Disordered periodic systems at the upper critical dimension

TL;DR

This paper investigates how weak point-like disorder affects various periodic systems at their upper critical dimension, revealing ultra-slow growth of correlations and implications for the Bragg glass phase.

Contribution

It applies Gaussian Variational and Functional Renormalisation Group methods to analyze disorder effects across different systems at their upper critical dimension.

Findings

Displacement correlations grow as loglog(x), indicating near-perfect translational order.

Results suggest stability of the Bragg glass phase under weak disorder.

Universal behavior observed across systems with different upper critical dimensions.

Abstract

The effects of weak point-like disorder on periodic systems at their upper critical dimension D_c for disorder are studied. The systems studied range from simple elastic systems with D_c=4 to systems with long range interactions with D_c=2 and systems with D_c=3 such as the vortex lattice with dispersive elastic constants. These problems are studied using the Gaussian Variational method and the Functional Renormalisation Group. In all the cases studied we find a typical ultra-slow loglog(x) growth of the asymptotic displacement correlation function, resulting in nearly perfect translational order. Consequences for the Bragg glass phase of vortex lattices are discussed.