{"id":"f35dc7c7-6117-4052-9525-230957da172e","arxiv_id":"2508.12080","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Fermionic instanton solutions are constructed for the Gross-Neveu and Gross-Neveu-Yukawa models, and their actions are matched to Hubbard-Stratonovich critical points in the large N limit.","lead":"This paper derives new fermionic instanton solutions in the Gross-Neveu and Gross-Neveu-Yukawa models, which describe interacting fermions in two and four dimensions. It shows their actions match special critical points of the large N effective action, connecting two calculational approaches.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The σ²/(2g) term dropped in Eq. (51) is leading order at the fixed point and removes the claimed σ=ik saddles, so the exact HS/instanton action matching is not established.","rationale":"The paper's explicit fermionic instanton solutions and their on-shell actions are derived with internally consistent algebra in Sections 2 and 3. The central advertised conclusion, however, is the identification with saddle points of the Hubbard-Stratonovich effective action. The reader's weakest-assumption analysis correctly locates the fragile step: Eq. (51) drops the tree-level σ²/(2g) term with a reference but no demonstration. A direct estimate confirms the term is leading order at the fixed point g* = 2πϵ/N: at the constant saddle σ=i on S² it contributes −N/ϵ, the same order as the determinant contribution in Eq. (64). More strongly, the full stationarity condition includes (Vol/g)σ, and because the determinant derivative vanishes at σ=ik, nonzero integer k cannot solve the full equation for finite g. The critical points used in Section 5 are therefore not critical points of the HS action once the quadratic term is retained. This is not a missing proof; an explicit leading-order term changes the saddle structure. The instanton solutions themselves may stand as a separate result, and a corrected HS comparison could restore part of the claim, but as written the identification is unjustified. This supports the reader's CONDITIONAL verdict, and since my concern coincides with the reader's weakest assumption, the verdict remains unchanged.","tokens_in":8184,"tokens_out":19281,"duration_ms":203004,"concrete_test":"Compute the stationarity condition of the full HS action (51) for constant σ on the unit S^d at the fixed-point coupling (15), retaining the σ² term: N ∂F_f/∂σ + (Vol/g)σ = 0. Evaluate at σ=i for d=2+ϵ; since ∂F_f/∂σ(i)=0 (Eqs. 59–62), the equation becomes (4π/g*)i = (2N/ϵ)i ≠ 0, so σ=i is not a saddle. Confirm the same for σ=2i in d=4−ϵ with the appropriate volume and fixed-point coupling; if no saddle exists at these points, the action matching in Eqs. (64), (68), (69) is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central identification rests on dropping the σ²/(2g) term in the Hubbard-Stratonovich effective action (51), justified only by a citation to [3] without demonstration. This term is not subleading: at the GN fixed point g* = 2πϵ/N (Eq. 15), the contribution of (1/(2g))∫_{S²} σ² at the claimed saddle σ=i is (1/(2g*))·4π·(−1) = −N/ϵ, identical in magnitude to the determinant contribution −N/ϵ in Eq. (64). More decisively, the full stationarity condition for constant σ is N ∂F_f/∂σ + (Vol/g)σ = 0. Since ∂F_f/∂σ vanishes at σ=ik (Eqs. 59–62), the full equation reduces to (Vol/g)σ = 0 for any finite g, so only σ=0 is a saddle. Thus σ=i (and σ=2i in d=4−ϵ) are not critical points of the HS action (51) when the tree-level quadratic term is retained. The claimed exact matching of actions in Eqs. (64), (68), (69) is therefore an artifact of neglecting a leading-order term, not a property of the HS effective action. The instanton solutions and on-shell actions in Sections 2 and 3 may still be valid, but the advertised identification with the large-N effective action is not established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies fermionic instantons in the Gross-Neveu model in d=2 and in the Gross-Neveu-Yukawa model in d=4. It constructs explicit smooth solutions with moduli C and x0, computes the on-shell actions (-2π/g in GN and 32π²/3 (g2/(6g1^4)-1/g1²) in GNY), evaluates them at the respective fixed points, and then maps the solutions to the sphere, where for C=1 and x0=0 they become constant. The paper also constructs additional solutions using Dirac operator eigenfunctions on S^d. In Section 5, the Hubbard-Stratonovich effective action is used to compute the large-N free energy for constant σ, and the critical points σ=i and σ=2i are claimed to reproduce the instanton actions. The central algebraic steps in Sections 2 and 3 appear internally consistent, but the claimed exact identification with the Hubbard-Stratonovich critical points is not established because a leading-order term in the effective action is dropped without justification.","tokens_in":8335,"tokens_out":30703,"duration_ms":327397,"significance":"If the identification with the large-N Hubbard-Stratonovich effective action were correct, the paper would provide a useful bridge between explicit fermionic instanton solutions and the large-N saddle-point