pith. sign in

arxiv: 2602.04313 · v2 · submitted 2026-02-04 · ❄️ cond-mat.stat-mech

Critical behavior of isotropic systems with strong dipole-dipole interaction from the functional renormalization group

Georgii Kalagov , Nikita Lebedev This is my paper

Pith reviewed 2026-05-16 07:33 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords functional renormalization groupdipole-dipole interactionscritical exponentsAharony fixed pointHeisenberg universalitythree-dimensional magnetslocal potential approximation
0
0 comments X

The pith

Functional renormalization group analysis shows that strong dipole-dipole interactions in three-dimensional magnets lead to critical exponents close to the Heisenberg O(3) universality class but governed by the distinct Aharony fixed point.

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

The paper applies the functional renormalization group method in the local potential approximation with wave-function renormalization to isotropic systems with strong dipole-dipole interactions. It locates the Aharony fixed point that governs the critical behavior in such systems. This fixed point is scale-invariant yet lacks conformal invariance. The computed critical exponents are numerically close to those of the Heisenberg O(3) class when both are calculated within the same approximation. This result indicates that the two universality classes are separate but their scaling properties are similar enough to require careful distinction in physical systems.

Core claim

The Aharony fixed point controls the critical behavior of three-dimensional magnets with strong dipole-dipole interactions. Nonperturbative FRG in the LPA' scheme identifies this fixed point and extracts its scaling behavior, producing critical exponents that are close to the values for the Heisenberg O(3) universality class obtained in the identical framework. This establishes the distinct yet numerically similar character of the Aharony universality class.

What carries the argument

The Aharony fixed point, the infrared fixed point for systems dominated by dipole-dipole interactions that is scale invariant but not conformally invariant.

If this is right

  • The critical exponents for these dipole systems can be predicted nonperturbatively using FRG.
  • Physical realizations of such magnets should display scaling laws distinct from pure short-range Heisenberg models.
  • Conformal invariance is absent, leading to different properties in correlation functions compared to standard classes.
  • The proximity to O(3) exponents suggests that high-precision measurements are needed to observe the difference.

Where Pith is reading between the lines

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

  • Materials with strong dipolar forces may appear to follow Heisenberg scaling in low-precision experiments due to the numerical closeness.
  • Extending the FRG calculation beyond LPA' could quantify the truncation error and confirm the proximity.
  • Similar analyses might apply to other long-range interaction systems where fixed points lack full conformal symmetry.

Load-bearing premise

The local potential approximation including wave-function renormalization is accurate enough to capture the essential features of the Aharony fixed point without large errors from higher-order terms.

What would settle it

A precise experimental determination of the critical exponent nu or beta in a three-dimensional isotropic magnet with dominant dipole-dipole interactions that shows a significant deviation from the FRG-computed value would falsify the result.

read the original abstract

We compute the critical exponents of three-dimensional magnets with strong dipole-dipole interactions using the functional renormalization group (FRG) within the local potential approximation including the wave function renormalization (LPA$^\prime$). The system is governed by the Aharony fixed point, which is scale-invariant but lacks conformal invariance. Our nonperturbative FRG analysis identifies this fixed point and determines its scaling behavior. The resulting critical exponents are found to be close to those of the Heisenberg $O(3)$ universality class, as computed within the same FRG/LPA$^\prime$ framework. This proximity confirms the distinct yet numerically similar nature of the two universality classes.

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

1 major / 0 minor

Summary. The manuscript computes the critical exponents of three-dimensional isotropic magnets with strong dipole-dipole interactions using the functional renormalization group (FRG) within the local potential approximation including wave-function renormalization (LPA'). It identifies the Aharony fixed point, which is scale-invariant but lacks conformal invariance, and reports that the resulting critical exponents are numerically close to those of the Heisenberg O(3) universality class when both are computed in the same FRG/LPA' framework. This is presented as confirming the distinct yet similar nature of the two universality classes.

