Cusps and shocks in the renormalized potential of glassy random
manifolds: How Functional Renormalization Group and Replica Symmetry Breaking
fit together
Pierre Le Doussal1, Markus Müller2, Kay Jörg Wiese1
1CNRS-Laboratoire de Physique Théorique de
l'Ecole Normale Supérieure,
24 rue Lhomond, 75005 Paris, France
2Department of Physics, Harvard University, Lyman
Laboratory, Cambridge MA 02138, USA
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, v2 ~ L-d
where Ld is the system volume and (ii) a
"thermodynamic" regime, v2 ~ 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.
arXiv:0711.3929 + hyperlinks [pdf]
Phys. Rev. B 77 (2008) 064203 [pdf]
Copyright (C) by Kay Wiese. Last edited March 17, 2008.