Functional Renormalization Group and the Field Theory of
Disordered Elastic Systems
Pierre Le Doussal1, Kay Jörg Wiese2, Pascal Chauve1
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 elastic systems such as interfaces
or lattices, pinned by quenched disorder. To escape triviality as a
result of dimensional reduction, we use the functional
renormalization group. Difficulties arise in the calculation of the
renormalization group functions beyond 1-loop order. Even worse,
observables such as the 2-point correlation function exhibit the same
problem already at 1-loop order. These difficulties are due to the
non-analyticity of the renormalized disorder correlator at zero
temperature, which is inherent to the physics beyond the Larkin
length, characterized by many metastable states. As a result, 2-loop
diagrams, which involve derivatives of the disorder correlator at the
non-analytic point, are naively ambiguous. We examine several routes
out of this dilemma, which lead to a unique renormalizable
field-theory at 2-loop order. It is also the only theory consistent
with the potentiality of the problem. The -function differs from
previous work and the one at depinning by novel anomalous terms. For
interfaces and random bond disorder we find a roughness exponent
ζ = 0.20829804 ε + 0.006858 ε2,
ε = 4-d. For random field disorder we find ζ=
ε/3 and compute universal amplitudes to order O(
ε2). For periodic systems we evaluate the universal
amplitude of the 2-point function. We also clarify the dependence of
universal amplitudes on the boundary conditions at large scale. All
predictions are in good agreement with numerical and exact results,
and an improvement over one loop. Finally we calculate higher
correlation functions, which turn out to be equivalent to those at
depinning to leading order in ε.
cond-mat/0304614 [pdf]
Phys. Rev. E 69 (2004) 026112 [pdf]
Copyright (C) by Kay Wiese. Last edited March 17, 2008.