Significance. If the LPA' truncation proves sufficient, the work supplies nonperturbative evidence for the existence and scaling properties of the Aharony fixed point in dipolar systems and offers a controlled comparison to the Heisenberg class within a single approximation scheme. Such a result would be useful for interpreting experiments on dipolar magnets and for clarifying the role of long-range interactions in breaking conformal invariance while preserving scale invariance.

major comments (1)
  1. The central claim that the Aharony fixed point is identified and its exponents are close to O(3) rests on the LPA' truncation. No explicit test of truncation error is reported (e.g., by extending to a next-order derivative expansion or by comparing regulators), despite the long-range 1/r^3 dipolar tail being expected to generate momentum-dependent corrections to the propagator and vertices that lie outside LPA'. This omission leaves open the possibility that the observed numerical proximity is an artifact of the truncation rather than a property of the universality class.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on truncation errors. We address the point below and indicate the changes planned for the revised manuscript.

read point-by-point responses

  1. Referee: The central claim that the Aharony fixed point is identified and its exponents are close to O(3) rests on the LPA' truncation. No explicit test of truncation error is reported (e.g., by extending to a next-order derivative expansion or by comparing regulators), despite the long-range 1/r^3 dipolar tail being expected to generate momentum-dependent corrections to the propagator and vertices that lie outside LPA'. This omission leaves open the possibility that the observed numerical proximity is an artifact of the truncation rather than a property of the universality class.

    Authors: We agree that the results rely on the LPA' truncation and that no explicit convergence test with respect to higher orders in the derivative expansion or alternative regulators is presented. This is a genuine limitation of the current work. Within LPA', however, the Aharony fixed point is unambiguously identified as the unique non-trivial infrared attractor of the flow equations for the effective potential and the wave-function renormalization, distinct from both the Gaussian and the Heisenberg fixed points. The numerical proximity of the critical exponents to those of the O(3) class is obtained under identical truncation and regulator choices, which at least ensures a controlled comparison inside the approximation. In the revised manuscript we will add a dedicated paragraph in the discussion section that (i) recalls the expected size of higher-order corrections for long-range dipolar interactions, (ii) references existing FRG benchmarks for the Heisenberg class at the same truncation level, and (iii) outlines how a next-order calculation could be performed. We do not claim that the present numbers are final; we present them as the first non-perturbative FRG estimate within a standard and well-tested approximation scheme. revision: partial

Circularity Check

0 steps flagged

No significant circularity in FRG fixed-point analysis

full rationale

The derivation proceeds by solving the functional renormalization group flow equations in the LPA' truncation to locate the Aharony fixed point and extract its critical exponents numerically. The comparison to the O(3) Heisenberg class is performed inside the identical truncation and regulator choice, which constitutes a controlled consistency check rather than a definitional reduction. No step equates a reported exponent to a fitted input by construction, nor does any load-bearing premise collapse to a self-citation whose validity is presupposed. The fixed-point equations and eigenvalue spectrum are independent of the final numerical values quoted; the observed proximity to Heisenberg exponents emerges from the solution of those equations. The analysis is therefore self-contained within the stated truncation and does not exhibit the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the LPA' truncation of the effective action in FRG and on the numerical identification of the Aharony fixed point; no explicit free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The LPA' truncation accurately describes the Aharony fixed point in three-dimensional dipolar systems
    Invoked to justify the reliability of the computed exponents.

pith-pipeline@v0.9.0 · 5408 in / 1169 out tokens · 34574 ms · 2026-05-16T07:33:42.427358+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A Monte Carlo Study of the Dipolar Universality Class in Three Dimensions

    hep-th 2026-05 unverdicted novelty 7.0

    Monte Carlo simulations on lattices up to 48 cubed produce estimates of critical exponents for the 3D dipolar universality class, confirm a continuous phase transition, and show restoration of rotation invariance.

Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages · cited by 1 Pith paper

  1. [1]

    L. P. KADANOFF, W. G ¨OTZE, D. HAMBLEN, R. HECHT, E. A. S. LEWIS, V . V . PALCIAUSKAS, M. RAYL, J. SWIFT, D. ASPNES, J. KANE, Static phenomena near critical points: Theory and experiment, Rev. Mod. Phys. 39 (1967) 395–431.doi:10.1103/RevModPhys. 39.395. URLhttps://link.aps.org/doi/10.1103/RevModPhys.39.395

    work page doi:10.1103/revmodphys 1967

  2. [2]

    M. E. Fisher, A. Aharony, Dipolar interactions at ferromagnetic critical points, Phys. Rev. Lett. 30 (1973) 559–562.doi:10.1103/ PhysRevLett.30.559. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.30.559

    work page doi:10.1103/physrevlett.30.559 1973

  3. [3]

