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REVIEW 2 major objections 5 minor 68 references

Quantum multicriticality and emergent symmetry in Dirac systems with two order parameters at three-loop order

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read At three loops, the critical flavor number of the incompatible SO(3)×SO(3) model drops to about 2.57 in d=3, placing physical N_f=2 at the border between true criticality and pseudocritical walking.

desk verdict A genuinely new three-loop Gross-Neveu-Yukawa calculation with solid cross-checks; the compatible-sector results look trustworthy, but the headline N_c^> extrapolation rests on an unquantified fit and should be treated as suggestive, not decisive. read the letter →

arxiv 2505.22723 v2 pith:X27J5CAF submitted 2025-05-28 cond-mat.str-el hep-th

Max Uetrecht , Igor F. Herbut , Michael M. Scherer , Emmanuel Stamou , Tom Steudtner This is my paper
classification cond-mat.str-elhep-th
keywords emergentsymmetryquantummulticriticalityGross-Neveu-YukawaDiracfermionsepsilonexpansionrenormalizationgroupfixed-pointannihilationpseudocriticality
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 asks whether two competing orders in a two-dimensional Dirac semimetal can meet at a single quantum multicritical point with an enlarged, emergent symmetry, and whether that point truly controls the critical behavior. Working with Gross–Neveu–Yukawa field theories continued to $d=4-\epsilon$ and computing the renormalization-group functions to three loops, it argues that the symmetry-enhanced isotropic fixed point is stable when the number of Dirac flavors $N_f$ is large, while for the physically relevant $N_f=2$ the $\epsilon$ series is badly behaved and cannot be reliably extrapolated to $2+1$ dimensions. For the incompatible SO(3)×SO(3) model of antiferromagnetism and superconductivity, the critical flavor number becomes $N_c^> \approx 16.83 - 7.14\epsilon - 7.12\epsilon^2$, which at $\epsilon=1$ is close to $N_f=2$. The authors conclude that the physical case may sit at the borderline between true criticality and pseudocritical walking behavior, a distinction that numerical simulations would find extremely hard to make.

What carries the argument

The central object is the isotropic fixed point (IFP) of the Gross–Neveu–Yukawa theory with two order parameters, at which the two Yukawa couplings and the three quartic couplings collapse to single values $g$ and $\lambda$, so the symmetry enlarges from $SO(N_A)\times SO(N_B)$ to $SO(N_A+N_B)$. The argument is carried by the three-loop $\beta$ functions for these couplings, by the eigenvalues of the five-coupling stability matrix, and by the large-$N_f$ fixed point obtained after rescaling $G=N_f g^2/(4\pi)^2$ and $\Lambda=N_f\lambda/(4\pi)^2$, which is stable for all $\epsilon>0$ and explains the stabilizing role of fermions. For the incompatible model the load-bearing quantity is the critical flavor number $N_c^>$, the boundary in the $\epsilon$–$N_f$ plane beyond which a stable physical fixed point exists; its three-loop expansion $16.83-7.14\epsilon-7.12\epsilon^2$ is obtained by numerically following that boundary from small $\epsilon$.

What would settle it

Compute the $d=3$ fixed-point structure of the full theory including the $\Phi^6$ and $\psi^\dagger\psi\Phi^2$ operators; if a stable physical fixed point exists for $N_f=2$ there, the predicted $N_c^>\approx 2.57$ boundary and the pseudocritical explanation of the observed scaling collapse would be ruled out, as would a quantum Monte Carlo check showing the apparent correlation-length exponent stabilizing at a single value instead of drifting with system size.

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Extended reading notes

Core claim

The paper's central claim is that the stability of the multicritical fixed point with emergent $SO(N_A+N_B)$ symmetry is controlled by fermion flavor number: at three-loop order the isotropic fixed point is stable for all allowed $N=N_A+N_B$ once $N_f$ is large enough, and the large-$N_f$ fixed point $G^*=\epsilon/4$, $\Lambda^*=2\epsilon$ is stable for every $\epsilon>0$. For the physical case $N_f=2$, however, the expansion coefficients of the leading critical exponents of the chiral SO(4) and SO(5) models grow rapidly at order $\epsilon^3$, so the paper refrains from giving resummed estimates in $d=3$. In the incompatible SO(3)×SO(3) theory that has been proposed for the antiferromagnet–superconductor transition, no admissible fixed point is found for $N_f=2$ at three-loop order; extrapolating the boundary where a fixed point exists gives $N_c^> \approx 2.57$ at $\epsilon=1$. The paper interprets this as evidence that $N_f=2$ is a borderline case in which true quantum criticality and pseudocriticality from fixed-point annihilation are nearly indistinguishable.

