Derivation of the Functional Renormalization Group Beta-Function at order 1/N for Manifolds Pinned by Disorder

TL;DR

This paper develops a method to compute the functional renormalization group beta-function at order 1/N for manifolds in disordered environments, extending previous large N results with novel diagrammatic and algebraic techniques.

Contribution

It introduces two new methods to perform complex resummations and derives the FRG beta-function at order 1/N, advancing understanding of disordered manifolds.

Findings

Derived the FRG beta-function at order 1/N

Introduced diagrammatic and algebraic resummation methods

Provided detailed effective action and flow equations

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=infinity and was shown to reproduce, with no need for ultrametric replica symmetry breaking, the predictions of the Mezard-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 beta-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.