renormalization group combined with a supersymmetric formalism, we find that such a critical value indeed exists, provided one chooses a specific relation between the decay exponents of the interactions and of the disorder correlations.