Replica Symmetry Breaking in Renormalization: Application to the Randomly Pinned Planar Flux Array

TL;DR

This paper extends renormalization group analysis of a vortex glass transition in a randomly pinned flux array by incorporating replica symmetry breaking, revealing new fixed points and correlation behaviors.

Contribution

It introduces a one-step replica symmetry breaking scheme into the RG analysis of the vortex glass transition, identifying new fixed points and correlation functions.

Findings

Unstable RG flow with respect to replica asymmetric perturbations.

Existence of new fixed points with broken replica symmetry.

Correlation functions diverge as ln(r) and ln^2(r) depending on parameters.

Abstract

The randomly pinned planar flux line array is supposed to show a phase transition to a vortex glass phase at low temperatures. This transition has been examined by using a mapping onto a 2D XY-model with random an\-iso\-tropy but without vortices and applying a renormalization group treatment to the replicated Hamiltonian based on the mapping to a Coulomb gas of vector charges. This renormalization group approach is extended by deriving renormalization group flow equations which take into account the possibility of a one-step replica symmetry breaking. It is shown that the renormalization group flow is unstable with respect to replica asymmetric perturbations and new fixed points with a broken replica symmetry are obtained. Approaching these fixed points the system can optimize its free energy contributions from fluctuations on large length scales; an optimal block size parameter $m$ can be found. Correlation functions for the case of a broken replica symmetry can be calculated. We obtain both correlations diverging as $\ln{r}$ and $\ln^2{r}$ depending on the choice of $m$.