    Aharony, M

    A. Aharony, M. E. Fisher, Critical Behavior of Magnets with Dipolar Interactions. I. Renormalization Group near Four Dimensions, Phys. Rev. B 8 (1973) 3323–3341.doi:10.1103/PhysRevB.8.3323. URLhttps://link.aps.org/doi/10.1103/PhysRevB.8.3323

    work page doi:10.1103/physrevb.8.3323 1973

  4. [4]

    Aharony, Critical Behavior of Magnets with Dipolar Interactions

    A. Aharony, Critical Behavior of Magnets with Dipolar Interactions. II. Feynman-Graph Expansion for Ferromagnets near Four Dimensions, Phys. Rev. B 8 (1973) 3342–3348.doi:10.1103/PhysRevB.8.3342. URLhttps://link.aps.org/doi/10.1103/PhysRevB.8.3342

    work page doi:10.1103/physrevb.8.3342 1973

  5. [5]

    Aharony, Critical Behavior of Magnets with Dipolar Interactions

    A. Aharony, Critical Behavior of Magnets with Dipolar Interactions. III. Antiferromagnets, Phys. Rev. B 8 (1973) 3349–3357.doi:10. 1103/PhysRevB.8.3349. URLhttps://link.aps.org/doi/10.1103/PhysRevB.8.3349

    work page doi:10.1103/physrevb.8.3349 1973

  6. [6]

    K ¨otzler, D

    J. K ¨otzler, D. G¨orlitz, F. Mezei, B. Farago, Depression of longitudinal fluctuations above tc of heisenberg ferromagnets observed by polarized neutrons, EPL (Europhysics Letters) 1 (12) (1986) 675–680.doi:10.1209/0295-5075/1/12/010. URLhttps://iopscience.iop.org/article/10.1209/0295-5075/1/12/010/meta

    work page doi:10.1209/0295-5075/1/12/010 1986

  7. [7]

    Kudlis, A

    A. Kudlis, A. Pikelner, Critical behavior of isotropic systems with strong dipole-dipole interaction: Three-loop study, Nuclear Physics B 985 (2022) 115990.doi:https://doi.org/10.1016/j.nuclphysb.2022.115990. URLhttps://www.sciencedirect.com/science/article/pii/S0550321322003418

    work page doi:10.1016/j.nuclphysb.2022.115990 2022

  8. [8]

    A. D. Bruce, A. Aharony, Critical exponents of ferromagnets with dipolar interactions: Second-orderϵexpansion, Phys. Rev. B 10 (1974) 2078–2087.doi:10.1103/PhysRevB.10.2078. URLhttps://link.aps.org/doi/10.1103/PhysRevB.10.2078

    work page doi:10.1103/physrevb.10.2078 1974

  9. [9]

    G. A. Baker, B. G. Nickel, M. S. Green, D. I. Meiron, Ising-model critical indices in three dimensions from the callan-symanzik equation, Phys. Rev. Lett. 36 (1976) 1351–1354.doi:10.1103/PhysRevLett.36.1351. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.36.1351

    work page doi:10.1103/physrevlett.36.1351 1976

  10. [10]

    J. C. Le Guillou, J. Zinn-Justin, Critical Exponents for then-Vector Model in Three Dimensions from Field Theory, Phys. Rev. Lett. 39 (1977) 95–98.doi:10.1103/PhysRevLett.39.95. URLhttps://link.aps.org/doi/10.1103/PhysRevLett.39.95

    work page doi:10.1103/physrevlett.39.95 1977

  11. [11]

    Henriksson, The criticalO(N) CFT: Methods and conformal data, Physics Reports 1002 (2023) 1–72.doi:https://doi.org/10.1016/ j.physrep.2022.12.002

    J. Henriksson, The criticalO(N) CFT: Methods and conformal data, Physics Reports 1002 (2023) 1–72.doi:https://doi.org/10.1016/ j.physrep.2022.12.002. URLhttps://www.sciencedirect.com/science/article/pii/S0370157322004057

    work page 2023

  12. [12]

    Schnetz, Numbers and functions in quantum field theory, Phys

    O. Schnetz, Numbers and functions in quantum field theory, Phys. Rev. D 97 (2018) 085018.doi:10.1103/PhysRevD.97.085018. URLhttps://link.aps.org/doi/10.1103/PhysRevD.97.085018

    work page doi:10.1103/physrevd.97.085018 2018

  13. [13]

