Chiral Long-Range Order in three Euclidean Lattice Gross-Neveu Models

Leonardo Goller; Simone Fabbri

arxiv: 2606.13075 · v1 · pith:DKR6CVW3new · submitted 2026-06-11 · 🧮 math-ph · cond-mat.stat-mech· hep-th· math.MP

Chiral Long-Range Order in three Euclidean Lattice Gross-Neveu Models

Simone Fabbri , Leonardo Goller This is my paper

Pith reviewed 2026-06-27 05:46 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.stat-mechhep-thmath.MP

keywords Gross-Neveu modellong-range orderreflection positivityHubbard-Stratonovich transformationlattice fermionsPeierls contour argumentchiral bilinearlarge-N limit

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The pith

Three lattice versions of the two-dimensional Gross-Neveu model exhibit long-range order in the chiral fermion-mass bilinear at weak coupling and large flavor number.

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

The paper maps each of three standard lattice fermion discretizations of the Gross-Neveu model to an equivalent bosonic system via Hubbard-Stratonovich transformation. Reflection positivity is established for the resulting measures, which permits chessboard estimates and a Peierls contour argument to produce a strictly positive lower bound on the expectation of the chirally charged bilinear. The bound holds uniformly across the discretizations once the coupling is small enough and the number of flavors is large enough, while the mass-mass correlator remains controlled by the minimizers of the effective potential. A sympathetic reader cares because the argument supplies a fully rigorous, non-perturbative confirmation that matches the large-N mean-field picture without relying on perturbation theory.

Core claim

By performing a Hubbard-Stratonovich transformation, we map the fermionic systems to bosonic ones and establish Reflection Positivity for the resulting measures. Exploiting this structure, we combine Chessboard Estimates with a Peierls-type contour argument to prove Long-Range Order for the chirally charged fermion-mass bilinear at sufficiently small coupling and sufficiently large flavor number. Our analysis is robust with respect to the choice of lattice discretization and applies uniformly across different realizations of the same underlying continuum model. Moreover, we obtain uniform pointwise bounds on the bosonic two-point function, equivalently on the fermionic mass-mass correlator,

What carries the argument

Hubbard-Stratonovich transformation to bosonic measures that satisfy uniform reflection positivity, which enables chessboard estimates and a Peierls contour argument to bound the chiral bilinear order parameter from below.

If this is right

Long-range order holds uniformly for naive, staggered, and a third standard lattice discretization.
The fermionic mass-mass correlator receives uniform pointwise bounds controlled by the effective-potential minimizers.
The result supplies a rigorous non-perturbative link between the lattice models and their large-N mean-field predictions.
The argument applies to any even number of flavors once that number is taken sufficiently large.

Where Pith is reading between the lines

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

The same reflection-positivity and contour technique could be tested on other two-dimensional fermionic models whose Hubbard-Stratonovich measures remain positive.
If the lower bound on the order parameter survives the continuum limit, it would give a lattice proof of chiral symmetry breaking in the corresponding quantum field theory.
The uniform control on two-point functions suggests that correlation lengths stay finite while spontaneous order persists in the infinite-volume limit.

Load-bearing premise

The bosonic measures obtained after the Hubbard-Stratonovich transformation satisfy reflection positivity uniformly for the three lattice discretizations.

What would settle it

An explicit computation or Monte Carlo measurement showing that the order parameter for the bilinear falls to zero as the number of flavors grows while the coupling is held fixed and small would falsify the claimed lower bound.

Figures

Figures reproduced from arXiv: 2606.13075 by Leonardo Goller, Simone Fabbri.

**Figure 1.** Figure 1: Distribution of Phases of the Anti-symmetrized Forward Dirac Operator γµ⊗Tµ in L = 2 Lattice. Solid Links represent a −1 sign. Notice that the set of edges intersecting ∂−(ΛL)+ are opposite of those intersecting ∂+(ΛL)+ because we have chosen an anti-symmetrization convention where the Z2-twisting is localized on ∂−(ΛL)+. By means of any Z2-Gauge Transformation, we can move such line wherever we desire, wi… view at source ↗

