Functional Renormalization Group at Large N for
Disordered Elastic Systems, and Relation to Replica Symmetry
Breaking
Pierre Le Doussal1, Kay Jörg Wiese2
1CNRS-Laboratoire de Physique Théorique de
l'Ecole Normale Supérieure,
24 rue Lhomond, 75005 Paris, France
2Kavli Institute of Theoretical Physics, University of California at Santa Barbara, Santa Barbara, CA 93106-4030, USA
Abstract
We study the replica field theory which describes the pinning of
elastic manifolds of arbitrary internal dimension d in a random
potential, with the aim of bridging the gap between mean field and
renormalization theory. The full effective action is computed exactly
in the limit of large embedding space dimension N. The second
cumulant of the renormalized disorder obeys a closed self-consistent
equation. It is used to derive a Functional Renormalization Group
(FRG) equation valid in any dimension d, which correctly matches the
Balents Fisher result to first order in ε = 4 - d. We analyze in
detail the solutions of the large-N FRG for both long-range and
short-range disorder, at zero and finite temperature. We find
consistent agreement with the results of Mezard Parisi (MP) from the
Gaussian variational method (GVM) in the case where full replica
symmetry breaking (RSB) holds there. We prove that the cusplike
non-analyticity in the large N FRG appears at a finite scale,
corresponding to the instability of the replica symmetric solution of
MP. We show that the FRG exactly reproduces, for any disorder
correlator and with no need to invoke Parisi's spontaneous RSB, the
non-trivial result of the GVM for small overlap. A formula is found
yielding the complete RSB solution for all overlaps. Since our
saddle-point equations for the effective action contain both the MP
equations and the FRG, it can be used to describe the crossover from
FRG to RSB. A qualitative analysis of this crossover is given, as well
as a comparison with previous attempts to relate FRG to GVM. Finally,
we discuss applications to other problems and new perspectives.
cond-mat/0305634 [pdf]
Phys. Rev. B 68 (2003) 174202 [pdf]
Copyright (C) by Kay Wiese. Last edited March 17, 2008.