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Spontaneous Space-Time Parity Breaking Without Thermal Restoration

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A 2+1-dimensional QFT is constructed whose parity symmetry breaks spontaneously at arbitrarily high temperatures.

desk verdict The zero-temperature RG flow is solid and the parity extension is genuinely new, but the all-temperature persistence claim is an extrapolation from μ∼T and one-loop thermal masses, not a computed finite-temperature result. read the letter →

arxiv 2507.19890 v1 pith:3SID6DV5 submitted 2025-07-26 hep-th cond-mat.str-el

Bilal Hawashin , Michael M. Scherer , Michael Smolkin , Lev Yung This is my paper
classification hep-thcond-mat.str-el PACS 11.30.Er11.10.Wx
keywords paritysymmetrybreakingpersistentthermalrestorationfunctionalrenormalizationgroupepsilonexpansionGross-Neveu-Yukawamodelbiconicalvectorrelevantdeformation
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 tries to establish that a specific, exactly defined quantum field theory in two space and one time dimension can have its parity symmetry spontaneously broken by heating, and that the broken phase survives at arbitrarily high temperature, making it the first known example for a space-time symmetry. The theory is built as a renormalization group trajectory between two known conformal field theories, with a relevant interaction coupling a scalar to a fermion sending the zero-temperature theory to a parity-symmetric infrared fixed point. The authors' key move is to treat temperature itself as the renormalization scale: at high temperature the theory is controlled by the ultraviolet fixed point, where a scalar field acquires a nonzero expectation value and breaks parity. If true, this would overturn the intuition that thermal fluctuations always restore symmetries, at least for discrete space-time symmetries.

What carries the argument

The load-bearing object is the renormalization group trajectory connecting two conformal fixed points, CFT_UV = (critical biconical vector model) $\times$ (free massless Dirac fermion) and CFT_IR = (critical Gross-Neveu-Yukawa model) $\times$ (critical $O(N_1)$ vector model), with the relevant Yukawa deformation $h\chi\bar\psi\psi$ as the trigger. At high temperature the authors invoke the identification $\mu\sim T$, so the thermal free energy is governed by CFT_UV; the one-loop thermal mass formula $m_\chi^2(T)=\frac{T^2}{24}\left(3\lambda_\chi+N_1\lambda_{\phi\chi}+N_2 h^2\right)$ makes the $\chi$-direction unstable once $\lambda_{\phi\chi}$ runs to its negative biconical value. In the functional renormalization group analysis in $d=3$, the same physics is reflected in the decoupling of fermionic Matsubara modes for $k<\pi T$, which leaves the bosonic scalar flow to run toward the biconical fixed point where $\langle\chi\rangle_T\neq 0$ is known to occur.

What would settle it

Integrate the finite-temperature FRG equations (10) and (11) directly in $d=3$ along the constructed trajectory, without the $\mu\sim T$ shortcut, and check whether the minimum $\kappa_\chi$ of the effective potential remains positive as $k\to 0$ for arbitrarily large $T$. If $\kappa_\chi$ returns to zero at any finite temperature, the claim of persistent parity breaking is falsified.

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Extended reading notes

Core claim

The paper's central claim is that a local, unitary, ultraviolet-complete quantum field theory in 2+1 dimensions can spontaneously break space-time parity at all temperatures: the symmetry is intact at zero temperature and breaks at sufficiently high temperature, remaining broken in the infinite-temperature limit. The theory is defined by a renormalization group trajectory starting at CFT_UV, the direct product of the critical biconical vector model and a free massless Dirac fermion, and deformed by a relevant Yukawa coupling $h\chi\bar\psi\psi$. The flow terminates in the infrared at CFT_IR, the decoupled product of the Gross-Neveu-Yukawa critical model and the critical $O(N_1)$ vector model, both parity symmetric. At finite temperature the authors identify the temperature with the renormalization group scale, so high-temperature physics is governed by CFT_UV; there the one-loop thermal mass of $\chi$ turns negative because $\lambda_{\phi\chi}^*<0$ for $N_1\ge 18$, giving $\langle\chi\rangle_T\neq 0$ and a parity-breaking fermion mass. The same conclusion is supported by a zero-temperature functional renormalization group computation in 2+1 dimensions that establishes the existence of the connecting trajectory.

Load-bearing premise

The argument rests on treating temperature as the energy scale that controls the theory's behavior, and on extrapolating a one-loop thermal-mass calculation made near four dimensions down to 2+1 dimensions; if either step is invalid, the high-temperature parity-broken phase could disappear.