**Figure 2.** Figure 2: Distribution of Phases of the Anti-symmetrized Forward Dirac Operator Γµ(x)Tµ in L = 2 Lattice. Solid Links represent a −1 sign. Notice that the set of edges intersecting ∂−(ΛL)+ are opposite of those intersecting ∂+(ΛL)+ because we have chosen an anti-symmetrization convention where the Z2-twisting is localized on ∂−(ΛL)+. By means of any Z2 Gauge Transformation, we can move such line wherever we desire, … view at source ↗

**Figure 3.** Figure 3: Blocking of the lattice ΛL (L = 4 in this example) in terms of the unit cell with points indexed by I. Lighter (resp. heavier) lines are used for denoting hoppings with a phase +1 (resp. −1). The model is constructed along the same lines as the SN one described above, but with a different interaction structure. We therefore adopt the same notation, adapted to the new labeling. Staggered fermions. At each l… view at source ↗

**Figure 4.** Figure 4: Graphical representation of the cut torus, with the cut plane highlighted in red (Credits: Davide Morgante) We aim to show that the three bosonic measures µα(φ), α ∈ {NN,SN,SP}, constructed in the previous section, are reflection positive [Frö+78] with respect to reflections across bond planes of the underlying lattice Λα L , where: Λ NN L = Λ SN L = ΛL Λ SP L = ΛeL. 21 [PITH_FULL_IMAGE:figures/full_fig_p… view at source ↗

**Figure 5.** Figure 5: By means of a Z2-Gauge Transformation supported on one side of the holonomy line, we can translate such line to the left making it overlap with the left boundary of (ΛL)+. An example of this construction for the NN and SN models is shown in the upper and lower panels, respectively. Performing the change of variables φ 7→ Θφ and using the reflection invariance of the measure: µα ◦ Θ −1 = µα, we obtain: ⟨A,B… view at source ↗

**Figure 6.** Figure 6: Sketch of the Effective Potential v α λ (ϕ) and can therefore be computed explicitly via Fourier transforms. The corresponding infinitevolume effective potential density: v α λ (ϕ) lim L→∞ 1 |Λα L | V α L,λ(φ) [PITH_FULL_IMAGE:figures/full_fig_p034_6.png] view at source ↗

**Figure 7.** Figure 7: deformation rule adopted for associating a set of Peierls’ contours to every sign configuration ζ ∈ {±}Λα L . 3. To each sign configuration ζ ∈ {±}Λα L , we associate its set of Peierls’ contours, namely the closed curves in the dual lattice (Λα L ) dual Λα L + ( 1 2 , 1 2 ) crossing precisely those edges ⟨x ′ ,y′ ⟩ ⊂ Λα L for which ζx ′ , ζy ′ . As usual, crossings are resolved according to the deform… view at source ↗

**Figure 8.** Figure 8: 4 Tassellation Patterns Used to Cover Lattices Λα L : notice that using all such covering one is able to recover all edges crossed by the contour γ. (In the Lattice Λ SP L all spacings are doubled) Using the standard contour counting bounds: #{γ around x : |γ| = k} ≤ k 2 3 k , #{γ : |γ| = k} ≤ L 2 3 k , we easily find, for N and L sufficiently large: X γaroundx e − 1 4N cα 3 (λ)|γ| ≤ 1 8 e − 1 4N cα 3 (λ) … view at source ↗

**Figure 9.** Figure 9: Periodic patterns generated by chessboard reflections of the gray boxes representing the covering configurations (h,e), (h,o), (v,e), and (v,o). 5. A further RP argument, closely analogous to Proposition 4.1, shows that the infimum over suppI (−++−) is attained by periodic configurations of the form: (ϕ−,ϕ+,ϕ+,ϕ−), ϕ+ ≥ 0, ϕ− ≤ 0, depending only on two real parameters (ϕ+,ϕ−). Accordingly: inf φ∈suppI (−++… view at source ↗

**Figure 10.** Figure 10: Graphical Representation of the Dirac Operator with periodic pattern produced by Chessboard’s Estimates. In Grey we have drawn the unit cells used to perform Fourier Transforms moreover, an explicit computation leads to ∆ SN(p;ϕ+,ϕ−) = 1 4 sin2 (2p0) + 1 4 (ϕ+ +ϕ−) 2 + (ϕ+ϕ−) 2 + sin2 (p1) 1 +ϕ 2 + +ϕ 2 − + sin2 (p1) . (5.25) Note that due to the periodicity of ∆ SN(·;ϕ+,ϕ−) under translations of π 2 … view at source ↗

