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arxiv: 2604.24675 · v1 · submitted 2026-04-27 · ❄️ cond-mat.stat-mech · hep-th

Conformal Invariance of the large-N limit of the O(N) universality class

Santiago Cabrera , Gonzalo De Polsi , Adam Ran\c{c}on , Nicol\'as Wschebor This is my paper

Pith reviewed 2026-05-07 17:53 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech hep-th
keywords conformal invarianceO(N) universality classlarge N limitnon-perturbative renormalization groupcritical phenomenaWard identities
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0 comments X

The pith

The large-N limit of the O(N) universality class realizes conformal invariance at criticality.

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

The paper provides two distinct proofs that the critical theory for the O(N) model in the large-N limit is conformally invariant, using the non-perturbative renormalization group approach. One proof is functional, treating the effective action as a whole, while the other examines individual vertices in Fourier space. A reader would care because this establishes conformal symmetry in a three-dimensional context, where it is less obvious than in two dimensions, and constrains how correlations decay and scale. The work also clarifies the structural requirements for any theory to exhibit this symmetry.

Core claim

Within the non-perturbative renormalization group, the large-N limit of the O(N) model at criticality satisfies the Ward identities for conformal transformations. This is demonstrated through a functional proof and a vertex-by-vertex verification in momentum space, revealing how the fixed-point theory is organized to support conformal invariance.

What carries the argument

The non-perturbative renormalization group flow equations in the large-N limit, used to prove that the fixed point is invariant under conformal transformations.

If this is right

  • The critical scaling behaviors must obey conformal constraints in addition to scale invariance.
  • The structure of the theory ensures that higher-order correlation functions are determined by conformal symmetry.
  • General principles for when a renormalization group fixed point will be conformally invariant are illuminated.
  • These results hold for the O(N) class in any dimension above two where the large-N limit applies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The proofs suggest that the large-N simplification makes conformal invariance more accessible to analytic treatment than in finite-N cases.
  • Similar structural analyses could be applied to other vector models or scalar theories to check for emergent conformal symmetry.
  • The vertex-by-vertex method provides a practical way to test conformal invariance in numerical or approximate RG studies.

Load-bearing premise

The non-perturbative renormalization group equations exactly describe the critical physics of the O(N) model once the large-N limit is taken, without needing further approximations.

What would settle it

A calculation showing that the four-point correlation function at the large-N critical point does not match the form required by conformal symmetry would disprove the claim.

Figures

Figures reproduced from arXiv: 2604.24675 by Adam Ran\c{c}on, Gonzalo De Polsi, Nicol\'as Wschebor, Santiago Cabrera.

Figure 1
Figure 1. Figure 1: FIG. 1. Diagrammatic representation of the view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Diagrammatic representation of two contributions view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Only diagram contributing to view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Independent diagrams contributing to the view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Independent diagrams contributing to view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Diagrammatic channel contribution to the flow of view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Diagrammatic channel contribution to the loop term view at source ↗
read the original abstract

Conformal symmetry is expected to be realized in many equilibrium statistical mechanical systems at criticality. Although this is certainly true in two-dimensional systems, the three-dimensional case is subtler, and only a few proofs exist, only so in very specific cases. In this work, we give two proofs for the large $N$ limit of the $O(N)$ universality class within the non-perturbative renormalization group framework: one functional, and one vertex-by-vertex in Fourier space. While doing so, we unveil how the theory is structured in order for conformal symmetry to be realized. As a consequence, we shed light on what to expect, on rather general grounds, for a theory to be conformally invariant.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript claims to provide two explicit proofs within the non-perturbative renormalization group (NPRG) framework that the large-N limit of the O(N) universality class is conformally invariant at criticality: one functional proof showing that the fixed-point solution satisfies the conformal Ward identities, and one vertex-by-vertex analysis in Fourier space. It additionally discusses the structural features of the theory required for conformal symmetry to hold.

