Dimensional reduction and its breakdown in the driven random field O(N) model

The critical behavior of the random field $O(N)$ model driven at a uniform velocity is investigated at zero-temperature. From naive phenomenological arguments, we introduce a dimensional reduction property, which relates the large-scale behavior of the $D$-dimensional driven random field $O(N)$ model to that of the $(D-1)$-dimensional pure $O(N)$ model. This is an analogue of the dimensional reduction property in equilibrium cases, which states that the large-scale behavior of $D$-dimensional random field models is identical to that of $(D-2)$-dimensional pure models. However, the dimensional reduction property breaks down in low enough dimensions due to the presence of multiple meta-stable states. By employing the non-perturbative renormalization group approach, we calculate the critical exponents of the driven random field $O(N)$ model near three-dimensions and determine the range of $N$ in which the dimensional reduction breaks down.