Editorial extensions

If this is right

  • If the construction holds, this is the first local, unitary, ultraviolet-complete QFT in 2+1 dimensions in which a space-time symmetry, parity, is spontaneously broken at all temperatures.
  • At zero temperature the theory sits in a parity-symmetric conformal vacuum, so the symmetry breaking is purely a finite-temperature effect generated by running to the ultraviolet fixed point.
  • The Dirac fermion acquires a mass proportional to $\langle\chi\rangle_T$ at high temperature without any explicit parity-violating term in the action.
  • The mechanism requires a large number of scalar flavors ($N_1\ge 18$ in the $\epsilon$-expansion analysis, $N_1=100$ in the FRG demonstration), suggesting the effect is controlled by large-$N$ physics.
  • Because only discrete symmetries can break spontaneously in 2+1 dimensions, the result does not contradict the no-go theorem for continuous symmetry breaking in two spatial dimensions, and it raises the open question of analogous persistent breaking of continuous symmetries in 3+1 dimensions.

Reading between the lines

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

  • The paper does not integrate a direct finite-temperature FRG flow; a natural next test is to do so in $d=3$ and map the full temperature dependence of $\kappa_\chi$, which would sharpen the prediction of a critical temperature.
  • The same $\mu\sim T$ logic suggests a general recipe: any ultraviolet CFT with a thermally unstable scalar coupled to fermions through a relevant Yukawa interaction should exhibit persistent parity breaking, so variants with different field content or in other dimensions may exist.
  • The high-temperature parity-broken phase gives the fermion a large mass and could serve as a toy model for cosmological mass generation or for ordered phases in Dirac materials, although the authors do not develop those applications.
  • The model implies a low-temperature parity-symmetric phase separated from a high-temperature broken phase by a transition; locating this transition quantitatively in the FRG framework would be a concrete follow-up calculation.
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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 / 3 minor

Summary. The paper constructs a 2+1-dimensional local, unitary quantum field theory as a renormalization-group trajectory connecting two conformal fixed points: a UV CFT consisting of the critical biconical vector model plus a decoupled free massless Dirac fermion, and an IR CFT consisting of a decoupled critical O(N1) vector model and a Gross-Neveu-Yukawa model. The relevant Yukawa deformation between the two fixed points is analyzed with one-loop 4−ε beta functions and with functional RG in the LPA′ truncation directly in d=3. The zero-temperature FRG flow is computed, and fixed-point data are compared with literature. The paper then argues that at sufficiently high temperature, the system is governed by the UV CFT, whose negative thermal mass for the pseudo-scalar χ leads to ⟨χ⟩≠0 and hence to spontaneous parity breaking that persists to arbitrarily high temperatures. The central claim is that this is the first local, UV-complete, unitary QFT with persistent breaking of a space-time symmetry.

Significance. If the finite-temperature claim is correct, this is a significant conceptual advance: all previous persistent-symmetry-breaking constructions concerned internal discrete symmetries, and several suffered from non-locality or UV incompleteness. The zero-temperature part of the paper is solid and valuable: the FRG flow connecting CFT_UV to CFT_IR is demonstrated with a concrete numerical trajectory, the LPA′8 results are checked against LPA′12, and Table II shows reasonable agreement with independent determinations of the critical exponents of the constituent CFTs. The paper also provides a detailed supplement with explicit one-loop beta-function derivations and the finite-temperature FRG expressions. However, the persistent-breaking claim is currently supported by inference and extrapolation rather than by a direct finite-temperature computation, so the significance of the paper as a proof of the main claim is not yet established.

