The $(2+1)$-dimensional Gross-Neveu-Yukawa model at finite temperature, density, and magnetic field within the Functional Renormalization Group

Dirk H. Rischke; Justin L.P. Mauldin

REVIEW 2 major objections 4 minor 1 cited by

The $(2+1)$-dimensional Gross-Neveu-Yukawa model at finite temperature, density, and magnetic field within the Functional Renormalization Group

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read In the (2+1)-dimensional Gross-Neveu-Yukawa model, a beyond-mean-field renormalization group computation finds de Haas–van Alphen oscillations in the chiral phase boundary and a tricritical point that moves with magnetic field.

desk verdict First FRG computation of the GNY model with magnetic field, but the sharp-regulator floor could be inflating the dHvA features — a smooth-regulator check is needed before the central claim is fully solid. read the letter →

arxiv 2608.02280 v1 pith:WCOOSU6C submitted 2026-08-03 hep-ph hep-thnucl-th

Justin L.P. Mauldin , Dirk H. Rischke This is my paper · ORCID

classification hep-phhep-thnucl-th

keywords Gross-Neveu-YukawamodelfunctionalrenormalizationgroupdeHaas–vanAlphenoscillationsmagneticcatalysisinversechiralphasediagramLandaulevelstricriticalpoint

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 studies the chiral phase diagram of the (2+1)-dimensional Gross-Neveu-Yukawa model at finite temperature, chemical potential, and background magnetic field, using the functional renormalization group in the local potential approximation. The authors find that, beyond the mean-field approximation, magnetic catalysis dominates at zero chemical potential, while inverse magnetic catalysis and de Haas–van Alphen oscillations appear at intermediate chemical potential and small magnetic field. They also report a tricritical point that shifts to higher temperature and lower chemical potential as the field grows, and a back-bending of the phase boundary distinct from the zero-field quark-meson-model effect. These features were previously seen in the Gross-Neveu model in mean field and optimized perturbation theory; the paper shows they persist in the bosonized GNY model with high numerical precision.

What carries the argument

The Landau-level truncated fermion loop in the flow equation for the effective potential, expressed by the floor function N_LL(k^2) = floor(k^2/(2|qB|)), which counts how many Landau levels contribute at a given renormalization-group scale k. This replacement converts the momentum integral into a sum over a finite number of levels that changes discretely as k flows, producing the oscillatory features. The boson loop is unaffected by the magnetic field, and the sharp momentum-space regulator makes the Landau-level sum finite.

What would settle it

A direct check is to solve the same FRG flow equation with a smooth regulator shape function (e.g., an exponential or power-law cutoff) and see whether the de Haas–van Alphen oscillations in the µ–|qB| phase diagram and the associated first-order transitions survive. Alternatively, a lattice simulation of the (2+1)-dimensional GNY model at finite density and magnetic field could test the existence and location of the first-order transitions and the tricritical-point shift.

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Extended reading notes

Core claim

The central claim is that the full phase diagram of the (2+1)-dimensional GNY model, computed beyond mean field via the FRG in the local potential approximation, contains: (i) magnetic catalysis at large field and small chemical potential; (ii) inverse magnetic catalysis at intermediate chemical potential and small field; (iii) de Haas–van Alphen oscillations in the µ–|qB| plane at small field and large µ, realized as a sequence of first-order transitions into the broken phase followed by second-order restoring transitions; and (iv) a tricritical point that moves to higher T and lower µ with increasing |qB|. The bosonization of the Gross-Neveu model with a dynamical scalar and a Yukawa coupl

Load-bearing premise

The sharp momentum-space regulator, which makes the number of Landau levels entering the flow a step function of k^2, is what produces the sharp oscillations; if a smooth regulator changes or washes out those transitions, the predicted de Haas–van Alphen structure could be an artifact of the regulator rather than a property of the model.