**Figure 11.** Figure 11: Graphical Representation of the Naive Dirac Operator with periodic pattern produced by Chessboard’s Estimates. In Grey we have drawn the unit cells used to perform Fourier Transforms Proof for the NN model. Proceeding in analogy with the case of the SN model (see [PITH_FULL_IMAGE:figures/full_fig_p053_11.png] view at source ↗

**Figure 12.** Figure 12: Graphical Representation of the Staggered Dirac Operator with Plaquette Interaction with periodic pattern produced by Chessboard Estimates. In Grey we have drawn the unit cells used to perform Fourier Transforms. w SP λ (ϕ+,ϕ−) = ϕ 2 + +ϕ 2 − 4λ − 8 " (− π 8 , π 8 ]×(− π 2 , π 2 ] d 2p (2π) 2 log∆ SP(p;ϕ+,ϕ−). (5.43) In this way, if ϕ+ = ϕ− = 0, the integrand has only one singularity at p = 0. Trivial tho… view at source ↗

**Figure 13.** Figure 13: Fundamental Cells: In absence of twistings of the boundary conditions one gets a manifestly translation invariant system. The set of allowed momenta is given by the dual of this reduced lattice: (Λ SN L ) ∗ −− = 2π L Z + 1 2 2 ∩ − π 2 , π 2 i × (−π,π] . Again the determinant can be computed starting from (2.7): det (D/ SN) − ΛL − Mφ [PITH_FULL_IMAGE:figures/full_fig_p067_13.png] view at source ↗

**Figure 14.** Figure 14: Picture of the action of the projector πL: each cell is sent to its center [PITH_FULL_IMAGE:figures/full_fig_p069_14.png] view at source ↗

read the original abstract

We prove the existence of Long-Range Order in a class of two-dimensional Euclidean lattice Gross-Neveu models with an even number of fermion flavors, covering three standard lattice discretizations, including naive and staggered fermions widely used in numerical studies. By performing a Hubbard-Stratonovich transformation, we map the fermionic systems to bosonic ones and establish Reflection Positivity for the resulting measures. Exploiting this structure, we combine Chessboard Estimates with a Peierls-type contour argument to prove Long-Range Order for the chirally charged fermion-mass bilinear $\overline{\psi}\psi$ at sufficiently small coupling and sufficiently large flavor number. Our analysis is robust with respect to the choice of lattice discretization and applies uniformly across different realizations of the same underlying continuum model. Moreover, we obtain uniform pointwise bounds on the bosonic two-point function, equivalently on the fermionic mass-mass correlator, showing that it is quantitatively controlled by the minimizers of the effective potential. This provides a fully rigorous and non-perturbative demonstration of Long-Range Order in lattice Gross-Neveu models and establishes a direct connection between the rigorous theory and its large-$N$ (mean-field) predictions.

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.

Desk Editor's Note private letter to a colleague

The paper proves chiral LRO for three lattice GN discretizations at weak coupling via HS, RP, chessboard estimates and Peierls contours, but the uniform RP claim for the post-HS bosonic measures is the part that needs explicit checking.

read the letter

The main result is a proof that the chirally charged bilinear has long-range order in three 2D Euclidean lattice Gross-Neveu models (naive, staggered, and one other) with even flavor number, at small coupling and large N. The argument goes through Hubbard-Stratonovich to a bosonic theory, reflection positivity of the resulting measure, chessboard estimates, and a Peierls contour to get a strictly positive lower bound on the order parameter, plus uniform bounds on the two-point function controlled by the effective potential.

What works is the uniform treatment across discretizations that are actually used in simulations, and the direct link to mean-field predictions without extra fitting parameters. That gives a concrete non-perturbative anchor for people running Monte Carlo on these models.

The soft spot is exactly the reflection positivity step for the bosonic measures after the HS transform. The abstract asserts it holds for all three choices, but the naive discretization can introduce sign issues under reflection that are not automatic. If the paper only sketches this without a clear verification that the quadratic form from the fermion determinant stays reflection-positive, the chessboard estimates do not launch and the contour argument fails. The stress-test note correctly identifies this as the load-bearing assumption. Everything else is standard once RP is granted.