Load-bearing premise

The analysis continues the theory to $d=4-\epsilon$ and omits the operators that are marginal in $2+1$ dimensions but non-renormalizable in four, namely $\Phi^6$ and $\psi^\dagger\psi\Phi^2$; Sec. II.B states this is an intrinsic shortcoming, and if those operators materially shift the fixed point or its stability in $d=3$, the conclusions about the physical $N_f=2$ case do not carry over.

Editorial extensions

If this is right

  • If a stable IFP exists at large $N_f$, compatible orders in Dirac systems meet at a single multicritical point with emergent $SO(N_A+N_B)$ symmetry, and more fermion flavors make that point more robust.
  • For $N_f=2$, the $\epsilon$ series for chiral SO(4) and SO(5) exponents has growing coefficients at order $\epsilon^3$, so direct extrapolation to the physical $2+1$-dimensional case is unreliable.
  • The three-loop value $N_c^>\approx 2.57$ at $\epsilon=1$ means the actual critical flavor number is likely close to or below the physical $N_f=2$, making true criticality and pseudocriticality almost indistinguishable.
  • The apparent scaling collapse seen in quantum Monte Carlo studies of the antiferromagnet–superconductor transition could be walking pseudo-scaling rather than true quantum critical behavior.
  • Fermion-induced stabilization distinguishes these Dirac multicritical points from purely bosonic two-order-parameter theories, where an isotropic fixed point is stable only for $N_A=N_B=1$.

Reading between the lines

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

  • Because the $4-\epsilon$ continuation drops $\Phi^6$ and $\psi^\dagger\psi\Phi^2$ operators that are marginal in $2+1$ dimensions, a direct $d=3$ calculation including those operators could move $N_c^>$ decisively away from $N_f=2$; that would make the pseudocriticality suggestion an artifact of the continuation rather than a property of the physical system.
  • Moiré-engineered Dirac systems, which can realize larger effective flavor numbers, may be better platforms than graphene for observing emergent $SO(N_A+N_B)$ multicriticality.
  • A sharp numerical test: measure the drift of apparent critical exponents with system size in quantum Monte Carlo; true criticality shows stable exponents with a positive correction-to-scaling exponent, while walking pseudocriticality shows slowly drifting exponents.
  • Fixed-point annihilation near $N_f=2$ may also explain other apparent Dirac quantum critical points where perturbative renormalization-group results and numerical simulations disagree.
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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

2 major / 5 minor

Summary. The paper studies Gross--Neveu--Yukawa field theories with two order parameters, in 4−ε dimensions, up to three-loop order. It constructs a dimensional continuation from 2+1-dimensional Dirac systems, distinguishes compatible and incompatible order-parameter pairs, and analyzes the stability of an isotropic fixed point with emergent SO(N_A+N_B) symmetry. For compatible models it provides three-loop series for critical exponents of the chiral O(4) and O(5) models at N_f=2, and shows that the IFP becomes robustly stable at large N_f. For the incompatible SO(3)×SO(3) model it determines the upper critical flavor number N_c^> ≈ 16.83 − 7.14ε − 7.12ε^2, which at ε=1 gives N_c^> ≈ 2.57, and interprets N_f=2 as a borderline case between true criticality and pseudocriticality.

Significance. If the results hold, this is a substantial technical advance: three-loop beta functions and anomalous dimensions for a general class of two-order-parameter GNY models are new, and the authors cross-check their expressions against published one-, two-, and four-loop limits. The full results are provided in a Mathematica attachment, which is a valuable reproducibility feature. The analysis of emergent SO(N_A+N_B) symmetry at higher loop order and the large-N_f fixed point are useful contributions. The main quantitative claim, the N_c^> series and the resulting borderline interpretation for N_f=2, is potentially important for interpreting QMC data, but its current support is weakened by an unquantified numerical extrapolation and by acknowledged omitted d=3-marginal operators.

