Derivation of the Functional Renormalization Group Beta-Function at order 1/N 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=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.

62 pages, 97 figures

References in corpus (8)

• Functional Renormalization Group and the Field Theory of Disordered Elastic Systems
• Universal interface width distributions at the depinning threshold
• Higher correlations, universal distributions and finite size scaling in the field theory of depinning
• 2-loop Functional Renormalization for elastic manifolds pinned by disorder in N dimensions
• Functional renormalization group for anisotropic depinning and relation to branching processes
• Broad relaxation spectrum and the field theory of glassy dynamics for pinned elastic systems
• Field theory of statics and dynamics of glasses: rare events and barrier distributions
• On a Dynamical-Like Replica-Symmetry-Breaking Scheme for the Spin Glass

Cited by in corpus (16)

• Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records
• Size distributions of shocks and static avalanches from the Functional Renormalization Group
• Non-Perturbative Functional Renormalization Group for Random Field Models and Related Disordered Systems. I: Effective Average Action Formalism
• Random-field Ising and $O(N)$ models: Theoretical description through the functional renormalization group
• 2-loop Functional Renormalization for elastic manifolds pinned by disorder in N dimensions
• Cusps and shocks in the renormalized potential of glassy random manifolds: How Functional Renormalization Group and Replica Symmetry Breaking fit together
• Large-n expansion for m-axial Lifshitz points
• Supersymmetry Breaking in Disordered Systems and Relation to Functional Renormalization and Replica-Symmetry Breaking
• Field theory of disordered systems -- Avalanches of an elastic interface in a random medium
• Field Theory of Disordered Elastic Interfaces at 3-Loop Order: Critical Exponents and Scaling Functions
• Spontaneous versus explicit replica symmetry breaking in the theory of disordered systems
• Functional renormalization-group approach to decaying turbulence
• Hierarchical Reference Theory of critical fluids in disordered porous media
• Effect of second-rank random anisotropy on critical phenomena of random field O(N) spin model in the large N limit
• A statistical mechanical description of metastable states and hysteresis in the 3D soft-spin random-field model at T=0
• Theory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles