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

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Pith's one-line read The paper argues that the SUSY/DR controlling fixed point of the random-field Ising model disappears at $d_{\mathrm{DR}}\approx 5.11\pm 0.09$ by annihilating with an unstable twin fixed point, and that the random-field $O(N>2)$ model…

desk verdict A credible, well-scoped FRG case that the RFIM SUSY/DR fixed point annihilates at d≈5.11±0.09, but the RFIM annihilation mechanism still depends on a generic-coalescence assumption that the authors honestly label non-rigorous. read the letter →

arxiv 2411.11147 v3 pith:TU2GHV3S submitted 2024-11-17 cond-mat.dis-nn hep-th

Gilles Tarjus , Matthieu Tissier , Ivan Balog This is my paper · ORCID
classification cond-mat.dis-nnhep-th MSC 82B2782B2882B44 PACS 11.10.Hi75.40.Cx
keywords random-fieldIsingmodelParisi-SourlassupersymmetrydimensionalreductionfunctionalrenormalizationgroupFeldmanoperatorsO(N)fixed-pointannihilationcuspyfixedpoint
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to establish that the breakdown of Parisi-Sourlas supersymmetry (SUSY) and dimensional reduction (DR) in random-field models is a fixed-point annihilation rather than a mere instability. The nonperturbative functional renormalization group (FRG) described here predicts that the SUSY/DR fixed point of the random-field Ising model disappears at $d_{\mathrm{DR}}\approx 5.11\pm 0.09$, at the exact dimension where the eigenvalue $\Lambda_2$ associated with Feldman's operator $F_4$ hits zero; below that dimension only a ``cuspy'' fixed point with broken SUSY and DR exists. The same mechanism is exhibited analytically for the random-field $O(N>2)$ model near $d=4$, where the SUSY/DR fixed point vanishes at $N=18$ at one-loop order. The annihilation follows from the nonlinearity of the FRG fixed-point equation for the second cumulant, which is why the perturbative $\epsilon=6-d$ and $1/N$ expansions cannot see it. If the prediction is correct, simulations in $d=4$ and $d=5$ should show at most transient SUSY/DR-like remnants, never an asymptotic SUSY/DR critical point.

What carries the argument

The load-bearing object is Feldman's operator $F_4$, the $p=2$ term in the expansion of the second cumulant in powers of $(\varphi_a-\varphi_b)^2$, whose 1-PI counterpart is the function $\delta_2(\varphi)$. Its fixed-point equation is nonlinear in $\delta_2(\varphi)$ itself---$A_*(\varphi)\,\delta_2(\varphi)^2 + L_*(\varphi,\partial_\varphi,\partial_\varphi^2)\,\delta_2(\varphi)+B_*(\varphi)=0$ at DE2, with up to cubic nonlinearities at DE4---while all higher-order $F_{2p}$ equations are linear. The eigenvalue $\Lambda_2$ is obtained by linearizing this equation around the fixed point, and the vanishing of $\Lambda_2$ coincides with the merging of two solutions and the disappearance of the fixed point below $d_{\mathrm{DR}}$. In the RFO($N>2$)M the same mechanism is carried by the polynomial $Q_{N,j}(R''(1))$ whose degree grows with loop order and whose derivative at the fixed point is $\Lambda_2(N)$. The functional, nonperturbative character of the flow is what allows the fixed point itself to be followed; the perturbative expansion around $d=6$ treats only the marginal coupling constant and cannot see the nonlinear coalescence.

What would settle it

Evaluate the product $R_{N_{\mathrm{DR}}}(X_{\ast 0})\,Q''_{N_{\mathrm{DR}}}(X_{\ast 0})$ at the candidate $N_{\mathrm{DR}}$ (or its DE4 analogue at the candidate $d_{\mathrm{DR}}$): if it vanishes, the fixed points cross rather than annihilate and the SUSY/DR fixed point could survive below the threshold. A direct numerical search for a cuspless fixed-point solution $\delta_{\ast,2}(\varphi)$ in $d=5$ below $d_{\mathrm{DR}}$, with the full functional dependence of the cumulants retained, would also decide the issue: finding one would falsify the claim, while in $d=4$ observing true asymptotic SUSY/DR scaling in a simulation with fine-tuned disorder would contradict the no-fixed-point conclusion.