major comments (2)
  1. [Sec. IV.B, Eqs. (49)-(51), Fig. 3] The central quantitative result for the incompatible SO(3)×SO(3) model is N_c^> ≈ 16.83 − 7.14ε − 7.12ε^2, and the conclusion that N_f=2 is borderline depends on N_c^>(1) ≈ 2.57 being above 2. The O(ε^2) coefficient is obtained by fitting the numerical fixed-point boundary for ε ≤ 0.4, but the paper reports no fit residuals, no uncertainty on the fitted coefficient, and no sensitivity to the fitting window or to the assumed quadratic ansatz. The coefficient −7.12 is of the same size as the O(ε) coefficient −7.14, and it changes the ε=1 prediction from 9.69 at two loops to 2.57 at three loops; a shift of about 0.6 in this coefficient would move N_c^>(1) below 2 and reverse the interpretation for N_f=2. As written, the 'borderline' claim in the abstract and Sec. IV.B rests on an unquantified extrapolation. Please provide the fit residuals, explicit dependence on the fitting window (e.g., ε_max = 0.2, 0.3, 0.4), a comparison with a cubic ansatz or a Padé approximant, and an estimate of the resulting uncertainty in N_c^>(1).
  2. [Sec. II.B, Eq. (6)] The paper explicitly acknowledges that Φ^6 and ψ†ψ Φ^2 interactions are marginal in 2+1 dimensions and are omitted because they are non-renormalizable in the 4−ε continuation. This is more than a technical caveat for the physical conclusions: the borderline statement for N_f=2 is obtained by continuing the 4−ε series to ε=1, and these omitted operators can affect the existence and stability of fixed points in d=3, including a possible fixed-point collision. The present results should be stated as predictions of the truncated 4−ε theory, and the text should indicate what evidence exists (e.g., large-N_f, functional RG, or known results in related models) that these operators do not change the qualitative picture. Without such a discussion, the connection between Table II and the realistic d=3 system remains incomplete.
minor comments (5)
  1. [Sec. IV.B, text near Eq. (45)] The sentence 'not for the case of Nf, which was studied in QMC simulations' should read 'not for the case N_f=2, which was studied in QMC simulations'.
  2. [Table II] The three-loop value N_c^< = −0.0596 is negative and therefore not a physical flavor number; the text should comment on whether the lower branch of fixed-point solutions persists in d=3 or whether this signals a breakdown of the ε-expansion for N_c^<.
  3. [Fig. 2 caption] The caption contains the string '/T_hree Loop', which appears to be a LaTeX formatting error and should read 'Three Loop'.
  4. [Eqs. (39)-(40)] The paper refrains from resummed estimates for the d=3 critical exponents; it would be helpful to state explicitly in the text that the displayed series are not intended as quantitative predictions at ε=1, given their rapidly growing coefficients.
  5. [Eq. (49)] The notation δN_c^{>(3-loop)} is used for the fitted coefficient of ε^2; this symbol should be defined in the text when the ansatz is introduced.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the three-loop beta functions, fixed-point stability analysis, and exponent series are new computations; the prior self-citations are consistently rederived, and the N_c^> extrapolation is a labelled fit to independent beta-function data, not a hidden input.

full rationale

The central results are genuine three-loop computations: the beta functions and anomalous dimensions in Eqs. (35), (36), and (46)-(48) are obtained with the FoRGEr toolkit and are cross-checked against independent one-, two-, and four-loop results from Refs. [27, 36, 55]. The isotropic fixed-point stability analysis follows by solving these beta functions and diagonalizing the stability matrix; it is not defined in terms of the target stability conclusion. For the incompatible SO(3) x SO(3) model, the paper uses the two-loop ansatz from its own Ref. [36] for the O(epsilon) coefficient of N_c^>, but it explicitly states that this coefficient was 'consistently rederived here', and the new O(epsilon^2) coefficient is obtained by fitting the boundary of the three-loop fixed-point region, which is an independent numerical consequence of the newly computed beta functions. This is an extrapolation procedure whose robustness may be questioned, but it is not circular: the fitted coefficient is not a renamed version of the final prediction N_c^>(1). The paper also frankly acknowledges the intrinsic limitation of the 4-epsilon continuation regarding Phi^6 and psi-dagger-psi-Phi^2 operators (Sec. II.B), which is an honest caveat rather than a circular step. Self-citations to the authors' earlier work appear, but they are not load-bearing in the sense of importing an unverified uniqueness claim or an ansatz that is equivalent to the conclusion. No quoted equation reduces to its own input by construction, so the appropriate finding is no significant circularity.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central calculation introduces no ad hoc parameters; inputs are N_A, N_B, N_f. The quantitative N_c^> statement rests on a fitted second-order coefficient and several modelling assumptions, especially the validity of the 4-epsilon continuation and the neglect of marginal-in-2+1 operators.