Editorial extensions

If this is right

Magnetic catalysis: at zero chemical potential the critical temperature grows monotonically with |qB|.
Inverse magnetic catalysis: at intermediate µ and small |qB|, the chemical potential at the second-order transition decreases before magnetic catalysis takes over at larger |qB|.
De Haas–van Alphen oscillations: the chiral condensate shows repeated first-order transitions into the broken phase and second-order restoring transitions when |qB| is varied at fixed µ in the large-µ small-|qB| region.
A tricritical point in the T–µ plane shifts to higher T and lower µ as |qB| increases, and a back-bending of the phase boundary appears at intermediate field strengths.
These phenomena are already present in the Gross-Neveu model; the GNY model reproduces them beyond mean field with higher numerical precision due to the improved finite-volume numerical solver.

Reading between the lines

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

If the oscillatory features are regulator-induced (as the appendix acknowledges that a smooth regulator shape would make the Landau-level transitions more gradual), the de Haas–van Alphen oscillations may not be a genuine property of the model; a test with a smooth regulator would settle this.
The finding that the lowest-Landau-level approximation becomes exact in the infrared (for k below the square root of 2|qB|) suggests LLL dominance is robust, but the first-order transitions at larger |qB| could depend on the sharp cutoff.
The finite-volume numerical scheme used here could be applied to the (3+1)-dimensional quark-meson model with pions to resolve the deep-infrared region and potentially improve earlier functional renormalization group calculations of the QCD-type phase diagram.
Since the chiral condensate is the order parameter, the predicted sequence of first-order transitions might be observable in condensed-matter analogues such as graphene in a magnetic field, if the GNY model captures the relevant universality class.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper does what it says: it computes the (2+1)-dimensional GNY phase diagram at finite T, mu, and B with the FRG in LPA, using the Kurganov–Tadmor scheme to solve the flow. The genuinely new piece is that this is the first beyond-mean-field FRG treatment of the GNY model in a magnetic field. The qualitative findings — MC, IMC at intermediate mu, dHvA oscillations, first-order lines in the broken phase, and the TCP shift — were already known from mean-field, OPT, and lattice studies of the GN model, and the authors say so plainly. So this is a useful confirmation in a renormalizable bosonized model, not a discovery of new physics.

What the paper does well: the derivation in the appendix is explicit, the numerical parameters are stated, and the presentation is honest. The citation pattern is fine — they credit the relevant mean-field, OPT, and lattice work and do not oversell the novelty. The footnote admitting that a smooth regulator would make the Landau-level transitions more gradual is a good sign of their own awareness of the issue.

The main soft spot is precisely that footnote. The floor function N_LL(k^2) in Eq. (17) makes the number of Landau levels jump discontinuously as k flows, and that non-analyticity enters directly into the flow of the effective potential. It is plausible that the sharp Litim regulator is at least enhancing the dHvA oscillations and the associated first-order transitions, and the authors do not test this with a smooth regulator. Since mean-field and OPT also see dHvA, the phenomenon is probably physical, but "probably" is doing real work here. A single smooth-regulator run, or even a plot showing the oscillations are robust, would remove the largest doubt. Second, the "unprecedented precision" claim lacks a quantitative baseline — no grid-convergence study, no comparison of the zero-field result with a different numerical scheme, no statement of truncation error in the LPA. Third, no code or data is provided, so independent reproduction requires reimplementation from scratch.

My overall take: this is a solid, careful calculation that extends known results to a new model. It deserves a serious referee. The referee should insist on a smooth-regulator cross-check and a more quantitative discussion of numerical uncertainty, but the core derivation and the honest framing make it publishable after that work.

Referee Report

2 major / 4 minor

Summary. The manuscript studies the (2+1)-dimensional Gross-Neveu-Yukawa model at finite temperature, chemical potential, and magnetic field using the Functional Renormalization Group in the local potential approximation. It derives the FRG flow equation for the effective potential, first without and then with a magnetic field, and solves it with a finite-volume Kurganov-Tadmor scheme, fixing the Yukawa coupling g^2 by requiring the vacuum condensate to sit at σ0=1. Results are presented as phase diagrams in the (T, μ), (T, |qB|), and (μ, |qB|) planes for N_f=2 and Λ=10^3. The main reported findings are magnetic catalysis at μ=0, inverse magnetic catalysis at intermediate μ and small |qB|, de Haas–van Alphen oscillations with sequences of first-order transitions, a TCP that shifts to higher T and lower μ with increasing |qB|, and a back-bending of the phase boundary.