Watch

Extended reading notes

Core claim

The central claim is that a SUSY/DR fixed point exists only above a critical dimension (or above a critical $N$ for the $O(N)$ model) and ceases to exist below it. In the FRG, the second cumulant of the renormalized random field is expanded around equal field arguments; the coefficient of the $(\varphi_1-\varphi_2)^2$ term, $\delta_2(\varphi)$, satisfies a fixed-point equation that is nonlinear in $\delta_2$ itself, with the nonlinearity growing from quadratic at DE2 to cubic at DE4. The stability eigenvalue $\Lambda_2$ is the derivative of that nonlinear function at the fixed point. Generically, when the eigenvalue vanishes the stable and unstable fixed-point branches merge with a square-root behavior and no real solution remains below $d_{\mathrm{DR}}$, so the full fixed point---including all cumulants at nonequal field arguments---disappears. The cuspy fixed point that replaces it has a nonanalytic dependence of the second cumulant on the field difference, and this cusp is the signature of broken SUSY and DR. For the RFO($N>2$)M in $d=4+\epsilon$ the same logic is carried by a polynomial equation for $R''(1)$, and the SUSY/DR fixed point disappears at $N=18$ at one-loop order.

Load-bearing premise

The load-bearing premise is that the coefficients controlling the fixed-point equation for $\delta_2(\varphi)$ (or for $R''(1)$ in the $O(N)$ model) do not vanish accidentally or by symmetry at the critical point, so the two fixed-point branches genuinely merge and disappear; if such a nongeneric cancellation occurred, the SUSY/DR fixed point could survive below $d_{\mathrm{DR}}$ as an unstable fixed point, which is the scenario the paper rejects. The rapid apparent convergence of the truncated FRG hierarchy is a separate additional assumption.

Editorial extensions

If this is right

  • If the claim is correct, no fine-tuning of the bare random-field distribution can produce a SUSY/DR critical point in $d=4$, because the fixed point does not exist there.
  • In $d=5$, only slightly above $d_{\mathrm{DR}}\approx 5.11$, simulations should see SUSY/DR-like behavior over finite sizes, but the asymptotic critical regime must be governed by the cuspy, SUSY/DR-broken fixed point.
  • The avalanche eigenvalue $\Lambda_{3/2}$ is still irrelevant at $d_{\mathrm{DR}}$, so the disappearance is not caused by avalanches; avalanches become the controlling mechanism only below $d_{\mathrm{DR}}$.
  • In the RFO($N>2$)M near $d=4$, the SUSY/DR fixed point ceases to exist for $N<18$ at one loop, so the cuspy fixed point is the only critical fixed point in that regime.
  • Near $d=6$ the nonperturbative FRG reproduces the 2-loop perturbative eigenvalues, so the two calculations agree where perturbation theory is controlled; they differ by the square-root annihilation below $d_{\mathrm{DR}}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • I infer that the annihilation mechanism implies an order-of-limits subtlety: extrapolating the 2-loop $\epsilon=6-d$ result gives $\Lambda_2=0$ near $d\approx 4.6$, but the nonperturbative zero sits at $d\approx 5.11$ because the eigenvalue has a square-root branch point.
  • A testable extension suggested by the same logic is to study long-range RFIM variants in which $d_{\mathrm{DR}}$ depends on the interaction range and could be shifted into dimensions accessible to simulation.
  • The authors leave open whether another unstable cuspy fixed point exists below $d_{\mathrm{DR}}$; finding it would complete the fixed-point diagram and sharpen the prediction for corrections to scaling.
  • I infer that the discrepancy with KRT about canonical dimensions of $F_6$ and higher operators affects subleading corrections but not the main conclusion, since all derivations agree on $F_4$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper revisits the Parisi-Sourlas supersymmetry (SUSY) and dimensional-reduction (DR) properties of random-field models, focusing on the random-field Ising model (RFIM) and the random-field O(N) model (RFO(N)M). The central claim is that, within the authors' nonperturbative functional renormalization group (FRG) framework, the SUSY/DR fixed point of the RFIM does not merely become unstable below some dimension but disappears entirely by coalescing with an unstable SUSY/DR fixed point at d_DR ≈ 5.11 ± 0.09. This disappearance is associated with the vanishing of the eigenvalue Λ2 of Feldman's operator F4. The paper also analyzes the analogous mechanism in the RFO(N>2)M near d=4, where the SUSY/DR fixed point disappears at N=18 at one-loop order. The authors compare their results with the perturbative ϵ=6−d calculations of Kaviraj, Rychkov, and Trevisani (KRT), argue that the perturbative expansion cannot capture the annihilation mechanism, and provide an error bar for d_DR from four truncation levels of their FRG scheme (LPA', LPA", DE2, and DE4).