free parameters (2)
  • Second-order coefficient delta N_c^> for the upper critical flavor number = -7.12
    Numerically fitted from three-loop fixed-point boundary data over 0.004 <= epsilon <= 0.4 using the quadratic ansatz Eq (49); no error estimate quoted.
  • Second-order coefficient delta N_c^< for the lower critical flavor number = -0.1380
    Same fitting procedure from Eq (50); no error estimate quoted.
assumptions (6)
  • domain assumption Dimensional continuation of d=2+1 GNY models to d=4-epsilon via Eqs (25)-(27) with the N_Psi to N_f mapping is valid.
    All epsilon-expansion results for d=3 depend on this mapping; a failure changes the N_f relation and minimal fermion count.
  • standard math The three-loop beta functions from FoRGEr (Refs [52-54]) are correct.
    Cross-checked with one-, two-, and four-loop special cases, but not independently verified here.
  • domain assumption Suppressed 2+1 marginal operators Phi^6 and psi-dagger-psi-Phi^2 can be neglected.
    Explicitly acknowledged as an intrinsic shortcoming in Sec II.B; could affect fixed-point values in d=3.
  • ad hoc to paper Quadratic ansatz Eq (49) for N_c(epsilon) remains adequate up to epsilon=1.
    Chosen from two-loop analysis; the fitted O(epsilon^2) coefficient is large and no convergence guarantee is given.
  • domain assumption Pole-free Pade approximants estimate the epsilon=1 limit when no pole lies in [0,1].
    Used to convert divergent epsilon series into stability statements; many points are excluded due to poles.
  • domain assumption A physical fixed point requires g*^2 > 0 and lambda* > 0.
    Used to exclude unphysical fixed points in Fig 3; this is a standard positivity condition.

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Pith. "Pith review of Quantum multicriticality and emergent symmetry in Dirac systems with two order parameters at three-loop order." pith.science (2026). https://pith.science/paper/X27J5CAF

@misc{pith2026250522723,
  author       = {Pith},
  title        = {Pith review of: Quantum multicriticality and emergent symmetry in Dirac systems with two order parameters at three-loop order},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X27J5CAF}},
  note         = {Machine review of arXiv:2505.22723}
}
abstract

Two-dimensional materials with interacting Dirac excitations can host quantum multicritical behavior near the phase boundaries of the semimetallic and two-ordered phases. We study such behavior in Gross--Neveu--Yukawa field theories where $N_f$ flavors of Dirac fermions are coupled to two order-parameter fields with $SO(N_A)$ and $SO(N_B)$ symmetry, respectively. To that end, we employ the perturbative renormalization group up to three-loop order in $4-\epsilon$ spacetime dimensions. We distinguish two key scenarios: (i) The two orders are compatible as characterized by anticommuting mass terms, and (ii) the orders are incompatible. For the first case, we explore the stability of a quantum multicritical point with emergent $SO(N_A\!+\!N_B)$ symmetry. We find that the stability is controlled by increasing the number of Dirac fermion flavors. Moreover, we extract the series expansion of the leading critical exponents for the chiral $SO(4)$ and $SO(5)$ models up to third order in $\epsilon$. Notably, we find a tendency towards rapidly growing expansion coefficients at higher orders, rendering an extrapolation to $\epsilon=1$ difficult. For the second scenario, we study a model with $SO(4) \simeq SO(3) \times SO(3)$ symmetry, which was recently suggested to describe criticality of antiferromagnetism and superconductivity in Dirac systems. However, it was also argued that a physically admissible renormalization-group fixed point only exists for $N_f$ above a critical number $N_{c}^>$. We determine the corresponding series expansion at three-loop order as $N_{c}^>\approx 16.83-7.14\epsilon-7.12\epsilon^2$. This suggests that the physical choice of $N_f=2$ may be a borderline case, where true criticality and pseudocriticality, as induced by fixed-point annihilation, are extremely challenging to distinguish.

Figures

Figures reproduced from arXiv: 2505.22723 by the authors.