calculus, and the explicit solutions themselves would be valuable for semiclassical studies of these models. The computations of the on-shell actions and the conformal mapping are clearly presented and appear correct. However, the advertised relation to the Hubbard-Stratonovich critical points is the main advertised conclusion, and that conclusion is not supported by the present analysis; the instanton solutions may still stand on their own, but the paper needs substantial revision before the identification claim can be accepted.","major_comments":[{"comment":"The neglect of the σ²/(2g) term in the Hubbard-Stratonovich effective action is load-bearing and is not demonstrated. At the Gross-Neveu fixed point g* = 2πϵ/N (Eq. (15)), the contribution of this term at the constant saddle σ=i on S² is (1/(2g*))∫_{S²} σ² = (4π/(2g*))·(−1) = −N/ϵ, the same order as the determinant contribution in Eq. (64). Moreover, the stationarity condition for constant σ with the term retained reads N ∂F_f/∂σ + (Vol/g) σ = 0; since ∂F_f/∂σ vanishes at σ=ik (Eqs. (59)–(62)), for any finite g the only solution is σ=0. Thus σ=i is not a critical point of the action (51), and the exact matching in Eq. (64) is an artifact of dropping a leading-order term. The citation to [3] is not a demonstration in this paper.","section":"Section 5, after Eq. (51)"},{"comment":"The d=4 identification uses the same truncation and also conflates the Gross-Neveu and Gross-Neveu-Yukawa effective actions. For the GNY model (17), the effective action at constant σ must also contain the scalar kinetic and g2 σ⁴/4! terms; these are subleading in N for σ=O(1), so the leading N/ϵ matching might survive, but this has to be shown rather than assumed. As written, the equality S_{σ=2i}−S_{σ=0}=2N/(3ϵ) in Eq. (69) is presented as a critical point of the determinant-only action, which is not the full effective action of the model whose instantons were computed in Section 3.","section":"Section 5, Eqs. (68)–(70)"},{"comment":"The statement that the derivative in Eq. (59) 'has zeros at σ=ik' needs qualification. For exactly d=2 the derivative reduces to πσ (Eq. (62)), whose only zero is σ=0, and for exactly d=4 it reduces to π(σ+σ³)/6 (Eq. (66)), whose zeros are σ=0, ±i. The zeros at all integer k are a property of the ϵ-regulated theory with ϵ>0, where the Gamma functions are finite at σ=ik and the zero comes from sinhπσ. Since the fixed-point results are obtained in d=2+ϵ and d=4−ϵ this is acceptable for the leading pole calculation, but the text should state the limiting prescription explicitly, because the use of the d=2 and d=4 simplified formulas to evaluate the action at σ=i and σ=2i is otherwise confusing.","section":"Section 5, Eq. (59)"}],"minor_comments":[{"comment":"An 'infrared fixed point' is mentioned, whereas Section 2 identifies g* in Eq. (15) as a UV fixed point of the Gross-Neveu model in d=2+ϵ; please reconcile the terminology.","section":"Section 5, text after Eq. (51)"},{"comment":"The symbol σ denotes both the scalar field and the Dirac matrices in the spinor ansatz; this overloaded notation should be clarified.","section":"Section 3, Eq. (27)"},{"comment":"The construction of the eigenfunction solutions assumes Ñ ≥ d_n (Eq. (41)); this condition should be stated explicitly.","section":"Section 4.1"},{"comment":"The abstract's phrase 'close to the fixed points' should be made precise in the body regarding whether the Hubbard-Stratonovich matching is claimed at leading order in ϵ or exactly.","section":"Section 1 and Abstract"}],"recommendation":"major_revision","confidential_remarks":"The reader's report and stress-test note correctly identify the σ²/(2g) issue. In my reading, the instanton solutions in Sections 2–3 are not affected by that issue, but the advertised exact identification with the Hubbard-Stratonovich critical points is not established as stated. I recommend asking the authors to either justify the truncation by an explicit large-N or contour-deformation argument, or to reframe the claims as a leading-order comparison of the fermion determinant with the instanton action. The zero-set subtlety in Eq. (59) should also be clarified before the paper can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Thanks for the report. The paper has two parts: explicit fermionic instanton solutions for GN and GNY, and an attempt to identify them with critical points of the large-N Hubbard-Stratonovich effective action. The first part is solid and genuinely new; the second, as your stress-test note correctly says, has a load-bearing flaw. The drop of the σ²/(2g) term in (51) is not a minor neglect at the fixed point: at g*=2πϵ/N this term contributes -N/ε at σ=i, the same order as the determinant contribution, and the full stationarity condition reduces to (Vol/g)σ=0 when ∂F_f/∂σ vanishes, so σ=ik are not saddles of the full action. The claimed exact matching in (64), (68), (69) is therefore not established.