Significance. If the reported phase structure is correct, this is a useful beyond-mean-field FRG confirmation that the GNY model reproduces features previously found in the Gross-Neveu model via mean-field, OPT, and lattice methods. The derivations in Appendix A are explicit, the numerical parameters are stated, and the coupling is fixed by a standard renormalization condition rather than by fitting the phase diagram, so there is no circularity. However, the central qualitative claims are not yet supported by an adequate regulator-sensitivity analysis: the non-analytic floor function in the Landau-level count enters directly into the fermionic loop, and the paper itself concedes that a smooth regulator would change the transition. For this reason the results are significant but currently conditional.

major comments (2)

[§II.B, Eqs. (17)–(18), and footnote 2] The dHvA oscillations and the associated sequences of first-order transitions in Fig. 2(a)–(c) occur in the small-|qB|, large-μ region where N_LL(k^2)=floor(k^2/(2|qB|)) discontinuously truncates the Landau-level sum. Because this floor function enters directly into the fermionic loop (Eq. (18)), the non-analytic k-dependence of the flow is inherited from the sharp Litim regulator. Footnote 2 acknowledges that a smooth regulator would make the transition 'more gradual,' but no smooth-regulator run is provided. Since the conclusion explicitly claims these features 'also exist in the Gross-Neveu-Yukawa model,' the paper needs to show that the oscillations and first-order segments survive, with comparable amplitude and period, for a smooth regulator shape function (e.g., exponential or power-law). If they weaken or disappear, the phase-diagram claims must be rephrased as regulator-dependent
[§III and Appendix B] The numerical setup is described in detail (Λ=10^3, k_IR=10^-2, Δσ=5×10^-3), but no convergence or truncation-error tests are reported. In particular, there is no variation of Δσ, k_IR, or Λ, and no test that the LPA truncation is adequate for the quantitatively claimed phase boundaries, such as the TCP location or the amplitude of the dHvA oscillations. The statement in §IV that the hydrodynamic scheme resolves features 'with unprecedented precision' is not supported by any quantitative comparison. Please add a convergence study and either provide numerical error estimates or soften the precision claim.

minor comments (4)

[Abstract and Introduction] Language issues: 'allows for go further' and similar phrasing should be corrected. Also, 'allows to resolve' appears in §IV and should be 'allows us to resolve'.
[Fig. 2(b)] The four curves are distinguished only by color in the text description. For print/accessibility, add explicit labels or markers for μ=0.92, 0.95, 1.01, 1.05.
[§III, Fig. 3] The claimed TCP shift would be easier to verify if the TCP coordinates were tabulated or if the three phase boundaries were overlaid in one panel. As written, the shift is only apparent by visual comparison of separate panels.
[Eq. (10)] The displayed equation '∂tU(t,σ) = −2' is incomplete in the text; presumably a diagrammatic term is missing in rendering. This should be fixed in the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the phase diagram is computed from the FRG flow with only a standard scale-setting condition for g^2.