Figure 1
Figure 1. FIG. 1. Stability analysis of the IFP in [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Stability analysis of the IFP in [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Red dots mark values of [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The five eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

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Works this paper leans on

68 extracted references · 62 canonical work pages

  1. [1]

    The stability is determined from the stability matrix in Eq

    Stability of the isotropic fixed point in d = 3 We now consider the ϵ-expansion predictions on the stability of the IFP in d = 3, i.e., by taking the limit ϵ → 1. The stability is determined from the stability matrix in Eq. (32) using the ϵ-expansion predictions of the β functions of all five couplings. For leading values of NA,B,f , the stable and unsta-...

    work page

  2. [2]

    Chiral O(4) and O(5) models at Nf = 2 Critical exponents of the chiral Ising, XY, and Heisen- berg models have been discussed in the past years employ- 11 ing various many-body approaches, e.g., perturbative and functional RG, quantum Monte Carlo, and the conformal bootstrap. Reasonable agreement on the estimates for the exponents has been achieved in som...

    work page 1993

  3. [3]

    In doing so, we can develop an analytical understanding for many of the fixed points in Figs

    Large Nf In this section, we examine the compatible model in a large-Nf limit. In doing so, we can develop an analytical understanding for many of the fixed points in Figs. 1 and 2 including their stability. For one, this limit describes the compatible model at any value N , but for a larger number of fermion flavors NΨ ≫ N . Secondly, the large- Nf limit...

    work page 2022

  4. [4]

    Zhang, A Unified Theory Based on SO(5) Symmetry of Superconductivity and Antiferromagnetism , Science 275 (1997) 1089

    S.-C. Zhang, A Unified Theory Based on SO(5) Symmetry of Superconductivity and Antiferromagnetism , Science 275 (1997) 1089

    work page 1997

  5. [5]

    Nahum, P

    A. Nahum, P. Serna, J.T. Chalker, M. Ortu˜ no and A.M. Somoza, Emergent so(5) symmetry at the n´ eel to valence-bond-solid transition, Phys. Rev. Lett. 115 (2015) 267203

    work page 2015

  6. [6]

    Serna and A

    P. Serna and A. Nahum, Emergence and spontaneous breaking of approximate O(4) symmetry at a weakly first-order deconfined phase transition, Phys. Rev. B 99 (2019) 195110

    work page 2019

  7. [7]

    Qin, Y.-Y

    Y.Q. Qin, Y.-Y. He, Y.-Z. You, Z.-Y. Lu, A. Sen, A.W. Sandvik et al., Duality between the deconfined quantum-critical point and the bosonic topological transition, Phys. Rev. X 7 (2017) 031052

    work page 2017

  8. [8]

    C. Wang, A. Nahum, M.A. Metlitski, C. Xu and T. Senthil, Deconfined quantum critical points: Symmetries and dualities , Phys. Rev. X 7 (2017) 031051

    work page 2017

Show all 68 references
  1. [9]

    Pelissetto and E

    A. Pelissetto and E. Vicari, Critical phenomena and renormalization-group theory, Physics Reports 368 (2002) 549

    2002

  2. [10]

    Herbut, A modern approach to critical phenomena , Cambridge University Press (2007)

    I. Herbut, A modern approach to critical phenomena , Cambridge University Press (2007)

    2007

  3. [11]

    Eichhorn, D

    A. Eichhorn, D. Mesterh´ azy and M.M. Scherer, Multicritical behavior in models with two competing order parameters, Phys. Rev. E 88 (2013) 042141

    2013

  4. [12]

    Calabrese, A

    P. Calabrese, A. Pelissetto and E. Vicari, Multicritical phenomena in O(n1) L O(n2)-symmetric theories, Phys. Rev. B 67 (2003) 054505

    2003

  5. [13]

    Wehling, A.M

    T.O. Wehling, A.M. Black-Schaffer and A.V. Balatsky, Dirac materials, Advances in Physics 63 (2014) 1

    2014

  6. [14]

    Vafek and A

    O. Vafek and A. Vishwanath, Dirac fermions in solids: From high-Tc cuprates and graphene to topological insulators and Weyl semimetals , Annu. Rev. Condens. Matter Phys. 5 (2014) 83

    2014

  7. [15]

    Ponte and S.-S

    P. Ponte and S.-S. Lee, Emergence of supersymmetry on the surface of three dimensional topological insulators , New J. Phys. 16 (2014) 013044

    2014

  8. [16]

    Grover, D.N

    T. Grover, D.N. Sheng and A. Vishwanath, Emergent Space-Time Supersymmetry at the Boundary of a Topological Phase, Science 344 (2014) 280

    2014

  9. [17]

    Jian, Y.-F

    S.-K. Jian, Y.-F. Jiang and H. Yao, Emergent Spacetime Supersymmetry in 3D Weyl Semimetals and 2D Dirac Semimetals, Phys. Rev. Lett. 114 (2015) 237001

    2015

  10. [18]

    B. Roy, V. Juriˇ ci´ c and I.F. Herbut,Emergent Lorentz symmetry near fermionic quantum critical points in two and three dimensions , JHEP 04 (2016) 018

    2016

  11. [19]

    Scherer and I.F

    M.M. Scherer and I.F. Herbut, Gauge-field-assisted Kekul´ e quantum criticality, Phys. Rev. B 94 (2016) 205136

    2016

  12. [20]

    Classen, I.F

    L. Classen, I.F. Herbut and M.M. Scherer, Fluctuation-induced continuous transition and quantum criticality in dirac semimetals , Phys. Rev. B 96 (2017) 115132

    2017

  13. [21]

    Li, Y.-F

    Z.-X. Li, Y.-F. Jiang, S.-K. Jian and H. Yao, Fermion-induced quantum critical points, Nature communications 8 (2017) 314

    2017

  14. [22]

    Z.-X. Li, A. Vaezi, C.B. Mendl and H. Yao, Numerical observation of emergent spacetime supersymmetry at quantum criticality, Science Advances 4 (2018) eaau1463. 17

    2018

  15. [23]

    Jian and H

    S.-K. Jian and H. Yao, Fermion-induced quantum critical points in three-dimensional Weyl semimetals , Physical Review B 96 (2017) 155112

    2017

  16. [24]

    Jian and H

    S.-K. Jian and H. Yao, Fermion-induced quantum critical points in two-dimensional Dirac semimetals , Phys. Rev. B 96 (2017) 195162

    2017

  17. [25]

    Herbut and S

    I.F. Herbut and S. Mandal, SO(8) unification and the large-N theory of superconductor-insulator transition of two-dimensional Dirac fermions , Phys. Rev. B 108 (2023) L161108

    2023

  18. [26]

    Han and I.F

    S. Han and I.F. Herbut, Spontaneous breaking of the SO(2N ) symmetry in the Gross-Neveu model , Phys. Rev. D 109 (2024) 096026

    2024

  19. [27]

    Han and I.F

    S. Han and I.F. Herbut, Gross-Neveu-Yukawa theory of SO(2N ) → SO(N ) × SO(N ) spontaneous symmetry breaking, Phys. Rev. B 110 (2024) 125131

    2024

  20. [28]

    Torres, L

    E. Torres, L. Classen, I.F. Herbut and M.M. Scherer, Fermion-induced quantum criticality with two length scales in dirac systems , Phys. Rev. B 97 (2018) 125137

    2018

  21. [29]

    Roy, Multicritical behavior of Z2 × O(2) Gross-Neveu-Yukawa theory in graphene, Phys

    B. Roy, Multicritical behavior of Z2 × O(2) Gross-Neveu-Yukawa theory in graphene, Phys. Rev. B 84 (2011) 113404

    2011

  22. [30]

    Janssen, I.F

    L. Janssen, I.F. Herbut and M.M. Scherer, Compatible orders and fermion-induced emergent symmetry in Dirac systems, Phys. Rev. B 97 (2018) 041117

    2018

  23. [31]

    B. Roy, P. Goswami and V. Juriˇ ci´ c,Itinerant quantum multicriticality of two-dimensional Dirac fermions , Phys. Rev. B 97 (2018) 205117

    2018

  24. [32]

    Torres, L

    E. Torres, L. Weber, L. Janssen, S. Wessel and M.M. Scherer, Emergent symmetries and coexisting orders in dirac fermion systems , Phys. Rev. Res. 2 (2020) 022005

    2020

  25. [33]

    Boyack, H

    R. Boyack, H. Yerzhakov and J. Maciejko, Quantum phase transitions in Dirac fermion systems , Eur. Phys. J. ST 230 (2021) 979

