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

by Gilles Tarjus, Matthieu Tissier, Ivan Balog

Submission summary

Authors (as registered SciPost users): Ivan Balog · Gilles Tarjus
Submission information
Preprint Link: https://arxiv.org/abs/2411.11147v2  (pdf)
Date submitted: 2024-11-27 13:39
Submitted by: Tarjus, Gilles
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties:
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Statistical and Soft Matter Physics
Approach: Theoretical

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.

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Current status:
In refereeing

Reports on this Submission

Report #1 by Anonymous (Referee 1) on 2025-1-25 (Invited Report)

Report

The critical point of random field Ising model (RFIM) is believed to have supersymmetry (SUSY) and the property of dimensional reduction (DR) thanks to a famous conjecture by Parisi and Sourlas. In the submitted paper the authors use the method of functional renormalization group (FRG) to discuss the failure of the SUSY/DR properties in various integer dimensions ($d=2,3,4$) below the upper critical dimension ($d=6$).

The paper revisits the FRG analysis on this topic, supported by new results/updates and compares it to the work of Kaviraj, Rychkov and Trevisani (KRT). The latter had predicted, in the framework of non-functional perturbative RG with replica fields, the presence of special SUSY-breaking (Feldman) operators that become relevant in low enough dimensions making the SUSY fixed point unstable. In the FRG framework the Feldman operators are argued to be analogous to the eigenvalues associated with the flow of derivatives of a function of fields $R(z)$, which describe the cumulants of the disorder interaction. The eigenvalue associated with the second derivative corresponds to the most dangerous Feldman operator in KRT analysis. However in FRG when this eigenvalue becomes marginal the SUSY fixed point ceases to exist due to nonlinearity of the flow equations, instead of being unstable as argued by KRT. There is a new “cuspy” fixed point that corresponds to the deformation by a nonanalytic component in the cumulant. The FRG mechanism can be demonstrated in two different examples: the RF O(N) model close to $d=4$
where SUSY/DR fails for $N<=18$, and the RFIM in where SUSY/DR fails for $d<=5$. The authors argue that the nonlinearity and cuspy features showing up in FRG are invisible in usual perturbative RG. To sum up, although the FRG method agrees with some of the results and predictions in KRT, there are still loose ends, since the two approaches do not agree on whether SUSY/DR fixed point is unstable or non-existent below a critical dimension (estimated around $d\approx 5.1$) and emergence of non-analytic operators in RG. These need to be addressed in the future.

I have a few questions that I request the authors to clarify: 


1. The main analysis of section III and IV shows that the number $p$, for derivatives of cumulant, can be analytically continued to non-integer values, which becomes relevant for $p=3/2$. Why should such a deformation (corresponding to non-integer p) be expected in RG? In particular, can they be related to a local operator in usual Wilsonian way? Or do they correspond to nonlocal deformations? 


2. Is there a selection rule that always lets us map a given FRG eigenvalue to a specific composite operator of replica fields? For instance, why should $\Lambda_p$ (say for $p=3$) correspond exactly to the $\mathcal{F}_6$ and not a derivative of $\mathcal{F}_4$. 

(This is also related to the apparent mismatch of scaling dimensions of Feldman operators with the corresponding FRG term, for $p \neq 2$.)


3. In section III, the RF O(N) model is studied at $d=4-\epsilon$ dimension. From the conclusion it seems there is a stable SUSY fixed point for $N \ge 19$ for any $\epsilon$. Does this mean the RF O(N) models, for high enough $N$, have a SUSY/DR fixed point at $d=5$?

In addition, I think it would be useful if the following points are clarified further in the paper:

A. In section III A. It would be nice to have some equations showing the action of the model under consideration, the definition of $R(z)$ from it, and some illustrative examples of how derivatives of $R(z)$ are equivalent to deformation by some replica field operator.

B. In section IV, I found it hard to follow the logic of dimension counting in FRG close to free theory, that is based on powers of inverse temperatures associated with cumulants. It would be useful to clarify the general formula with some equation(s).


These point may have been explained in previous papers of the authors. But since the current paper has a broader take on the RG mechanism, there is a need for completeness especially for an interested reader who may not be fully familiar with the FRG developments.



The RFIM and related models have an interesting physics and its SUSY/DR properties is a fascinating topic. The intriguing connection between two quite different renormalization proposals for the model: the FRG and a perturbative Wilsonian approach, have been nicely discussed in the paper. This definitely clarifies a lot of points in the mysterious RG flow of the model. However, I request the author to address my above queries and requests. I would then recommend the paper for publication.

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