full rationale

The derivation chain is self-contained. The only fitted parameter is g^2, tuned via Newton-Raphson so that the IR effective potential has its minimum at sigma_0 = 1 (Appendix B, Eq. B4). This is a standard renormalization/scale-setting condition, not a fit to any target phase-diagram feature: the dHvA oscillations, TCP shift, and back-bending do not enter as inputs or constraints. All results are reported in units of h sigma_0, so this scale choice does not inject the predicted structure. The dHvA oscillations and first-order transitions are computed from the Landau-level sum in Eq. (18), with N_LL(k^2) = floor(k^2/(2|qB|)) in Eq. (17); the discrete Landau-level spectrum is part of the model input, and the floor function follows from the sharp Litim regulator rather than from a fitted or self-cited premise. The paper explicitly notes in footnote 2 that a smooth regulator would make the transition 'more gradual', which flags a genuine regulator-robustness concern for the sharpness of the oscillations, but that is a correctness/validity risk, not circularity: the output is not equivalent to the input by construction. The comparisons with mean-field, OPT, and lattice results in Refs. [20-22] are external benchmarks from independent groups, and the hydrodynamic solver is cited to Refs. [27,28], neither of which is a load-bearing self-citation by the authors. The LPA truncation and Litim regulator are stated approximations, not smuggled-in ans"atze. Overall, no step reduces to its own input, and no central claim is forced by a self-citation chain.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central calculation rests on standard FRG identities and a set of explicitly stated model/truncation choices. No new particles or entities are introduced. The only tuned parameter is g^2, set by a renormalization condition. The main unverified assumption is regulator independence of the sharp-regulator-induced Landau-level discontinuities.

free parameters (1)

g^2 (four-fermion/Yukawa UV coupling) = not quoted; tuned to give σ0=1
Newton-Raphson tuning of Eq. (B4) sets the vacuum condensate to 1; this is a renormalization condition, not a fit to external data, but it is an input to all predicted scales.

assumptions (5)

standard math The Wetterich flow equation (Eq. 3) is exact; LPA is the only truncation.
Standard FRG identity; not independently proved in the paper. Truncation to the local potential approximation is invoked in Eq. (6).
domain assumption The Litim regulator shape functions (Eqs. 7-8) produce physical results; sharp-cutoff discontinuities do not generate spurious phase structure.
The floor-function Landau-level truncation (Eq. 17) creates non-analytic jumps; footnote 2 admits a smooth regulator would make transitions more gradual but no smooth-regulator test is provided.
domain assumption In a 2+1-dimensional model with a reducible 4-component Clifford representation, N_f=2 and dγ=4 define the physical flavor content.
Sec. II.B chooses dγ=4 to describe magnetic catalysis; this doubles the fermion degrees of freedom relative to a two-component representation and affects quantitative comparison with earlier N_f=2 GN studies.
domain assumption Fermions move only in the transverse plane (p_z=0) with all flavors carrying the same charge q.
Stated in Sec. II.B and footnote 1; needed to reduce the Dirac equation to Landau levels and to keep a single condensate.
domain assumption Adding a scalar kinetic term by hand in the Hubbard-Stratonovich bosonization yields the GNY model whose phase structure is representative of QCD-like chiral symmetry breaking.
Sec. II model construction; the GNY model is renormalizable in 2+1 dimensions, which is the stated motivation for studying it.

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Pith. "Pith review of The $(2+1)$-dimensional Gross-Neveu-Yukawa model at finite temperature, density, and magnetic field within the Functional Renormalization Group." pith.science (2026). https://pith.science/paper/WCOOSU6C

@misc{pith2026260802280,
  author       = {Pith},
  title        = {Pith review of: The $(2+1)$-dimensional Gross-Neveu-Yukawa model at finite temperature, density, and magnetic field within the Functional Renormalization Group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WCOOSU6C}},
  note         = {Machine review of arXiv:2608.02280}
}

read the original abstract

We investigate the phase diagram of the (2+1)-dimensional Gross-Neveu-Yukawa (GNY) model at finite temperature, density, and magnetic field beyond mean-field, using the Functional Renormalization Group (FRG) in the local potential approximation. Large magnetic fields result in magnetic catalysis, a dimensional reduction of the system, and enhancement of chiral symmetry breaking. We employ a hydrodynamical algorithm to solve the FRG flow equation for the effective potential, which allows for go further into the infrared region than with previously used methods. We find that the chiral condensate exhibits non-trivial behavior in various regions of the phase diagram: several first-order phase transitions and de Haas -- van Alphen oscillations at small magnetic field and large chemical potential, as well as a critical endpoint which shifts to higher temperature with increasing magnetic field.

