Cusps and shocks in the renormalized potential of glassy random manifolds: How functional renormalization group and replica symmetry breaking fit together
Pierre Le Doussal, Markus M\"{u}ller, and Kay J\ " { o} rg Wiese

We compute the Functional Renormalization Group (FRG) disorder-correlator function $R(v)$ 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 $2 \leq d \leq 4$), 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 1-step and non-marginal (e.g., $d &lt 2$ for SR disorder), the correlator $R(v)$ in regime (ii) is considerably reduced, and exhibits no cusp. Solutions of the FRG flow corresponding to non-equilibrium states are discussed as well. In all cases the regime (i) exhibits a cusp non-analyticity at $T=0$, whose form and thermal rounding at finite $T$ is 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.

© 2008 The American Physical Society.