major comments (3)
  1. [Thermal effects; Supplementary Materials, FRG] The central high-temperature claim is inferred, not computed. Equation (10) is the finite-temperature flow of the effective average action, but no finite-temperature FRG flow is integrated; the supplement states explicitly that "numerical studies are performed at T=0, and all the thermal factors are set to unity." The effective potential at scale k∼T is an intermediate object, not the physical free energy, and a negative curvature there does not by itself determine the free energy after the flow is integrated from k=Λ down to k=0 with thermal kernels. The argument that for k<πT fermions decouple and the bosonic flow near CFT_UV forces ⟨χ⟩≠0 is a qualitative inference from the T=0 phase diagram of [11], not a calculation for the specific trajectory used in Fig. 3. The identification μ∼T is also adopted, not derived. Please provide a direct finite-T FRG integration, or an equivalent free-energy calculation, to support the persistent-breaking claim.
  2. [Thermal effects, Eq. (14)] The one-loop thermal masses in Eq. (14) are derived in d=4−ε and then used at ε=1 in d=3. At ε=1 the fixed-point couplings are not small, and the sign of m_χ^2(T) is not protected beyond one loop. In particular, the combination 3λχ+N1λφχ+N2h² is evaluated at CFT_UV couplings obtained from the ε-expansion fixed point; the FRG fixed-point values in Table I (λφχ*=-0.29, λχ*=0.31 for N1=100) give the same sign for N1=100, but this is still an evaluation of the same one-loop formula. The extrapolation from d=4−ε to d=3 needs to be justified or replaced by a direct d=3 finite-temperature calculation.
  3. [Thermal effects, paragraphs after Eq. (14)] The argument that the high-temperature phase is governed by CFT_UV and that this CFT exhibits ⟨χ⟩_T≠0 is transferred from Refs. [5,6,11], but CFT_UV is itself in the SSB-SSB phase at T=0 (Table I: κφ=1.93, κχ=0.26). Since CFT_UV is a multicritical point where all four phase regimes meet (Fig. 2), it is not automatic that the particular relevant deformations Δg2, Δg3, and Δh defining the trajectory of Fig. 3, once fermions decouple for k<πT, lie in the basin of attraction of the χ-broken phase. The text only states that "sufficiently small deformations lead to ⟨χ⟩_T≠0"; this needs to be checked for the specific trajectory used in the paper.
minor comments (3)
  1. [Model, Eq. (2)] The text says fractional dimensions with 0<ϵ<1 are treated via analytic continuation, but Eq. (14) is later used at d=3, i.e., ϵ=1. Please state explicitly that this is an extrapolation outside the analytic-continuation regime.
  2. [Fig. 2 caption] The phase diagram in Fig. 2 is discussed but the axes and sign conventions for Δg2 and Δg3 are not defined in the caption. Please specify the plotted variables and the normalization of the deformations.
  3. [Supplementary Materials, Thermal Masses] The derivation of L_S in Eq. (33) relies on the divergent zero mode being excluded via dimensional regularization, as noted in footnote [70]; for the main-text reader this point should be repeated in the main text where Eq. (14) is introduced.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the high-temperature parity-breaking sign follows from the paper's own beta functions and one-loop thermal masses; the FRG corroboration cites prior work by overlapping authors, but that cited work is independent published support rather than an assumed conclusion.

full rationale

The derivation is not circular. The high-temperature parity-breaking conclusion follows from the one-loop thermal masses in Eq. (14), mχ²(T) = T²/24 [3λχ + N1λϕχ + N2h²], evaluated at the CFT_UV fixed point whose λ*ϕχ < 0 is obtained from the paper's own beta functions in Eq. (4), not assumed from the conclusion. The deformation construction and the T=0 FRG flow to CFT_IR are independent inputs, and the negative sign of mχ² is a computed output of those inputs. The finite-temperature FRG corroboration invokes the authors' earlier Ref. [11] for the phase diagram of the biconical fixed point; this is a self-citation and is load-bearing for the direct 2+1 statement, but Ref. [11] is an externally published, independently corroborated result (large-N, FRG, and epsilon-expansion studies) and is not equivalent to the present paper's parity-breaking claim. The main limitations — the adopted identification μ∼T, the extrapolation from 4−ε to 3 dimensions, and the fact that the FRG numerical studies are performed at T=0 with thermal factors set to unity — are assumptions and approximation concerns about the strength of the inference, not reductions of the output to the input. No equation in the paper assumes ⟨χ⟩_T≠0; the thermal mass sign is derived, not fitted. Hence there is no significant circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the existence of the two CFTs and on the finite-T behavior of the biconical fixed point. The FRG truncation and the mu~T identification are the main assumptions; no new particles or forces are introduced.

free parameters (4)
  • N1 = 100
    Flavor number of the O(N1) scalar sector, chosen large for convergence of the FRG truncation; not fitted to data.
  • N2 = 8
    Flavor number of the Dirac fermion sector, chosen for comparability with condensed matter literature; not fitted to data.
  • RG deformation sizes = Delta g2 = 1.12437360545e-05, Delta g3 = 0.0032287264705453, Delta h = 0.001
    Chosen by hand to place the theory on the RG trajectory connecting CFT_UV to CFT_IR; not inferred from data.
  • FRG truncation order = LPA'8, with LPA'12 cross-check
    Approximation order for the effective potential expansion; convergence is tested but not rigorously proven.
assumptions (5)
  • domain assumption The LPA' truncation of the Wetterich equation faithfully represents the RG flow of the model.
    The FRG analysis is performed in the extended local potential approximation and is not exact.
  • domain assumption Temperature acts as an RG scale, mu ~ T, so increasing T drives the theory toward the UV fixed point.
    Adopted in the 'Thermal effects' section to identify the high-T phase with CFT_UV.
  • domain assumption The one-loop thermal masses computed in 4-epsilon dimensions determine the sign of m_chi^2 in 2+1 dimensions.
    The central thermal mechanism is established perturbatively near d=4 and extrapolated to d=3.
  • domain assumption The biconical fixed point exhibits spontaneous chi condensation at finite temperature, as established in [5,6,11].
    The finite-T phase diagram of the biconical model is imported from prior work, not recalculated here.
  • standard math Minimal subtraction renormalization and standard Feynman-integral identities are valid.
    Used for the one-loop beta functions in the Supplement.