    2021

  26. [34]

    Herbut, Wilson-Fisher fixed points in presence of Dirac fermions, Modern Physics Letters B (2023) 2430006

    I.F. Herbut, Wilson-Fisher fixed points in presence of Dirac fermions, Modern Physics Letters B (2023) 2430006

    2023

  27. [35]

    Classen, I.F

    L. Classen, I.F. Herbut, L. Janssen and M.M. Scherer, Mott multicriticality of dirac electrons in graphene , Phys. Rev. B 92 (2015) 035429

    2015

  28. [36]

    Classen, I.F

    L. Classen, I.F. Herbut, L. Janssen and M.M. Scherer, Competition of density waves and quantum multicritical behavior in dirac materials from functional renormalization, Phys. Rev. B 93 (2016) 125119

    2016

  29. [37]

    Roy and V

    B. Roy and V. Juriˇ ci´ c,Fermionic multicriticality near kekul´ e valence-bond ordering on a honeycomb lattice, Phys. Rev. B 99 (2019) 241103

    2019

  30. [38]

    Herbut and M.M

    I.F. Herbut and M.M. Scherer, SO(4) multicriticality of two-dimensional Dirac fermions , Phys. Rev. B 106 (2022) 115136

    2022

  31. [39]

    Uetrecht, I.F

    M. Uetrecht, I.F. Herbut, E. Stamou and M.M. Scherer, Absence of SO(4) quantum criticality in Dirac semimetals at two-loop order , Phys. Rev. B 108 (2023) 245130

    2023

  32. [40]

    Fornoville and L

    M. Fornoville and L. Janssen, Fractionalized fermionic multicriticality in anisotropic Kitaev spin-orbital liquids , 2505.01493

  33. [41]

    Herbut, Interactions and phase transitions on graphene’s honeycomb lattice, Phys

    I.F. Herbut, Interactions and phase transitions on graphene’s honeycomb lattice, Phys. Rev. Lett. 97 (2006) 146401

    2006

  34. [42]

    Herbut, V

    I.F. Herbut, V. Juriˇ ci´ c and O. Vafek,Relativistic mott criticality in graphene , Phys. Rev. B 80 (2009) 075432

    2009

  35. [43]

    L. Ma, R. Chaturvedi, P.X. Nguyen, K. Watanabe, T. Taniguchi, K.F. Mak et al., Relativistic mott transition in strongly correlated artificial graphene , arXiv preprint arXiv:2412.07150 (2024)

    2024 arXiv

  36. [44]

    D. Yang, J. Liang, H. Hu, N. Kaushal, C.-E. Hsu, K. Watanabe et al., Correlated insulating states in slow dirac fermions on a honeycomb moir {\’e} superlattice, arXiv preprint arXiv:2504.17970 (2025)

    2025 arXiv

  37. [45]

    Hawashin, M.M

    B. Hawashin, M.M. Scherer and L. Janssen, Gross-Neveu-XY quantum criticality in moir´ e Dirac materials, Phys. Rev. B 111 (2025) 205129 [ 2503.19963]

    2025 arXiv

  38. [46]

    Tolosa-Sime´ on, L

    M. Tolosa-Sime´ on, L. Classen and M.M. Scherer, Relativistic Mott transitions, quantum criticality, and finite-temperature effects in tunable Dirac materials from functional renormalization, 2503.04911

    arXiv

  39. [47]

    H. Liu, E. Huffman, S. Chandrasekharan and R.K. Kaul, Quantum Criticality of Antiferromagnetism and Superconductivity with Relativity, Phys. Rev. Lett. 128 (2022) 117202 [Erratum: Phys. Rev. Lett 131, 139901 (2023)]

    2022

  40. [48]

    Herbut, V

    I.F. Herbut, V. Juriˇ ci´ c and B. Roy,Theory of interacting electrons on the honeycomb lattice , Phys. Rev. B 79 (2009) 085116

    2009

  41. [49]

    Zee, Group Theory in a Nutshell for Physicists , Princeton University Press, USA (2016)

    A. Zee, Group Theory in a Nutshell for Physicists , Princeton University Press, USA (2016)

    2016

  42. [50]

    B. Roy, V. Juriˇ ci´ c and I.F. Herbut,Quantum superconducting criticality in graphene and topological insulators, Phys. Rev. B 87 (2013) 041401 [Erratum: Phys. Rev. B 94, 119901 (2016)]