This is for people who need rigorous control on lattice fermionic models in condensed-matter or lattice QFT. A reader who cares about non-perturbative existence results and their relation to large-N approximations will find it useful. It deserves peer review because the claim is a specific, falsifiable existence theorem even if the RP verification requires close reading.

Referee Report

1 major / 2 minor

Summary. The manuscript proves the existence of long-range order (LRO) for the chirally charged bilinear ψ̅ψ in three Euclidean lattice discretizations (naive, staggered, and one additional) of the two-dimensional Gross-Neveu model with even number of flavors. After a Hubbard-Stratonovich transformation mapping the fermionic theory to a bosonic one, reflection positivity (RP) of the resulting measures is established; chessboard estimates and a Peierls contour argument then yield a strictly positive lower bound on the order parameter at sufficiently small coupling and large flavor number. Uniform pointwise bounds on the bosonic two-point function (equivalently the fermionic mass-mass correlator) are also obtained, controlled by the minimizers of the effective potential.

Significance. If the reflection-positivity step holds uniformly, the result supplies a fully rigorous, non-perturbative existence proof of chiral LRO that is discretization-independent and directly links the lattice models to their large-N mean-field predictions. This is a substantive advance for constructive quantum field theory and lattice fermion models, as it furnishes the first such proof covering multiple standard discretizations used in numerical work.

major comments (1)

[Hubbard-Stratonovich transformation and reflection-positivity section] The uniform reflection positivity of the post-Hubbard-Stratonovich bosonic measures is the load-bearing step for the entire argument (abstract and the section deriving the bosonic action). The manuscript asserts that the quadratic form induced by the fermion determinant satisfies the reflection-positivity condition for naive, staggered, and the third discretization, but must supply explicit verification that no additional sign factors arise under lattice reflection for any of the three choices; without this, the chessboard estimates cannot be launched and the Peierls contour argument fails to produce the claimed strictly positive lower bound.

minor comments (2)

Notation for the three discretizations should be introduced with explicit definitions of the Dirac operators before the HS step to make the uniformity claim easier to follow.
The statement that the two-point function is 'quantitatively controlled by the minimizers of the effective potential' would benefit from a precise inequality relating the correlator to the potential minima.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the single major comment below.

read point-by-point responses

Referee: [Hubbard-Stratonovich transformation and reflection-positivity section] The uniform reflection positivity of the post-Hubbard-Stratonovich bosonic measures is the load-bearing step for the entire argument (abstract and the section deriving the bosonic action). The manuscript asserts that the quadratic form induced by the fermion determinant satisfies the reflection-positivity condition for naive, staggered, and the third discretization, but must supply explicit verification that no additional sign factors arise under lattice reflection for any of the three choices; without this, the chessboard estimates cannot be launched and the Peierls contour argument fails to produce the claimed strictly positive lower bound.

Authors: We agree that an explicit, case-by-case verification of reflection positivity (including confirmation that no extraneous sign factors appear under lattice reflections) is required to make the argument fully rigorous. The current manuscript states that the quadratic form satisfies the condition for each of the three discretizations but does not display the detailed sign checks. In the revised version we will add a dedicated subsection that carries out these calculations explicitly for the naive, staggered, and additional discretizations, thereby justifying the subsequent chessboard estimates and Peierls contour argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a self-contained mathematical proof.

full rationale

The paper derives long-range order for the chiral bilinear via Hubbard-Stratonovich mapping to a bosonic measure, followed by an explicit claim to establish reflection positivity, then chessboard estimates and a Peierls contour argument. These steps are presented as direct applications of standard lattice techniques to the model definitions, with no reduction of any central quantity to a fitted parameter, self-referential definition, or load-bearing self-citation. The result is obtained from the assumptions on the measures and the contour argument without the output being equivalent to the input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; the paper invokes reflection positivity after Hubbard-Stratonovich and standard chessboard/Peierls machinery, but no explicit free parameters, ad-hoc axioms or invented entities are visible.

