Two-loop Functional Renormalization Group of the Random Field and Random Anisotropy O(N) Models

TL;DR

This paper uses a two-loop perturbative Functional Renormalization Group approach to analyze the critical behavior of Random Field and Random Anisotropy O(N) models near four dimensions, revealing nonanalytic fixed points and the breakdown of dimensional reduction.

Contribution

It provides the first two-loop order beta functions for these models, demonstrating their perturbative renormalizability despite nonanalyticities, and connects perturbative results with non-perturbative FRG predictions.

Findings

Confirmation of nonanalytic fixed points controlling long-distance physics.

Demonstration of the breakdown of dimensional reduction at criticality.

Stability of quasi-long range order in dimensions below four.

Abstract

We study by the perturbative Functional Renormalization Group (FRG) the Random Field and Random Anisotropy O(N) models near $d=4$, the lower critical dimension of ferromagnetism. The long-distance physics is controlled by zero-temperature fixed points at which the renormalized effective action is nonanalytic. We obtain the beta functions at 2-loop order, showing that despite the nonanalytic character of the renormalized effective action, the theory is perturbatively renormalizable at this order. The physical results obtained at 2-loop level, most notably concerning the breakdown of dimensional reduction at the critical point and the stability of quasi-long range order in $d<4$, are shown to fit into the picture predicted by our recent non-perturbative FRG approach.