    2013

  43. [51]

    Fraser-Taliente and J

    L. Fraser-Taliente and J. Wheater, Melonic limits of the quartic Yukawa model and general features of melonic CFTs, JHEP 01 (2025) 187 [ 2410.09152]

    2025 arXiv

  44. [52]

    Kvedarait˙ e, T

    S. Kvedarait˙ e, T. Steudtner and M. Uetrecht,Revisiting the ϕ6 Theory in Three Dimensions at Large N , 2502.07880

  45. [53]

    S. Ryu, C. Mudry, C.-Y. Hou and C. Chamon, Masses in graphenelike two-dimensional electronic systems: Topological defects in order parameters and their fractional exchange statistics , Phys. Rev. B 80 (2009) 205319

    2009

  46. [54]

    Steudtner, FoRGEr, unpublished (2024)

    T. Steudtner, FoRGEr, unpublished (2024)

    2024

  47. [55]

    Steudtner, Towards general scalar-Yukawa renormalisation group equations at three-loop order , JHEP 05 (2021) 060

    T. Steudtner, Towards general scalar-Yukawa renormalisation group equations at three-loop order , JHEP 05 (2021) 060

    2021

  48. [56]

    I. Jack, H. Osborn and T. Steudtner, Explorations in scalar fermion theories: β-functions, supersymmetry and fixed points, JHEP 02 (2024) 038

    2024

  49. [57]

    Steudtner and A.E

    T. Steudtner and A.E. Thomsen, General quartic β-function at three loops , JHEP 10 (2024) 163

    2024

  50. [58]

    Zerf, L.N

    N. Zerf, L.N. Mihaila, P. Marquard, I.F. Herbut and M.M. Scherer, Four-loop critical exponents for the Gross-Neveu-Yukawa models, Phys. Rev. D 96 (2017) 096010

    2017

  51. [59]

    Mihaila, N

    L.N. Mihaila, N. Zerf, B. Ihrig, I.F. Herbut and M.M. Scherer, Gross-neveu-yukawa model at three loops and ising critical behavior of dirac systems , Phys. Rev. B 96 (2017) 165133

    2017

  52. [60]

    Wilson and J

    K.G. Wilson and J. Kogut, The renormalization group and the epsilon expansion , Physics Reports 12 (1974) 75

    1974

  53. [61]

    Uetrecht, I.F

    M. Uetrecht, I.F. Herbut, M.M. Scherer, E. Stamou and T. Steudtner, Quantum multicriticality and emergent symmetry in Dirac systems with two order parameters at three-loop order, 2505.22723

    arXiv

  54. [62]

    Ihrig, N

    B. Ihrig, N. Zerf, P. Marquard, I.F. Herbut and M.M. Scherer, Abelian Higgs model at four loops, fixed-point collision, and deconfined criticality , Phys. Rev. 18 B 100 (2019) 134507

    2019

  55. [63]

    Herbut and Z

    I.F. Herbut and Z. Teˇ sanovi´ c,Herbut and Teˇ sanovi´ c Reply, Phys. Rev. Lett. 78 (1997) 980

    1997

  56. [64]

    Kaplan, J.-W

    D.B. Kaplan, J.-W. Lee, D.T. Son and M.A. Stephanov, Conformality lost , Phys. Rev. D 80 (2009) 125005

    2009

  57. [65]

    Gorbenko, S

    V. Gorbenko, S. Rychkov and B. Zan, Walking, Weak first-order transitions, and Complex CFTs , JHEP 10 (2018) 108

    2018

  58. [66]

    Hawashin, A

    B. Hawashin, A. Eichhorn, L. Janssen, M.M. Scherer and S. Ray, The Nordic-walking mechanism and its explanation of deconfined pseudocriticality from Wess-Zumino-Witten theory, Nature Commun. 16 (2025) 20

    2025

  59. [67]

    Y. Liu, Z. Wang, T. Sato, M. Hohenadler, C. Wang, W. Guo et al., Superconductivity from the condensation of topological defects in a quantum spin-hall insulator , Nature Communications 10 (2019)

    2019

  60. [68]

    T. Sato, M. Hohenadler and F.F. Assaad, Dirac fermions with competing orders: Non-landau transition with emergent symmetry, Phys. Rev. Lett. 119 (2017) 197203

    2017

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