Significance. If the central claim is correct, the paper resolves a long-standing question about the mechanism of SUSY/DR breakdown in random-field systems: below d_DR there is no SUSY/DR fixed point at all, so no fine-tuning of the disorder distribution can restore dimensional reduction. The paper provides a concrete, falsifiable prediction (d_DR ≈ 5.11 ± 0.09) and supports it by comparing several successive truncation levels and by matching the exact two-loop perturbative results near d=6 to within 20–30% at the DE4 level. The RFO(N)M analysis is particularly valuable because there the coalescence mechanism can be checked explicitly at one- and two-loop order. However, the generality of the mechanism for the RFIM rests on a generic-coalescence assumption that the authors themselves label as 'not rigorous' and 'indicative' in Appendices E and G, and no explicit verification of the required non-vanishing coefficients is provided for the RFIM. This limits the certainty of the central disappearance claim, although the numerical evidence from multiple truncations is substantial.

major comments (3)
  1. [Sec. IV.C and Appendices E, G] The claim that the SUSY/DR fixed point disappears for d < d_DR depends on the generic-coalescence assumption that certain coefficient functions do not vanish accidentally or by symmetry. In Appendix E, the argument assumes A*(φ) ≠ 0 and in Appendix G the absence of 'accidental (or symmetry-induced) cancellation' is assumed. The authors explicitly state in Appendix E that the argument 'is not rigorous but is indicative of what the generic behavior should be,' and Appendix G similarly relies on generic expectations. For the RFO(N)M, Appendix B verifies R_NDR(X*0)Q''_NDR(X*0) ≠ 0 at one- and two-loop order, but no analogous explicit check is performed for the RFIM. If a nongeneric cancellation occurred, the SUSY/DR fixed point would survive below d_DR as an unstable fixed point, which is precisely the scenario of KRT that the paper rejects. This is a load-bearing point that needs to be strengthened, for example by numerically verifying that the relevant products of coefficients are nonzero with controlled error bars in the DE2 and DE4 truncations near d_DR, or by providing an independent argument ruling out such cancellations.
  2. [Sec. V.C and Eq. (44)] The critical dimension d_DR is determined as the location where the eigenvalue Λ2(d) vanishes, and the disappearance of the fixed point is then identified with this same value. This makes the 'coincidence' of the vanishing of Λ2 and the annihilation of the fixed point partly built into the procedure. An independent determination would be preferable: for instance, one could solve the fixed-point equations for δ*,2(φ) (and the additional functions at DE4) directly and demonstrate that no real solution exists below some dimension, or measure the divergence of the Larkin time as d → d_DR⁻. The authors mention this alternative in Sec. IV.C but say it is 'less accurate' than the eigenvalue criterion; however, given that the central claim is about existence of the fixed point, an independent existence check, even if less precise, would provide decisive evidence that the disappearance is not an artifact of the chosen diagnostic.
  3. [Sec. V.C, Table I] The convergence evidence is presented as 'apparent rapid convergence,' but the numerical sequence d_DR = 5.2005 (LPA'), 5.0180 (LPA"), 5.1307 (DE2), 5.0678 (DE4) oscillates rather than monotonically converging, and the two-loop coefficients at DE4 differ from the exact values by about 20% for Λ2 and Λ3 and about 30% for Λ3/2. The quoted error bar of ±0.09 in Eq. (44) is based on the spread of these four values, but the regulator dependence is stated to be within this range rather than being explicitly demonstrated. Since the central quantitative prediction is d_DR ≈ 5.11 ± 0.09, the paper should either provide a more transparent error estimate that separates truncation and regulator uncertainties, or explicitly state that the error bar is a heuristic estimate rather than a rigorous bound.
minor comments (4)
  1. [Sec. IV.A] The discussion of the discrepancy in canonical dimensions of the F_{2p} operators for p ≥ 3 is clear, but the sentence 'which differs by an additive term −2(p−2) from the result of [1]' would benefit from an explicit statement that for p=2 both formalisms agree, since that agreement is crucial for the main conclusions.
  2. [Fig. 1 caption] The dashed curve is described as a '2-loop calculation in d=6−ϵ, together with a plausible extrapolation,' but the extrapolation procedure is not specified. Please state how the extrapolated curve is obtained, since the comparison with the nonperturbative results depends on this choice.
  3. [Sec. V.A] There is a typo in the phrase 'the explanantion of the SUSY/DR breakdown' in the first sentence of Sec. V.A; it should read 'explanation.'
  4. [Appendix F] The text states that a Mathematica notebook 'can be made available' for the DE4 flow equations. Since the equations are too long to display, the paper should either include the notebook as supplementary material or provide a stable repository link, so that the numerical results can be reproduced independently.