Significance. If the derivations hold, the result is significant because explicit proofs of conformal invariance remain rare for three-dimensional critical theories. By taking the large-N limit first, the NPRG hierarchy closes exactly (reducing to the spherical model), yielding a controlled, parameter-free demonstration that the fixed-point solution is annihilated by the conformal generators for a symmetry-preserving regulator choice. The work is credited for its explicit derivations, the absence of uncontrolled truncations, and the general insights it offers into conditions for conformal invariance.

minor comments (3)
  1. [Abstract] The abstract states that two proofs are given but does not indicate the central technical step in each; a single sentence per proof would improve accessibility.
  2. [Vertex-by-vertex analysis] In the vertex-by-vertex proof, the assumption that the regulator preserves the necessary momentum-space symmetries is stated but not verified equation-by-equation for the special conformal transformations; an explicit check would strengthen the presentation.
  3. [Conclusion] The discussion of general conditions for conformal invariance in the conclusion is insightful but would benefit from a concise enumerated list of the necessary structural properties identified in the large-N case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. The referee's summary correctly identifies the two explicit proofs (functional and vertex-by-vertex) of conformal invariance for the large-N O(N) fixed point within the NPRG, as well as the structural insights into the conditions for conformal symmetry. We appreciate the recognition that the exact closure of the hierarchy in the large-N limit provides a controlled demonstration without truncations.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper derives conformal invariance of the large-N O(N) fixed point from the NPRG Wetterich equation after taking N→∞ first, which exactly closes the hierarchy into a known spherical-model equation. Two explicit proofs (functional and vertex-by-vertex in Fourier space) are given showing that the fixed-point solution is annihilated by the conformal generators when the regulator preserves the symmetries. No parameter is fitted to the target conformal property and then renamed a prediction; no self-citation chain is invoked to justify the central result; the structure for conformal symmetry is extracted from the closed equations rather than inserted by definition. The work is therefore independent of its inputs beyond the standard NPRG setup and the exact large-N limit.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the exactness of the NPRG flow equations in the large-N limit and on standard properties of the O(N) model; no free parameters or new entities are mentioned in the abstract.

axioms (2)
  • domain assumption The non-perturbative renormalization group equations provide an exact description of the critical fixed point in the large-N limit.
    This is the framework in which both proofs are constructed.
  • standard math Conformal invariance can be checked by examining the structure of the effective action or its vertices at the fixed point.
    Invoked when moving from the functional proof to the vertex-by-vertex proof.

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discussion (0)

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Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

    Furthermore, one can easily show that[∂2, Dx] = 2∂2, with∂ 2 =∂ ν∂ν

    Useful identities It is useful to note that R x f(x)D xg(x) = R x g(x)(−d−D x)f(x). Furthermore, one can easily show that[∂2, Dx] = 2∂2, with∂ 2 =∂ ν∂ν. This implies the first identity I1 ≡ Z x f(x)∂ 2(Dxf(x)), = Z x f(x) 2 +D x]∂ 2f(x), = Z x (2−d−D x f(x)]∂ 2f(x), = (2−d) Z x f(x)∂ 2f(x)−I 1, (B1) and thus I1 =− d−2 2 Z x f(x)∂ 2f(x).(B2) This, in turn,...

    work page

  2. [2]

    Proof of Eq.(57) With this preamble, we are now in a position to show that Z y,z G[x, y]∂ tRk(y, z)G[z, x] =−D G[x, x]−G 0(x, x) .(B12) 18 To this end, we start from T1 ≡ Z y,z G[x, y]∂ tRk(y, z)G[z, x] = Z y,z G[x, y] 2d−2D φ +D y +D z Rk(y, z)G[z, x], =− Z y,z G[x, y]R k(y, z) 2Dφ + 2Dz G[z, x]. (B13) ReplacingR k(y, z)byG −1[y, z]−δ(y−z)[−∂ 2 z +σ(z)],...

    work page

  3. [3]

    Then we have used thatDx acts on both arguments of G[x, x], and thus R zδ(x−z)D z G[z, x] = 1 2 DxG[x, x], and similarly forG0(x, x) Using Eqs

    The subtraction is necessary to ensure UV convergence. Then we have used thatDx acts on both arguments of G[x, x], and thus R zδ(x−z)D z G[z, x] = 1 2 DxG[x, x], and similarly forG0(x, x) Using Eqs. (B3) and (B5), we have Z z G[x, z] [−∂2 z +σ(z)] 2Dφ + 2Dz G[z, x] =− Z z G[x, z] (d−2D φ +D z)σ(z) G[x, z], = Z z (d−2D φ +D z)σ(z) δG[x, x] δσ(z) , = Z z (d...

