(4+N)-Dimensional Elastic Manifolds in Random Media: a Renormalization-Group Analysis

TL;DR

This paper employs a real-space functional renormalization group approach to analyze a 4+N-dimensional elastic manifold in quenched disorder, deriving key scaling relations for pinned domain size and critical force relevant to vortex lattices in superconductors.

Contribution

It introduces a two-loop correction to the RG analysis of elastic manifolds in disorder, providing refined expressions for pinning domain size and critical force.

Findings

Derived RG equations up to two-loop order for the model

Calculated the scaling relations for R_c and F_c with disorder strength

Identified the importance of two-loop corrections over lowest-order perturbation

Abstract

Motivated by the problem of weak collective pinning of vortex lattices in high-temperature superconductors, we study the model system of a four-dimensional elastic manifold with N transverse degrees of freedom (4+N-model) in a quenched disorder environment. We assume the disorder to be weak and short-range correlated, and neglect thermal effects. Using a real-space functional renormalization group (FRG) approach, we derive a RG equation for the pinning-energy correlator up to two-loop correction. The solution of this equation allows us to calculate the size R_c of collectively pinned elastic domains as well as the critical force F_c, i.e., the smallest external force needed to drive these domains. We find R_c prop. to delta_p^alpha_2 exp(alpha_1/delta_p) and F_c prop. to delta_p^(-2 alpha_2) exp(-2 alpha_1/delta_p), where delta_p <<1 parametrizes the disorder strength, alpha_1=(2/pi)^(N/2) 8 pi^2/(N+8), and alpha_2=2(5N+22)/(N+8)^2. In contrast to lowest-order perturbation calculations which we briefly review, we thus arrive at determining both alpha_1 (one-loop) and alpha_2 (two-loop).