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Cite this review

Pith. "Pith review of Spontaneous Space-Time Parity Breaking Without Thermal Restoration." pith.science (2026). https://pith.science/paper/3SID6DV5

@misc{pith2026250719890,
  author       = {Pith},
  title        = {Pith review of: Spontaneous Space-Time Parity Breaking Without Thermal Restoration},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3SID6DV5}},
  note         = {Machine review of arXiv:2507.19890}
}
abstract

We construct an ultraviolet-complete, local, and unitary quantum field theory in 2+1 dimensions that exhibits spontaneous breaking of space-time parity, persisting to arbitrarily high temperatures. The theory is defined by a renormalization group trajectory, triggered by a relevant deformation of a conformal field theory, consisting of a critical biconical vector model and a free massless Dirac fermion. This deformation couples the fermion to the scalar sector, generating a renormalization group flow that terminates at a nontrivial infrared fixed point described by a conformal Gross-Neveu-Yukawa model and a decoupled critical vector model. By construction, the quantum field theory is parity invariant at zero temperature. However, we show that at sufficiently high temperatures, parity symmetry is spontaneously broken and remains so even in the infinite-temperature limit. Our analysis relies on both, perturbative renormalization group techniques in $4\!-\!\epsilon$ dimensions and functional renormalization group techniques directly in 2+1 dimensions.

Figures

Figures reproduced from arXiv: 2507.19890 by the authors.

Figure 1
Figure 1. FIG. 1. RG flow from the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. FRG flow connecting CFT [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Infrared phases of deformed CFT [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. One-loop diagrams contributing to the wave function renormalization for [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. One-loop diagram contributing to the renormalization of the Yukawa coupling [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. One-loop diagrams contributing to [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. One-loop diagrams contributing to the renormalization of the coupling [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

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Reference graph

Works this paper leans on

78 extracted references · 41 canonical work pages · cited by 2 Pith papers

  1. [11]

    Hawashin, J

    B. Hawashin, J. Rong, and M. M. Scherer, Ultraviolet- complete local field theory of persistent symmetry break- 7 ing in 2 + 1 dimensions, Phys. Rev. Lett.134, 041602 (2025)

    work page 2025

  2. [1]

    L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 3rd ed., Course of The- oretical Physics, Vol. 5 (Pergamon Press, Oxford, 1980)

    work page 1980

  3. [2]

    I. Y. Pomeranchuk, On the theory of liquid helium-3, Zh. Eksp. Teor. Fiz.20, 919 (1950), in Russian

    work page 1950

  4. [3]

    Weinberg, Gauge and global symmetries at high tem- perature, Physical Review D9, 3357 (1974)

    S. Weinberg, Gauge and global symmetries at high tem- perature, Physical Review D9, 3357 (1974)

    1974

  5. [4]

    Y. Han, X. Huang, Z. Komargodski, A. Lucas, and F. K. Popov, Entropic Order, (2025), arXiv:2503.22789 [cond- mat.stat-mech]

    arXiv 2025

  6. [5]

    N. Chai, S. Chaudhuri, C. Choi, Z. Komargodski, E. Rabinovici, and M. Smolkin, Thermal Order in Con- formal Theories, Phys. Rev. D102, 065014 (2020), arXiv:2005.03676 [hep-th]

    arXiv 2020

  7. [6]

    N. Chai, S. Chaudhuri, C. Choi, Z. Komargodski, E. Ra- binovici, and M. Smolkin, Symmetry Breaking at All Temperatures, Phys. Rev. Lett.125, 131603 (2020)

    work page 2020

  8. [7]

    N. Chai, E. Rabinovici, R. Sinha, and M. Smolkin, The bi-conical vector model at 1/N, JHEP05, 192, arXiv:2011.06003 [hep-th]

    work page Pith review arXiv 2011

Show all 78 references
  1. [8]

