Cusps and shocks in the renormalized potential of glassy random manifolds: How Functional Renormalization Group and Replica Symmetry Breaking fit together

TL;DR

This paper analyzes the functional renormalization group (FRG) and replica symmetry breaking (RSB) in glassy random manifolds, revealing two distinct scaling regimes and the conditions under which cusps and shocks appear in the disorder correlator.

Contribution

It unifies FRG and RSB approaches for glassy manifolds, showing how different RSB schemes affect the disorder correlator and its non-analytic features across regimes.

Findings

Identifies two scaling regimes: single shock and thermodynamic.

Proves the large-N FRG function matches previous results for continuous RSB.

Shows RSB type influences cusp presence and amplitude in the correlator.

Abstract

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 ~ 1/L^d where L^d is the system volume and (ii) a "thermodynamic" regime, v^2 ~ 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 1-step and non- marginal (e.g., d < 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.