    A. M. Shalaby, Critical exponents of theO(N)-symmetricϕ 4 model from theε 7 hypergeometric-Meijer resummation, The European Physical Journal C 81 (1) (Jan. 2021).doi:10.1140/epjc/s10052-021-08884-5. URLhttp://dx.doi.org/10.1140/epjc/s10052-021-08884-5

    work page doi:10.1140/epjc/s10052-021-08884-5 2021

  14. [14]

    Abhignan, R

    V . Abhignan, R. Sankaranarayanan, Continued Functions and Perturbation Series: Simple Tools for Convergence of Diverging Se- ries inO(n)-Symmetricϕ 4 Field Theory at Weak Coupling Limit, Journal of Statistical Physics 183 (1) (Mar. 2021).doi:10.1007/ s10955-021-02719-z. URLhttp://dx.doi.org/10.1007/s10955-021-02719-z

    work page doi:10.1007/s10955-021-02719-z 2021

  15. [15]

    Nickel, D

    B. Nickel, D. Meiron, G. Baker Jr, University of guelph report, Guelph, Ontario (1977)

    work page 1977

  16. [16]

    H. Mera, T. G. Pedersen, B. K. Nikoli ´c, Hypergeometric resummation of self-consistent sunset diagrams for steady-state electron-boson quantum many-body systems out of equilibrium, Phys. Rev. B 94 (2016) 165429.doi:10.1103/PhysRevB.94.165429. URLhttps://link.aps.org/doi/10.1103/PhysRevB.94.165429

    work page doi:10.1103/physrevb.94.165429 2016

  17. [17]

    H. Mera, T. G. Pedersen, B. K. Nikoli ´c, Fast summation of divergent series and resurgent transseries from meijer-gapproximants, Phys. Rev. D 97 (2018) 105027.doi:10.1103/PhysRevD.97.105027. URLhttps://link.aps.org/doi/10.1103/PhysRevD.97.105027

    work page doi:10.1103/physrevd.97.105027 2018

  18. [18]

    M. V . Kompaniets, E. Panzer, Minimally subtracted six-loop renormalization ofO(n)-symmetricϕ 4 theory and critical exponents, Phys. Rev. D 96 (2017) 036016.doi:10.1103/PhysRevD.96.036016. URLhttps://link.aps.org/doi/10.1103/PhysRevD.96.036016

    work page doi:10.1103/physrevd.96.036016 2017

  19. [19]

    F. Kos, D. Poland, D. Simmons-Duffin, A. Vichi, Precision islands in the Ising andO(N) models, Journal of High Energy Physics 2016 (8) (2016) 1–16.doi:10.1007/JHEP08(2016)036

    work page doi:10.1007/jhep08(2016)036 2016

  20. [20]

    Gimenez-Grau, Y

    A. Gimenez-Grau, Y . Nakayama, S. Rychkov, Scale without conformal invariance in dipolar ferromagnets, Phys. Rev. B 110 (2024) 024421. doi:10.1103/PhysRevB.110.024421. URLhttps://link.aps.org/doi/10.1103/PhysRevB.110.024421

    work page doi:10.1103/physrevb.110.024421 2024

  21. [21]

    Non-perturbative renormalization flow in quantum field theory and statistical physics

    J. Berges, N. Tetradis, C. Wetterich, Non-perturbative renormalization flow in quantum field theory and statistical physics, Physics Reports 363 (4) (2002) 223–386.doi:https://doi.org/10.1016/S0370-1573(01)00098-9. URLhttps://www.sciencedirect.com/science/article/pii/S0370157301000989

    work page doi:10.1016/s0370-1573(01)00098-9 2002

  22. [22]

    J. M. Pawlowski, Aspects of the functional renormalisation group, Annals of Physics 322 (12) (2007) 2831–2915.doi:https://doi.org/ 16 10.1016/j.aop.2007.01.007. URLhttps://www.sciencedirect.com/science/article/pii/S0003491607000097

    work page doi:10.1016/j.aop.2007.01.007 2007

  23. [23]

    Dupuis, L

    N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. Pawlowski, M. Tissier, N. Wschebor, The nonperturbative functional renormalization group and its applications, Physics Reports 910 (2021) 1–114.doi:https://doi.org/10.1016/j.physrep.2021.01.001. URLhttps://www.sciencedirect.com/science/article/pii/S0370157321000156

    work page doi:10.1016/j.physrep.2021.01.001 2021

  24. [24]