    N. Chai, A. Dymarsky, and M. Smolkin, Model of Per- sistent Breaking of Discrete Symmetry, Phys. Rev. Lett. 128, 011601 (2022), arXiv:2106.09723 [hep-th]

    2022 arXiv

  2. [9]

    N. Chai, A. Dymarsky, M. Goykhman, R. Sinha, and M. Smolkin, A model of persistent breaking of continuous symmetry, SciPost Phys.12, 181 (2022), arXiv:2111.02474 [hep-th]

    2022 arXiv

  3. [10]

    Liendo, J

    P. Liendo, J. Rong, and H. Zhang, Spontaneous breaking of finite group symmetries at all temperatures, SciPost Phys.14, 168 (2023), arXiv:2205.13964 [hep-th]

    2023 arXiv

  4. [12]

    277146847 (SFB 1238, project C02) and Project No

    MMS acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foun- dation) under Project No. 277146847 (SFB 1238, project C02) and Project No. 452976698 (Heisenberg program). ∗a hawashin@tp3.rub.de, bscherer@tp3.rub.de, cmichael.smolkin@mail.huji.ac....

  5. [13]

    Komargodski and F

    Z. Komargodski and F. K. Popov, Temperature- Resistant Order in 2+1 Dimensions, (2024), arXiv:2412.09459 [hep-th]

    2024 arXiv

  6. [14]

    Hogervorst, S

    M. Hogervorst, S. Rychkov, and B. C. van Rees, Unitarity violation at the Wilson-Fisher fixed point in 4-ϵdimen- sions, Phys. Rev. D93, 125025 (2016), arXiv:1512.00013 [hep-th]

    2016 arXiv

  7. [15]

    A. A. Belavin and A. A. Migdal, Calculation of anoma- lous dimensions in non-abelian gauge field theories, Pisma Zh. Eksp. Teor. Fiz.19, 317 (1974)

    1974

  8. [16]

    W. E. Caswell, Asymptotic behavior of non-abelian gauge theories to two-loop order, Phys. Rev. Lett.33, 244 (1974)

    1974

  9. [17]

    Banks and A

    T. Banks and A. Zaks, On the Phase Structure of Vector- Like Gauge Theories with Massless Fermions, Nucl. Phys. B196, 189 (1982)

    1982

  10. [18]

    Chaudhuri, C

    S. Chaudhuri, C. Choi, and E. Rabinovici, Thermal or- der in large N conformal gauge theories, JHEP04, 203, arXiv:2011.13981 [hep-th]

    2011 arXiv

  11. [19]

    B. Bajc, A. Lugo, and F. Sannino, Asymptotically free and safe fate of symmetry nonrestoration, Phys. Rev. D 103, 096014 (2021), arXiv:2012.08428 [hep-th]

    2021 arXiv

  12. [20]

    Chaudhuri and E

    S. Chaudhuri and E. Rabinovici, Symmetry breaking at high temperatures in large N gauge theories, JHEP08, 148, arXiv:2106.11323 [hep-th]

    arXiv

  13. [21]

    Moshe and J

    M. Moshe and J. Zinn-Justin, Quantum field theory in the large N limit: A Review, Phys. Rept.385, 69 (2003), arXiv:hep-th/0306133

    2003 arXiv

  14. [22]

    D. R. Nelson, J. M. Kosterlitz, and M. E. Fisher, Renormalization-group analysis of bicritical and tetra- critical points, Phys. Rev. Lett.33, 813 (1974)

    1974

  15. [23]

    Calabrese, A

    P. Calabrese, A. Pelissetto, and E. Vicari, Multicritical phenomena in O(n1) L O(n2)-symmetric theories, Phys. Rev. B67, 054505 (2003)

    2003

  16. [24]

    Eichhorn, D

    A. Eichhorn, D. Mesterh´ azy, and M. M. Scherer, Mul- ticritical behavior in models with two competing order parameters, Phys. Rev. E88, 042141 (2013)

    2013

  17. [25]

    Meade and H

    P. Meade and H. Ramani, Unrestored Electroweak Symmetry, Phys. Rev. Lett.122, 041802 (2019), arXiv:1807.07578 [hep-ph]

    2019 arXiv

  18. [26]

    Y. Bai, S. J. Lee, M. Son, and F. Ye, Global electroweak symmetric vacuum, JHEP07, 225, arXiv:2103.09819 [hep-ph]

    arXiv

  19. [27]

    Ramazanov, F

    S. Ramazanov, F. R. Urban, and A. Vikman, Observing primordial magnetic fields through Dark Matter, JCAP 02, 011, arXiv:2010.03383 [astro-ph.CO]