Circularity Check

3 steps flagged · score 4.0 of 10

Annihilation at d_DR ≈ 5.11 is computed within the authors' own NP-FRG truncation and is anchored externally near d=6 by agreement with Feldman/KRT 2-loop results, but the rejection of the KRT scenario rests on an admitted generic-coalescence assumption and on the self-cited nonlinear structure of Eq. (24), so the eigenvalue/disappearance coincidence is partly structural.

  1. self citation load bearing [Sec. IV C, Eq. (24); Sec. III A; Refs. [19,26,27]]
    "one can derive a closed FRG equation for δk,2(φ), which is of the form ∂tδk,2(φ) = A∗(φ)δk,2(φ)2 + L∗(φ, ∂φ, ∂2φ)δk,2(φ) + B∗(φ) (24) where L∗ is a linear operator ... The expressions are given in [27]."

    The annihilation mechanism is driven entirely by the nonlinear (quadratic) structure of Eq. (24), whose explicit form the paper imports from the authors' own prior work: 'The expressions are given in [27]' (M. Baczyk, G. Tarjus, M. Tissier, I. Balog, J. Stat. Mech. 2014). Λ2 is then defined by linearizing that self-cited equation, so the central conclusion — fixed point disappears exactly when Λ2 = 0 — is a property of the assumed (self-cited) truncation structure rather than an externally established result. The self-citation is mitigated by independent anchors: near d=6 the computed Λ2 coincides with Feldman's and KRT's 2-loop result, and Table I checks the ϵ² coefficients against the exact 2-loop values, so the central quantitative claim does have independent content.

  2. other [Sec. III A (after Eq. 9); Appendix E; Appendix G]
    "Generically, and in the absence of an additional symmetry, this corresponds to the collapse of the SUSY/DR solution with another solution, such that the SUSY/DR solution disappears altogether for N < NDR: see Appendix B. Although not a rigorous proof, this strongly supports the fact that the SUSY/DR fixed point no longer exists below the value NDR at which the eigenvalue Λ2 associated with the operator F4 vanishes. ... The above argument is not rigorous but is indicative of what the generic behavior should be."

    The central difference from KRT — annihilation instead of mere instability below d_DR — is conditional on the generic-coalescence assumption that the relevant coefficient products (R_NDR(X*0)Q''_NDR(X*0) for the RFO model; A_*(φ) and a sign-changing discriminant for the RFIM) do not vanish accidentally or by symmetry. The paper verifies this condition explicitly for the RFO(N)M at 1- and 2-loop orders, but for the RFIM it is only supported by numerical solutions of the truncated equations; no analytic check is provided. If a nongeneric cancellation occurred, the SUSY/DR fixed point would persist below d_DR as an unstable point — exactly the KRT scenario this paper rejects.

1 more flagged steps
  1. self definitional [Sec. II B; Sec. V C; Sec. III A (Λ2 = Q′N,j(R′′∗(1)))]
    "The critical dimension dDR can then be located in two ways: either looking at the appearance of a cusp in the fixed-point function ..., or by studying the vanishing of the eigenvalue Λ2(d), as we did in [10,27] and in the present paper. ... As explained before, we determine the critical dimension dDR by looking at the location where the most dangerous eigenvalue Λ2(d) vanishes."

    By construction, d_DR is defined as the zero of Λ2, and Λ2 is the eigenvalue of the linearization of the fixed-point equation (Eq. 25) that defines δ_{*,2}; for the quadratic structure of Eq. (24), the double-root collapse of the two fixed-point branches and the vanishing of the linearized eigenvalue are the same event (Λ2 = Q'(X*0) = 0 whenever the fixed-point polynomial has a double root). The 'coincidence' of Λ2 = 0 with the disappearance is therefore partly structural — a consequence of how both quantities are defined from the same assumed nonlinear equation — rather than an emergent finding. The specific value d_DR ≈ 5.11 is nonetheless an emergent numerical result, not fitted to a target, so this step is partially, not wholly, circular.