axioms (2)

domain assumption The Hubbard-Stratonovich transformed measures are reflection positive for the three lattice discretizations
Stated as established in the abstract; required for chessboard estimates
domain assumption Chessboard estimates and Peierls contour argument apply at large even flavor number and small coupling
Core of the existence proof; location not specified beyond abstract

pith-pipeline@v0.9.1-grok · 5745 in / 1417 out tokens · 18629 ms · 2026-06-27T05:46:20.714223+00:00 · methodology

discussion (0)

Reference graph

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Resurgence analysis of the Nambu-Jona-Lasinio model at large charge

[Ber+25] Jahmall Bersini et al. “Resurgence analysis of the Nambu-Jona-Lasinio model at large charge”. In:JHEP09 (2025), p

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work page doi:10.1007/978- 2005

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Fermion Interactions and Universal Behavior in Strongly Interacting Theories

[Bra12] Jens Braun. “Fermion Interactions and Universal Behavior in Strongly Interacting The- ories”. In:J. Phys. G39 (2012), p. 033001. doi:10.1088/0954-3899/39/3/033001. arXiv:1108.4449 [hep-ph]. [BTW02] J. Berges, N. T etradis, and C. Wetterich. “Non-perturbative renormalizationﬂow in quantumﬁeld theory and statistical physics”. In:Phys. Rept.363 (2002...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0954-3899/39/3/033001 2012

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work page doi:10.1007/bf01608557 2024

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Spontaneous Mass Generation and Chiral Symmetry Breaking in a Lattice Nambu–Jona-Lasinio Model

[GK23] Y ukimi Goto and T ohru Koma. “Spontaneous Mass Generation and Chiral Symmetry Breaking in a Lattice Nambu–Jona-Lasinio Model”. In:Communications in Mathematical Physics404.3 (Oct. 2023), pp. 1463–1493. issn: 1432-0916. doi:10.1007/s00220-023- 04858-8. url:http://dx.doi.org/10.1007/s00220-023-04858-8. [GK80] F. Guerin and R.D. Kenway. “A strong-cou...

work page doi:10.1007/s00220-023- 2023

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Gross-Neveu model through convergent per- turbation expansions

url:https : / / www . sciencedirect . com / science / article / pii / 0550321380900693. [GK85] Krysztof Gawçdzki and Antti Kupiainen. “Gross-Neveu model through convergent per- turbation expansions”. In:Communications in mathematical physics102.1 (1985), pp. 1–

1985

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QCD Phenomenology based on a Chiral Effective Lagrangian

isbn: 978-3-642-01849-7, 978-3-642-01850-3. doi: 10.1007/978-3-642-01850-3. [GMP12] Alessandro Giuliani, Vieri Mastropietro, and Marcello Porta. “Universality of conduc- tivity in interacting graphene”. In:Communications in Mathematical Physics311.2 (2012), pp. 317–355. 79 [GMP17] Alessandro Giuliani, Vieri Mastropietro, and Marcello Porta. “Universality ...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/978-3-642-01850-3 2012

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The Nambu–Jona-Lasinio model of quantum chromodynamics

url:https : / / www . sciencedirect . com / science/article/pii/0370269386914346. [Kle92] S. P. Klevansky. “The Nambu–Jona-Lasinio model of quantum chromodynamics”. In: Rev. Mod. Phys.64 (1992), pp. 649–708. doi:10.1103/RevModPhys.64.649. [KMR89] C. Kopper, J. Magnen, and V . Rivasseau. “Mass generation in the large-NGross–Neveu model”. In:Commun. Math. P...

work page doi:10.1103/revmodphys.64.649 1992

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Inhomogeneous phases in the Gross-Neveu model in1+1dimensions atﬁnite number ofﬂavors

url:https://link.aps.org/doi/10.1103/PhysRevD.11.395. [Len+20] Julian Lenz et al. “Inhomogeneous phases in the Gross-Neveu model in1+1dimensions atﬁnite number ofﬂavors”. In:Phys. Rev. D101 (9 May 2020), p. 094512. doi:10.1103/ PhysRevD.101.094512. url:https://link.aps.org/doi/10.1103/PhysRevD. 101.094512. [LL93] Elliott H. Lieb and Michael Loss. “Fluxes,...

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