    2010 arXiv

  20. [28]

    Ramazanov, E

    S. Ramazanov, E. Babichev, D. Gorbunov, and A. Vikman, Beyond freeze-in: Dark matter via inverse phase transition and gravitational wave signal, Phys. Rev. D105, 063530 (2022), arXiv:2104.13722 [hep-ph]

    2022 arXiv

  21. [29]

    N. D. Mermin and H. Wagner, Absence of ferromag- netism or antiferromagnetism in one- or two-dimensional isotropic heisenberg models, Phys. Rev. Lett.17, 1133 (1966)

    1966

  22. [30]

    P. C. Hohenberg, Existence of long-range order in one and two dimensions, Phys. Rev.158, 383 (1967)

    1967

  23. [31]

    S. R. Coleman, There are no Goldstone bosons in two- dimensions, Commun. Math. Phys.31, 259 (1973)

    1973

  24. [32]

    Buchel, Thermal order in holographic CFTs and no- hair theorem violation in black branes, Nucl

    A. Buchel, Thermal order in holographic CFTs and no- hair theorem violation in black branes, Nucl. Phys. B 967, 115425 (2021), arXiv:2005.07833 [hep-th]

    2021 arXiv

  25. [33]

    Buchel, Holographic conformal order in supergravity, Phys

    A. Buchel, Holographic conformal order in supergravity, Phys. Lett. B814, 136111 (2021), arXiv:2007.09420 [hep- th]

    2021 arXiv

  26. [34]

    Buchel, Fate of the conformal order, Phys

    A. Buchel, Fate of the conformal order, Phys. Rev. D 103, 026008 (2021), arXiv:2011.11509 [hep-th]

    2021 arXiv

  27. [35]

    Buchel, Compactified holographic conformal order, Nucl

    A. Buchel, Compactified holographic conformal order, Nucl. Phys. B973, 115605 (2021), arXiv:2107.05086 [hep-th]

    2021 arXiv

  28. [36]

    Buchel, The quest for a conifold conformal order, JHEP08, 080, arXiv:2205.00612 [hep-th]

    A. Buchel, The quest for a conifold conformal order, JHEP08, 080, arXiv:2205.00612 [hep-th]

    arXiv

  29. [37]

    Buchel, Holographic conformal order with higher derivatives, Nucl

    A. Buchel, Holographic conformal order with higher derivatives, Nucl. Phys. B1004, 116578 (2024), arXiv:2312.15764 [hep-th]

    2024 arXiv

  30. [38]

    Buchel, The ordered phase of charged N=4 SYM plasma, (2025), arXiv:2501.01856 [hep-th]

    A. Buchel, The ordered phase of charged N=4 SYM plasma, (2025), arXiv:2501.01856 [hep-th]

    2025 arXiv

  31. [39]

    For brevity, we omit the explicit flavor indices ofϕandψ; summation over flavor contractions is implied through- out

  32. [40]

    Zinn-Justin, Four fermion interaction near four- dimensions, Nucl

    J. Zinn-Justin, Four fermion interaction near four- dimensions, Nucl. Phys. B367, 105 (1991)

    1991

  33. [41]

    Hands, A

    S. Hands, A. Kocic, and J. B. Kogut, Four Fermi theo- ries in fewer than four-dimensions, Annals Phys.224, 29 (1993), arXiv:hep-lat/9208022

    1993 arXiv

  34. [42]

    Karkkainen, R

    L. Karkkainen, R. Lacaze, P. Lacock, and B. Peters- son, Critical behavior of the three-dimensional Gross- Neveu and Higgs-Yukawa models, Nucl. Phys. B415, 781 (1994), [Erratum: Nucl.Phys.B 438, 650–650 (1995)], arXiv:hep-lat/9310020

    1994 arXiv

  35. [43]

    GRACEY, Computation ofβ ′(gc) at o(1/n2) in the o(n) gross-neveu model in arbitrary dimensions, Inter- national Journal of Modern Physics A09, 567 (1994), arXiv:hep-th/9306106

    J. GRACEY, Computation ofβ ′(gc) at o(1/n2) in the o(n) gross-neveu model in arbitrary dimensions, Inter- national Journal of Modern Physics A09, 567 (1994), arXiv:hep-th/9306106

    1994 arXiv

  36. [44]

    GRACEY, Computation of critical exponentηat o(1/n3) in the four-fermi model in arbitrary dimen- sions, International Journal of Modern Physics A09, 727 (1994), arXiv:hep-th/9306107