    Nakayama, Functional renormalization group approach to dipolar fixed point which is scale invariant but nonconformal, Phys

    Y . Nakayama, Functional renormalization group approach to dipolar fixed point which is scale invariant but nonconformal, Phys. Rev. D 110 (2024) 025020.doi:10.1103/PhysRevD.110.025020. URLhttps://link.aps.org/doi/10.1103/PhysRevD.110.025020

    work page doi:10.1103/physrevd.110.025020 2024

  25. [25]

    El-Showk, Y

    S. El-Showk, Y . Nakayama, S. Rychkov, What Maxwell theory ind,4 teaches us about scale and conformal invariance, Nuclear Physics B 848 (3) (2011) 578–593.doi:https://doi.org/10.1016/j.nuclphysb.2011.03.008. URLhttps://www.sciencedirect.com/science/article/pii/S0550321311001465

    work page doi:10.1016/j.nuclphysb.2011.03.008 2011

  26. [26]

    Mauri, M

    A. Mauri, M. I. Katsnelson, Scale without conformal invariance in membrane theory, Nuclear Physics B 969 (2021) 115482.doi:https: //doi.org/10.1016/j.nuclphysb.2021.115482. URLhttps://www.sciencedirect.com/science/article/pii/S0550321321001796

    work page doi:10.1016/j.nuclphysb.2021.115482 2021

  27. [27]

    D. F. Litim, Optimisation of the exact renormalisation group, Physics Letters B 486 (1) (2000) 92–99.doi:https://doi.org/10.1016/ S0370-2693(00)00748-6. URLhttps://www.sciencedirect.com/science/article/pii/S0370269300007486

    work page 2000

  28. [28]

    D. F. Litim, Optimized renormalization group flows, Phys. Rev. D 64 (2001) 105007.doi:10.1103/PhysRevD.64.105007. URLhttps://link.aps.org/doi/10.1103/PhysRevD.64.105007

    work page doi:10.1103/physrevd.64.105007 2001

  29. [29]

    Tetradis, C

    N. Tetradis, C. Wetterich, Critical exponents from the effective average action, Nuclear Physics B 422 (3) (1994) 541–592.doi:https: //doi.org/10.1016/0550-3213(94)90446-4. URLhttps://www.sciencedirect.com/science/article/pii/0550321394904464

    work page doi:10.1016/0550-3213(94)90446-4 1994

  30. [30]

    T. R. Morris, On truncations of the exact renormalization group, Physics Letters B 334 (3) (1994) 355–362.doi:https://doi.org/10. 1016/0370-2693(94)90700-5. URLhttps://www.sciencedirect.com/science/article/pii/0370269394907005

    work page arXiv 1994

  31. [31]

    Codello, Scaling solutions in a continuous dimension, Journal of Physics A: Mathematical and Theoretical 45 (46) (2012) 465006

    A. Codello, Scaling solutions in a continuous dimension, Journal of Physics A: Mathematical and Theoretical 45 (46) (2012) 465006. doi:10.1088/1751-8113/45/46/465006. URLhttps://doi.org/10.1088/1751-8113/45/46/465006

    work page doi:10.1088/1751-8113/45/46/465006 2012

  32. [32]

    Borchardt, B

    J. Borchardt, B. Knorr, Global solutions of functional fixed point equations via pseudospectral methods, Phys. Rev. D 91 (2015) 105011. doi:10.1103/PhysRevD.91.105011. URLhttps://link.aps.org/doi/10.1103/PhysRevD.91.105011

    work page doi:10.1103/physrevd.91.105011 2015

  33. [33]

    Murgana, A

    F. Murgana, A. Koenigstein, D. H. Rischke, Reanalysis of critical exponents for theO(N) model via a hydrodynamic approach to the functional renormalization group, Phys. Rev. D 108 (2023) 116016.doi:10.1103/PhysRevD.108.116016. URLhttps://link.aps.org/doi/10.1103/PhysRevD.108.116016

    work page doi:10.1103/physrevd.108.116016 2023

  34. [34]

    K. I. Aoki, K. Morikawa, W. Souma, J. I. Sumi, H. Terao, Rapidly converging truncation scheme of the exact renormalization group, Progress of Theoretical Physics 99 (1998) 451–466.doi:10.1143/PTP.99.451. URLhttps://academic.oup.com/ptp/article/99/3/451/1845807

    work page doi:10.1143/ptp.99.451 1998

  35. [35]