Figures

Figures reproduced from arXiv: 2608.02280 by the authors.

**Figure 2.** FIG. 2: (a) Close-up of the phase diagram in the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3: Contour plots of the chiral condensate in the [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

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Reference graph

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Zero Magnetic Field In LPA, only the effective potentialUdepends on the RG scale, such that the Wetterich equation (3) reads V2∂tU(t, σ) =∂tΓt[ ¯ψ, ψ, φ] φ=σ, ¯ψ=ψ=0 =STr 1 2 ∂tRt Γ(2) t [Φ] +R t −1 φ=σ, ¯ψ=0,ψ=0 ,(A1) whereV 2 =β(2π) 2δ(2)(0)is the two-dimensional volume element. For vanishing magnetic field, Fourier-transforming the right-hand side of E...
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P. Virtanen, R. Gommers, T. E. Oliphant, M. Haber- land, T. Reddy, D. Cournapeau, E. Burovski, P. Pe- terson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, ˙I. Po- lat, Y...

2020

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

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For vanishing magnetic field, Fourier-transforming the right-hand side of Eq

Zero Magnetic Field In LPA, only the effective potentialUdepends on the RG scale, such that the Wetterich equation (3) reads V2∂tU(t, σ) =∂tΓt[ ¯ψ, ψ, φ] φ=σ, ¯ψ=ψ=0 =STr 1 2 ∂tRt Γ(2) t [Φ] +R t −1 φ=σ, ¯ψ=0,ψ=0 ,(A1) whereV 2 =β(2π) 2δ(2)(0)is the two-dimensional volume element. For vanishing magnetic field, Fourier-transforming the right-hand side of E...

[2] [2]

Therefore, for nonvanishing magnetic field, only the cal- culation of the fermion loop changes

Nonzero Magnetic Field In this model, only fermions carry electric charge. Therefore, for nonvanishing magnetic field, only the cal- culation of the fermion loop changes. To introduce the magnetic field we employ the replacements from Eqs. (15), (16), −2 =d γ Nf V2 |qB| 4π ∞X n=−∞ ∞X l=0 αl k2θ k2 −2|qB|l (νn +iµ) 2 +k 2 +m 2 f (t, σ) =d γ Nf βV2 |qB| 8π ...

[3] [3]

This allows to use methods from numerical hydrodynamics to solve the flow equations

FRG Flow Equation in Conservative F orm Recently it has been found that the flow equation for the effective potential can be cast into conservative form [27]. This allows to use methods from numerical hydrodynamics to solve the flow equations. Here, we employ the Kurganov-Tadmor (KT) scheme [32], a high- resolution finite-volume method for solving hyperbo...

[4] [4]

We fit the parameterg 2 using a Newton- Raphson root-finding algorithm so that the resulting ef- fective potential in the IR has a minimum atσ 0 = 1

Initial and Boundary Conditions The initial condition (UV limit) for the effective po- tential is given by its classical value in vacuum, UΛ(σ) = (hσ)2 2g2 .(B4) We seth= 1, and measure all dimensionful quantities in units ofσ 0. We fit the parameterg 2 using a Newton- Raphson root-finding algorithm so that the resulting ef- fective potential in the IR ha...

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Grasso and H

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Skokov, A

V. Skokov, A. Y. Illarionov, and V. Toneev, Estimate of the magnetic field strength in heavy-ion collisions, Int. J. Mod. Phys. A24, 5925 (2009), arXiv:0907.1396 [nucl-th]

arXiv 2009

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Tuchin, Time and space dependence of the electro- magnetic field in relativistic heavy-ion collisions, Phys

K. Tuchin, Time and space dependence of the electro- magnetic field in relativistic heavy-ion collisions, Phys. Rev. C88, 024911 (2013), arXiv:1305.5806 [hep-ph]

arXiv 2013