    work page

  4. [4]

    channel 1

    Proof of Eq.(61) Finally, we wish to prove that Z y,z G[x, y] K y µ +K z µ −(2d−2D φ)(yµ +z µ) Rk(y, z)G[z, x] =−K µ G[x, x]−G 0(x, x) .(B16) The proof is similar to that just above. We start from T2 =− Z y,z G[x, y] K y µ +K z µ −2(2d−2D φ)(yµ +z µ) Rk(y, z)G[z, x], =− Z y,z G[x, y]R k(y, z) −2K z µ + 4Dφzµ G[z, x], =− Z y,z G[x, y] G−1[y, z] +δ(y−z)∂ 2 ...

    work page

  5. [5]

    In the first case, ther momentum is at the beginning: ∂rµ I(n+1)(r, p1,

    The left-hand side term∂ rµ I(n+1) When computing the∂ rµ contribution, in general, we have three different possibilities. In the first case, ther momentum is at the beginning: ∂rµ I(n+1)(r, p1, . . . , pn−1)|r=0 = ∂rµ Z q G(q)G(q+r) n−1Y i=1 G q+r+ iX j=1 pj r=0 =∂ 1µ Z q G(q)2 n−1Y i=1 G q+r+ iX j=1 pj +1 2 Z q ∂qµ G2(q) n−1Y i=1 G q+r+ iX j=1 pj = 1 2 ...

    work page

  6. [6]

    ∂iµ∂jν”; ii) a structure proportional to “∂iν∂jµ

    The right-hand side term(∂ qµ +∂ q′µ)H(n) j To evaluate these contributions we start by considering the cases where theimplicitexternal legp n is right next to the regulator insertion in the diagrams. There are two such contributions which correspond toJ(n) 0,µ andJ (n) n−1,µ. These are J(n) 0,µ = Z q ∂tRk(q2)G2(q) × ∂qµ +∂ q′µ n n−1Y i=1 G q+ iX j=1 pj o...

    work page

  7. [7]

    Polchinski, Scale and conformal invariance in quantum field theory, Nucl

    J. Polchinski, Scale and conformal invariance in quantum field theory, Nucl. Phys. B303, 226 (1988)

    work page 1988

  8. [8]

    A. B. Zamolodchikov, Irreversibility of the flux of the renormalization group in a 2d field theory, JETP Lett. 43, 730 (1986)

    work page 1986

  9. [9]

    Di Francesco, P

    P. Di Francesco, P. Mathieu, and D. Senechal,Conformal 25 Field Theory, Graduate Texts in Contemporary Physics (Springer-Verlag, New York, 1997)

    work page 1997

  10. [10]

    Delamotte, M

    B. Delamotte, M. Tissier, and N. Wschebor, Scale invariance implies conformal invariance for the three- dimensional ising model, Phys. Rev. E93, 012144 (2016)

    work page 2016

  11. [11]

    De Polsi, M

    G. De Polsi, M. Tissier, and N. Wschebor, Conformal Invariance and Vector Operators in the O(N) Model, J. Stat. Phys.177, 1089 (2019)

    work page 2019

  12. [12]

    Riva and J

    V. Riva and J. Cardy, Scale and conformal invariance in field theory: a physical counterexample, Phys. Lett. B 622, 339 (2005)

    work page 2005

  13. [13]

    K. G. Wilson and J. B. Kogut, The Renormalization group and the epsilon expansion, Phys. Rept.12, 75 (1974)

    work page 1974

  14. [14]

    Zinn-Justin,Quantum Field Theory and Critical Phenomena(Oxford University Press, 2021)

    J. Zinn-Justin,Quantum Field Theory and Critical Phenomena(Oxford University Press, 2021)

    work page 2021

  15. [15]

    Pelissetto and E

    A. Pelissetto and E. Vicari, Critical phenomena and renormalization-group theory, Physics Reports368, 549 (2002)

    work page 2002

  16. [16]

    Poland, S

    D. Poland, S. Rychkov, and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques, and Applications, Rev. Mod. Phys.91, 015002 (2019)

    work page 2019

  17. [17]

    Pagani and H

    C. Pagani and H. Sonoda, Explicit construction of the energy-momentum tensor in the largenlimit, Phys. Rev. D113, 025012 (2026)

    work page 2026

  18. [18]

    Polchinski, Renormalization and Effective Lagrangians, Nucl

    J. Polchinski, Renormalization and Effective Lagrangians, Nucl. Phys.B231, 269 (1984)

    work page 1984

  19. [19]

    Wetterich, Exact evolution equation for the effective potential, Phys

    C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B301, 90 (1993)

    work page 1993

  20. [20]

