We compute the functional renormalization group (FRG) disorder-correlator function $R\left(v\right)$ for $d$-dimensional elastic manifolds pinned by a random potential in the limit of infinite embedding space dimension $N$. It measures the equilibrium response of the manifold in a quadratic potential well as the center of the well is varied from 0 to $v$. We find two distinct scaling regimes: (i) a “single shock” regime, $v^2\sim L^{-d}$, where $L^d$ is the system volume, and (ii) a “thermodynamic” regime, $v^2\sim N$. In regime (i), all the equivalent replica symmetry breaking (RSB) saddle points within the Gaussian variational approximation contribute, while in regime (ii), the effect of RSB enters only through a single anomaly. When the RSB is continuous (e.g., for short-range disorder, in dimension $2⩽d⩽4$), we prove that regime (ii) yields the large-$N$ FRG function obtained previously. In that case, the disorder correlator exhibits a cusp in both regimes, though with different amplitudes and of different physical origin. When the RSB solution is one step and nonmarginal (e.g., $d<2$ for short-range disorder), the correlator $R\left(v\right)$ in regime (ii) is considerably reduced and exhibits no cusp. Solutions of the FRG flow corresponding to nonequilibrium states are discussed as well. In all cases, regime (i) exhibits a cusp nonanalyticity at $T=0$, whose form and thermal rounding at finite $T$ are obtained exactly and interpreted in terms of shocks. The results are compared with previous work, and consequences for manifolds at finite $N$ as well as extensions to spin glasses and related models are discussed.

• Received 29 November 2007

DOI:https://doi.org/10.1103/PhysRevB.77.064203