    J. GRACEY, Computation of critical exponentηat o(1/n3) in the four-fermi model in arbitrary dimen- sions, International Journal of Modern Physics A09, 727 (1994), arXiv:hep-th/9306107

    1994 arXiv

  37. [45]

    Hasenbusch, Finite size scaling study of lattice mod- els in the three-dimensional Ising universality class, Phys

    M. Hasenbusch, Finite size scaling study of lattice mod- els in the three-dimensional Ising universality class, Phys. Rev. B82, 174433 (2010), arXiv:1004.4486 [cond- mat.stat-mech]

    2010 arXiv

  38. [46]

    F. Kos, D. Poland, and D. Simmons-Duffin, Boot- strapping theO(N) vector models, JHEP06, 091, arXiv:1307.6856 [hep-th]

    arXiv

  39. [47]

    Janssen and I

    L. Janssen and I. F. Herbut, Antiferromagnetic crit- ical point on graphene’s honeycomb lattice: A func- tional renormalization group approach, Phys. Rev. B 89, 205403 (2014), [Addendum: Phys.Rev.B 102, 199902 (2020)], arXiv:1402.6277 [cond-mat.str-el]

    2014 arXiv

  40. [48]

    Knorr, Ising and gross-neveu model in next-to-leading order, Phys

    B. Knorr, Ising and gross-neveu model in next-to-leading order, Phys. Rev. B94, 245102 (2016)

    2016

  41. [49]

    Ihrig, L

    B. Ihrig, L. N. Mihaila, and M. M. Scherer, Critical be- havior of dirac fermions from perturbative renormaliza- tion, Phys. Rev. B98, 125109 (2018)

    2018

  42. [50]

    R. S. Erramilli, L. V. Iliesiu, P. Kravchuk, A. Liu, D. Poland, and D. Simmons-Duffin, The Gross-Neveu- Yukawa archipelago, JHEP02, 036, arXiv:2210.02492 [hep-th]

    arXiv

  43. [51]

    Wang and Z

    T.-T. Wang and Z. Y. Meng, Quantum monte carlo cal- culation of critical exponents of the gross-neveu-yukawa on a two-dimensional fermion lattice model, Phys. Rev. B108, L121112 (2023)

    2023

  44. [52]

    See Supplementary Materials for details

  45. [53]

    Wetterich, Exact evolution equation for the effective potential, Phys

    C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B301, 90 (1993), arXiv:1710.05815 8 [hep-th]

    1993 arXiv

  46. [54]

    Berges, N

    J. Berges, N. Tetradis, and C. Wetterich, Nonperturba- tive renormalization flow in quantum field theory and sta- tistical physics, Phys. Rept.363, 223 (2002), arXiv:hep- ph/0005122

    2002

  47. [55]

    J. M. Pawlowski, Aspects of the functional renormalisa- tion group, Annals Phys.322, 2831 (2007), arXiv:hep- th/0512261

    2007

  48. [56]

    Baldazzi, R

    A. Baldazzi, R. Percacci, and L. Zambelli, Functional renormalization and the ms scheme, Phys. Rev. D103, 076012 (2021)

    2021

  49. [57]

    Hofling, C

    F. Hofling, C. Nowak, and C. Wetterich, Phase tran- sition and critical behavior of the D = 3 Gross-Neveu model, Phys. Rev. B66, 205111 (2002), arXiv:cond- mat/0203588

    2002

  50. [58]

    D. D. Scherer, J. Braun, and H. Gies, Many-flavor Phase Diagram of the (2+1)d Gross-Neveu Model at Finite Temperature, J. Phys. A46, 285002 (2013), arXiv:1212.4624 [hep-ph]

    2013 arXiv

  51. [59]

    Janssen and I

    L. Janssen and I. F. Herbut, Antiferromagnetic criti- cal point on graphene’s honeycomb lattice: A functional renormalization group approach, Physical Review B89, 205403 (2014)

    2014

  52. [60]

    Classen, I

    L. Classen, I. F. Herbut, and M. M. Scherer, Fluctuation- induced continuous transition and quantum criticality in dirac semimetals, Physical Review B96, 115132 (2017)

    2017

  53. [61]

    Tolosa-Sime´ on, L

    M. Tolosa-Sime´ on, L. Classen, and M. M. Scherer, Rela- tivistic Mott transitions, quantum criticality, and finite- temperature effects in tunable Dirac materials from func- tional renormalization, (2025), arXiv:2503.04911 [cond- mat.str-el]

    2025 arXiv

  54. [62]

    D. F. Litim, Optimized renormalization group flows, Phys. Rev. D64, 105007 (2001), arXiv:hep-th/0103195

    2001 arXiv

  55. [63]