full rationale

The core quantitative result is not a fit: d_DR ≈ 5.11 emerges from locating where the fixed-point equation for δ_{*,2} (Eq. 25) loses real solutions, the four truncation levels (LPA', LPA'', DE2, DE4) bracket the value, and near d=6 the computed Λ2 reproduces the independent 2-loop results of Feldman and KRT, with Table I showing the ϵ² coefficients oscillating about the exact values. The paper is also honest about its central assumption: the disappearance mechanism relies on the generic-coalescence condition (R_NDR(X*0)Q''_NDR(X*0) ≠ 0; A_*(φ) ≠ 0 with sign-changing discriminant), which is verified analytically only for the RFO(N)M, and the paper labels the RFIM arguments in Appendices E/G 'not rigorous' and 'indicative'. Weighing against the paper: (i) the annihilation mechanism is inherited from the authors' own formalism — Eq. (24), the quadratic equation whose nonlinearity drives the conclusion, is taken from Ref. [27], a prior paper by the same authors — so the truncation's assumed form largely determines the outcome; (ii) d_DR is defined as the zero of Λ2, and since Λ2 is the linearization of the same equation that defines the fixed point's existence, the coincidence of Λ2 = 0 with disappearance is structural for the quadratic ansatz (Λ2 = Q'(X*) at a double root), not an independent empirical discovery, although the numerical value 5.11 is emergent; and (iii) the central rejection of the KRT scenario is conditional on an assumption whose failure would restore exactly the KRT picture. No fitted parameter is renamed as a prediction and no external benchmark is evaded, so the circularity is partial: score 4.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central d_DR result is not fitted to experimental or simulation data; it is computed from truncated FRG equations. The main input assumptions are the truncation scheme, SUSY-preserving regulator, zero-temperature scaling, sector decoupling, and the generic-coalescence condition. No new particles, forces, or conserved quantities are introduced.

free parameters (1)
  • Exponential regulator prefactor(s) = optimized by principle of minimum sensitivity near d_DR at each truncation level
    The IR regulator shape and prefactor are not fixed by the model; the d_DR values depend on truncation and regulator choice. The authors fold this into the ±0.09 error bar but do not provide a regulator scan.
assumptions (6)
  • domain assumption The exact FRG hierarchy truncated at finite derivative and cumulant order (LPA', LPA'', DE2, DE4) describes the physical fixed-point structure.
    The paper's central calculation solves these truncated equations; no proof is given that higher-order terms would not alter the annihilation mechanism.
  • domain assumption Parisi-Sourlas SUSY Ward identities are preserved by the regulator and hold in the equal-field sector.
    Used to relate δ(φ,φ) to z(φ) and to define the DR fixed point, for example in Eq. (23) and Sec. IV C.
  • domain assumption A zero-temperature fixed point exists with the dimensionless temperature flowing to zero, and the cumulant hierarchy scales with powers of inverse temperature.
    This is needed to set the scaling dimensions in Eqs. (17)-(18) and to derive the fixed-point equations for the cumulants.
  • ad hoc to paper Generic absence of symmetry-induced or accidental cancellation in the coalescence argument.
    Appendices B, E, and G rely on products such as R_NDR(X*0)Q''_NDR(X*0) being nonzero; if they vanish, the fixed point could cross instead of annihilate.
  • domain assumption The equal-field and difference-field sectors decouple for sufficiently regular field dependence of the cumulants.
    This decoupling is used to derive closed equations for δ_{k,2} and related functions, especially in the DE4 calculation.
  • standard math The Wetterich flow equation and the cumulant expansion in free replica sums are the correct starting point.
    Taken as background formal framework from Ref. 73 and previous papers; not re-proven here.

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Cite this review

Pith. "Pith review of On the breakdown of dimensional reduction and supersymmetry in random-field models." pith.science (2026). https://pith.science/paper/TU2GHV3S

@misc{pith2026241111147,
  author       = {Pith},
  title        = {Pith review of: On the breakdown of dimensional reduction and supersymmetry in random-field models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TU2GHV3S}},
  note         = {Machine review of arXiv:2411.11147}
}
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.

Figures

Figures reproduced from arXiv: 2411.11147 by the authors.

Figure 1
Figure 1. FIG. 1: Eigenvalue Λ [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Stability of the (most stable) SUSY/DR fixed point [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Stability of the SUSY/DR fixed points in the [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Schematic phase diagram of the RFO( [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 6
Figure 6. Figure 6: • The validity of the truncation beyond the pertur￾bative regime can be assessed by studying differ￾ent levels of the nonperturbative approximation scheme. This is what we do and detail in the next section. There, we show that the apparent conver￾gence of the approxima…
Figure 6
Figure 6. Figure 6: FIG. 6: Determination of the critical dimension [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Evidence for a subcusp in the SUSY/DR fixed point [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: One-loop Feynman diagrams with three 4-point ver [PITH_FULL_IMAGE:figures/full_fig_p025_8.png]

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