By applying the recently developed nonperturbative functional renormalization group (FRG) approach, we study the interplay between ferromagnetism, quasi-long-range order (QLRO), and criticality in the $d$-dimensional random-field $O(N)$ model in the whole $(N,d)$ diagram. Even though the “dimensional reduction” property breaks down below some critical line, the topology of the phase diagram is found similar to that of the pure $O(N)$ model, with, however, no equivalent of the Kosterlitz-Thouless transition. In addition, we obtain that QLRO, namely, a topologically ordered “Bragg glass” phase, is absent in the 3-dimensional random-field $XY$ model. The nonperturbative results are supplemented by a perturbative FRG analysis to two loops around $d=4$.

• Received 14 November 2005

DOI:https://doi.org/10.1103/PhysRevLett.96.087202