Derivation of the functional renormalization group β-function at order [formula omitted] for manifolds pinned by disorder

Abstract

In an earlier publication, we have introduced a method to obtain, at large N, the effective action for d-dimensional manifolds in a N-dimensional disordered environment. This allowed to obtain the functional renormalization group (FRG) equation for N = ∞ and was shown to reproduce, with no need for ultrametric replica symmetry breaking, the predictions of the Mézard–Parisi solution. Here we compute the corrections at order 1 / N . We introduce two novel complementary methods, a diagrammatic and an algebraic one, to perform the complicated resummation of an infinite number of loops, and derive the β-function of the theory to order 1 / N . We present both the effective action and the corresponding functional renormalization group equations. The aim is to explain the conceptual basis and give a detailed account of the novel aspects of such calculations. The analysis of the FRG flow, comparison with other studies, and applications, e.g., to the strong-coupling phase of the Kardar–Parisi–Zhang equation are examined in a subsequent publication.