    Kuipers, T

    J. Kuipers, T. Ueda, J. Vermaseren, J. V ollinga, Form version 4.0, Computer Physics Communications 184 (5) (2013) 1453–1467.doi: https://doi.org/10.1016/j.cpc.2012.12.028. URLhttps://www.sciencedirect.com/science/article/pii/S0010465513000052

    work page doi:10.1016/j.cpc.2012.12.028 2013

  36. [36]

    Kalagov, Maple file, includes (.mw)-file with explicit expressions for the scalar coefficients used in Eq

    G. Kalagov, Maple file, includes (.mw)-file with explicit expressions for the scalar coefficients used in Eq. (22). (2025).doi:10.5281/ zenodo.18403539. URLhttps://doi.org/10.5281/zenodo.18403539

    work page doi:10.5281/zenodo.18403539 2025

  37. [38]

    Boettcher, Scaling relations and multicritical phenomena from functional renormalization, Phys

    I. Boettcher, Scaling relations and multicritical phenomena from functional renormalization, Phys. Rev. E 91 (2015) 062112.doi:10.1103/ PhysRevE.91.062112. URLhttps://link.aps.org/doi/10.1103/PhysRevE.91.062112

    work page doi:10.1103/physreve.91.062112 2015

  38. [39]

    A. I. Sokolov, M. A. Nikitina, Pseudo-ϵexpansion and renormalized coupling constants at criticality, Phys. Rev. E 89 (2014) 052127. doi:10.1103/PhysRevE.89.052127. URLhttps://link.aps.org/doi/10.1103/PhysRevE.89.052127

    work page doi:10.1103/physreve.89.052127 2014

  39. [40]

    Sokolov, A

    A. Sokolov, A. Kudlis, M. Nikitina, Effective potential of the three-dimensional Ising model: The pseudo-ϵexpansion study, Nuclear Physics B 921 (2017) 225–235.doi:https://doi.org/10.1016/j.nuclphysb.2017.05.019. URLhttps://www.sciencedirect.com/science/article/pii/S0550321317301931

    work page doi:10.1016/j.nuclphysb.2017.05.019 2017

  40. [41]

    Kudlis, A

    A. Kudlis, A. Sokolov, Universal effective couplings of the three-dimensional n-vector model and field theory, Nuclear Physics B 950 (2020) 114881.doi:https://doi.org/10.1016/j.nuclphysb.2019.114881. URLhttps://www.sciencedirect.com/science/article/pii/S0550321319303670

    work page doi:10.1016/j.nuclphysb.2019.114881 2020

  41. [42]

    De Polsi, G

    G. De Polsi, G. Hern ´andez-Chifflet, N. Wschebor, Precision calculation of universal amplitude ratios inO(N) universality classes: Derivative expansion results at orderO(∂ 4), Physical Review E 104 (6) (2021) 064101.doi:https://doi.org/10.1103/PhysRevE.104.064101. URLhttps://journals.aps.org/pre/abstract/10.1103/PhysRevE.104.064101

    work page doi:10.1103/physreve.104.064101 2021

  42. [43]

    Campostrini, M

    M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, E. Vicari, Critical exponents and equation of state of the three-dimensional heisenberg universality class, Phys. Rev. B 65 (2002) 144520.doi:10.1103/PhysRevB.65.144520. URLhttps://link.aps.org/doi/10.1103/PhysRevB.65.144520

    work page doi:10.1103/physrevb.65.144520 2002

  43. [44]

    De Polsi, I

    G. De Polsi, I. Balog, M. Tissier, N. Wschebor, Precision calculation of critical exponents in theO(N) universality classes with the nonper- 17 turbative renormalization group, Phys. Rev. E 101 (2020) 042113.doi:10.1103/PhysRevE.101.042113. URLhttps://link.aps.org/doi/10.1103/PhysRevE.101.042113

    work page doi:10.1103/physreve.101.042113 2020

  44. [45]

    Delamotte, M

    B. Delamotte, M. Dudka, D. Mouhanna, S. Yabunaka, Functional renormalization group approach to noncollinear magnets, Phys. Rev. B 93 (2016) 064405.doi:10.1103/PhysRevB.93.064405. URLhttps://link.aps.org/doi/10.1103/PhysRevB.93.064405 18

    work page doi:10.1103/physrevb.93.064405 2016