    D’Attanasio and T

    M. D’Attanasio and T. R. Morris, Large N and the renormalization group, Phys. Lett. B409, 363 (1997), arXiv:hep-th/9704094

    1997 arXiv

  56. [64]

    I. F. Herbut, V. Juriˇ ci´ c, and B. Roy, Theory of interact- ing electrons on the honeycomb lattice, Physical Review B—Condensed Matter and Materials Physics79, 085116 (2009)

    2009

  57. [65]

    A. B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett.43, 730 (1986)

    1986

  58. [66]

    J. L. Cardy, Is There a c Theorem in Four-Dimensions?, Phys. Lett. B215, 749 (1988)

    1988

  59. [67]

    Komargodski and A

    Z. Komargodski and A. Schwimmer, On Renormaliza- tion Group Flows in Four Dimensions, JHEP12, 099, arXiv:1107.3987 [hep-th]

    arXiv

  60. [68]

    I. R. Klebanov, S. S. Pufu, and B. R. Safdi, F-Theorem without Supersymmetry, JHEP10, 038, arXiv:1105.4598 [hep-th]

    arXiv

  61. [69]

    Casini and M

    H. Casini and M. Huerta, On the RG running of the entanglement entropy of a circle, Phys. Rev. D85, 125016 (2012), arXiv:1202.5650 [hep-th]

    2012 arXiv

  62. [70]

    Casini, E

    H. Casini, E. Test´ e, and G. Torroba, Markov Property of the Conformal Field Theory Vacuum and the a Theorem, Phys. Rev. Lett.118, 261602 (2017), arXiv:1704.01870 [hep-th]

    2017 arXiv

  63. [71]

    The contribution of the divergent zero mode is excluded using dimensional regularization

  64. [72]

    Braun, The QCD Phase Boundary from Quark- Gluon Dynamics, Eur

    J. Braun, The QCD Phase Boundary from Quark- Gluon Dynamics, Eur. Phys. J. C64, 459 (2009), arXiv:0810.1727 [hep-ph]

    2009 arXiv

  65. [73]

    Classen, I

    L. Classen, I. F. Herbut, L. Janssen, and M. M. Scherer, Competition of density waves and quantum multicritical behavior in Dirac materials from functional renormaliza- tion, Phys. Rev. B93, 125119 (2016), arXiv:1510.09003 [cond-mat.str-el]

    2016 arXiv

  66. [74]

    De Polsi, I

    G. De Polsi, I. Balog, M. Tissier, and N. Wschebor, Precision calculation of critical exponents in theO(N) universality classes with the nonperturbative renor- malization group, Phys. Rev. E101, 042113 (2020), arXiv:2001.07525 [cond-mat.stat-mech]. 9 SUPPLEMENT AR Y MA TERIALS...

    2020 arXiv

  67. [75]

    SYM-SYM: ∂t¯λnm = (∂tu)(n,m) ¯ρϕ=0 ¯ρχ=0 ,(55) 15

  68. [76]

    SYM-SSB: ∂t¯λnm = (∂tu)(n,m) + ¯λn,m+1∂t¯κχ ¯ρϕ=0 ¯ρχ=¯κχ , ∂ t¯κχ =− (∂tu)(0,1) ¯λ0,2 ¯ρϕ=0 ¯ρχ=¯κχ ,(56)

  69. [77]

    SSB-SYM: ∂t¯λnm = (∂tu)(n,m) + ¯λn+1,m∂t¯κϕ ¯ρϕ=¯κϕ ¯ρχ=0 , ∂ t¯κϕ =− (∂tu)(1,0) ¯λ2,0 ¯ρϕ=¯κϕ ¯ρχ=0 ,(57)

  70. [78]

    SSB-SSB: ∂t¯λnm = (∂tu)(n,m) + ¯λn+1,m∂t¯κϕ + ¯λn,m+1∂t¯κχ ¯ρϕ=¯κϕ ¯ρχ=¯κχ ,(58) ∂t¯κϕ = ¯λ0,2 (∂tu)(1,0) − ¯λ1,1(∂tu)(0,1) ¯λ2 1,1 − ¯λ2,0¯λ0,2 ¯ρϕ=¯κϕ ¯ρχ=¯κχ , ∂ t¯κχ = ¯λ2,0 (∂tu)(0,1) − ¯λ1,1(∂tu)(1,0) ¯λ2 1,1 − ¯λ2,0¯λ0,2 ¯ρϕ=¯κϕ ¯ρχ=¯κχ .(59) Anomalous dimensions and th...

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