On the breakdown of dimensional reduction and supersymmetry in random-field models

TL;DR

This paper investigates the conditions under which supersymmetry and dimensional reduction break down in random-field models, using functional renormalization group methods to identify critical dimensions and analyze nonperturbative effects.

Contribution

It provides a comprehensive nonperturbative FRG analysis of supersymmetry and dimensional reduction breakdown in random-field models, refining the critical dimension estimate and clarifying the limitations of perturbative approaches.

Findings

Critical dimension for SUSY/DR breakdown around 5.11

Nonperturbative FRG captures breakdown mechanisms missed by perturbation

Disappearance of SUSY/DR fixed point linked to FRG nonlinearity

Abstract

We discuss the breakdown of the Parisi-Sourlas supersymmetry (SUSY) and of the dimensional-reduction (DR) property in the random field Ising and O($N$) models as a function of space dimension $d$ and/or number of components $N$. The functional renormalization group (FRG) predicts that this takes place below a critical line $d_{\rm DR}(N)$. We revisit the perturbative FRG results for the RFO($N$)M in $d=4+\epsilon$ and carry out a more comprehensive investigation of the nonperturbative FRG approximation for the RFIM. In light of this FRG description, we discuss the perturbative results in $\epsilon=6-d$ recently derived for the RFIM by Kaviraj, Rychkov, and Trevisani. We stress in particular that the disappearance of the SUSY/DR fixed point below $d_{\rm DR}$ arises as a consequence of the nonlinearity of the FRG equations and cannot be found via the perturbative expansion in $\epsilon=6-d$ (nor in $1/N$). We also provide an error bar on the value of the critical dimension $d_{\rm DR}$ for the RFIM, which we find around $5.11\pm0.09$, by studying several successive orders of the nonperturbative FRG approximation scheme.