    Ellwanger, Collective fields and flow equations, Z

    U. Ellwanger, Collective fields and flow equations, Z. Phys.C58, 619 (1993)

    work page 1993

  21. [21]

    T. R. Morris, The Exact renormalization group and approximate solutions, Int. J. Mod. Phys.A9, 2411 (1994)

    work page 1994

  22. [22]

    Berges, N

    J. Berges, N. Tetradis, and C. Wetterich, Nonperturbative renormalization flow in quantum field theory and statistical physics, Phys. Rept.363, 223 (2002)

    work page 2002

  23. [23]

    Delamotte, An Introduction to the nonperturbative renormalization group, Lect

    B. Delamotte, An Introduction to the nonperturbative renormalization group, Lect. Notes Phys.852, 49 (2012)

    work page 2012

  24. [24]

    Dupuis, L

    N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier, and N. Wschebor, The nonperturbative functional renormalization group and its applications, Phys. Rept.910, 1 (2021)

    work page 2021

  25. [25]

    Delamotte, G

    B. Delamotte, G. De Polsi, M. Tissier, and N. Wschebor, Conformal invariance and composite operators: A strategy for improving the derivative expansion of the nonperturbative renormalization group, Phys. Rev. E 109, 064152 (2024)

    work page 2024

  26. [26]

    Cabrera, G

    S. Cabrera, G. De Polsi, and N. Wschebor, Conformal invariance constraints in theo(n)models: A study within the nonperturbative renormalization group, Phys. Rev. E 111, 054126 (2025)

    work page 2025

  27. [27]

    T. R. Morris, Derivative expansion of the exact renormalization group, Phys. Lett.B329, 241 (1994)

    work page 1994

  28. [28]

    Balog, G

    I. Balog, G. De Polsi, M. Tissier, and N. Wschebor, Conformal invariance in the nonperturbative renormalization group: A rationale for choosing the regulator, Phys. Rev. E101, 062146 (2020)

    work page 2020

  29. [29]

    D’Attanasio and T

    M. D’Attanasio and T. R. Morris, Large n and the renormalization group, Phys. Lett. B409, 363 (1997)

    work page 1997

  30. [30]

    J.P.Blaizot, R.Méndez-Galain,andN.Wschebor,Anew method to solve the non-perturbative renormalization group equations, Phys. Lett. B632, 571 (2006)

    work page 2006

  31. [31]

    Moshe and J

    M. Moshe and J. Zinn-Justin, Quantum field theory in the large N limit: A Review, Phys. Rept.385, 69 (2003)

    work page 2003

  32. [32]

    Knorr, Exact solutions and residual regulator dependence in functional renormalisation group flows, J

    B. Knorr, Exact solutions and residual regulator dependence in functional renormalisation group flows, J. Phys. A54, 275401 (2021)

    work page 2021

  33. [33]

    Sonoda, Exact Renormalization Group in LargeN, (2023), arXiv:2302.09914

    H. Sonoda, Exact Renormalization Group in LargeN, (2023), arXiv:2302.09914

    work page arXiv 2023

  34. [34]

    Yabunaka and B

    S. Yabunaka and B. Delamotte, Surprises in O(N) models: nonperturbative fixed points, large N limits, and multicriticality, Phys. Rev. Lett.119, 191602 (2017)

    work page 2017

  35. [35]

    Yabunaka and B

    S. Yabunaka and B. Delamotte, Why Might the Standard LargeNAnalysis Fail in the O(N) Model: The Role of Cusps in the Fixed Point Potentials, Phys. Rev. Lett. 121, 231601 (2018)

    work page 2018

  36. [36]

    Yabunaka and B

    S. Yabunaka and B. Delamotte, One Fixed Point Can Hide Another One: Nonperturbative Behavior of the Tetracritical Fixed Point of O(N) Models at LargeN, Phys. Rev. Lett.130, 261602 (2023)

    work page 2023

  37. [37]

    S. Cabrera,Grupo de renormalización no perturbativo de los modelosO(N): explorando la simetría conforme, Master’s thesis, Universidad de la República, Facultad de Ciencias (PEDECIBA), Montevideo, Uruguay (2024)

    work page 2024

  38. [38]

    Polchinski, Scale and conformal invariance in quantum field theory, Nuclear Physics B303, 226 (1988)

    J. Polchinski, Scale and conformal invariance in quantum field theory, Nuclear Physics B303, 226 (1988)

    work page 1988