\n\nThe instanton part holds up. The ansatz in Sections 2 and 3 satisfies the equations of motion, the on-shell actions -2π/g and (32π²/3)(g₂/(6g₁⁴)-1/g₁²) are computed correctly, and the fixed-point values -N/ε, -N/(12ε), 2N/(3ε) are new. The reflection-Hermiticity discussion is thoughtful, and the Dirac eigenfunction construction in 4.1 is a nice extension. The moduli counting and the C, x₀ independence of the action are also clean.\n\nSoft spots beyond the HS issue: the 4D spinor ansatz in (27) is under-specified; the numerator structure is not defined for the four-dimensional Clifford algebra, and the solution is not derived in detail. The paper also gives no demonstration of why the σ² term is irrelevant, and citing [3] without checking the parametric balance is not enough. If the HS connection is dropped or fixed, the paper becomes a shorter but still useful note on fermionic instantons in these models.\n\nWho is this for? Subfield practitioners: large-N vector models, instanton calculus, GN/GNY relation. The instanton part deserves referee time; the HS identification should not be accepted as is. My recommendation: send to peer review, ask the authors to either repair Section 5 with a correct treatment of the σ² term or remove the identification claim, and clean up the spinor notation. That is a workable path.","headline":"New fermionic instanton solutions are worth a look, but the Hubbard-Stratonovich identification in Section 5 is undone by a leading-order dropped term.","tokens_in":9022,"tokens_out":4236,"would_cite":false,"duration_ms":43969,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that Gross-Neveu and Gross-Neveu-Yukawa models admit smooth fermionic instanton solutions whose on-shell actions exactly match the constant saddle points of the large-N Hubbard-Stratonovich effective action.","keywords":["Gross-Neveu model","Gross-Neveu-Yukawa model","fermionic instantons","Hubbard-Stratonovich method","large N limit","fixed points","reflection Hermiticity","conformal mapping to sphere"],"falsifier":"Evaluate the full Hubbard-Stratonovich effective action on $S^d$ keeping the term $\\sigma^2/(2g)$, at the constant saddles $\\sigma=i$ in $d=2+\\epsilon$ with $g_* = 2\\pi\\epsilon/N$ and $\\sigma=2i$ in $d=4-\\epsilon$ with the fixed-point couplings of (22); if either quadratic term contributes at order $N/\\epsilon$ rather than being suppressed, the claimed exact action equalities fail.","tokens_in":7857,"feed_emoji":"⚛️","tokens_out":12508,"duration_ms":110213,"temperature":0.7,"pith_summary":"This paper tries to establish that the Gross-Neuveu model in two dimensions and the Gross-Neveu-Yukawa model in four dimensions have genuine smooth fermionic instanton solutions with finite, coupling-dependent actions. It claims the on-shell actions computed directly from the instanton ansatz, at the renormalization-group fixed points, take the values $-N/\\epsilon$ and $2N/(3\\epsilon)$. The paper then maps the instantons onto spheres and shows that at a particular choice of moduli they become constant, with the same action values appearing as critical points of the large-$N$ Hubbard-Stratonovich effective action. If this identification is right, nonperturbative instanton effects in these models can be studied in the simpler large-$N$ saddle-point language.","feed_headline":"Fermionic instanton actions match large-N saddles","feed_subtitle":"Gross-Neveu and Gross-Neveu-Yukawa instantons reproduce fixed-point actions exactly.","key_machinery":"The argument is carried by three pieces. The first is the instanton ansatz itself, a rational spinor profile $(C+i\\sigma\\cdot(x-x_0))/(C^2+(x-x_0)^2)$ whose squared norm is proportional to the same denominator, so that the quartic interaction collapses to a known integral and the equations of motion reduce to one algebraic constraint on the spinor moduli. The second is the stereographic mapping to $S^d$: because the equations of motion are conformally invariant, the flat-space solutions remain solutions on the sphere, and for the moduli choice $C=1$, $x_0=0$ they become constant fields. The third is the large-$N$ Hubbard-Stratonovich effective action $F_f(\\sigma)=-\\frac{1}{\\mathrm{tr}\\,1}\\log\\det(\\nabla\\!\\!\\! /+\\sigma)$; using a Gamma-function summation identity, its derivative is proportional to $\\Gamma(d/2+i\\sigma)\\Gamma(d/2-i\\sigma)\\sinh(\\pi\\sigma)$, a function whose zeros at $\\sigma=ik$ produce the polynomial actions matching the instanton actions. Reflection Hermiticity is what justifies treating $\\psi$ and $\\bar\\psi$ as independent fields in the Euclidean construction.","core_discovery":"The paper claims a precise two-way correspondence. On the one hand, the Gross-Neveu model (with the quartic fermion interaction) and the Gross-Neveu-Yukawa model (with an added scalar) admit smooth spinor field configurations, built from the two-dimensional scalar function $(C \\pm i\\sigma\\cdot(x-x_0))/(C^2+(x-x_0)^2)$ raised to the appropriate power, whose on-shell actions are $-2\\pi/g$ and $\\frac{32\\pi^2}{3}\\left(\\frac{g_2}{6g_1^4}-\\frac{1}{g_1^2}\\right)$. At the fixed points these become $-N/\\epsilon$ and $2N/(3\\epsilon)$. On the other hand, when these configurations are stereographically mapped to the sphere and the moduli are set to $C=1$, $x_0=0$, they become constant fields, and exactly the same action values appear as the $\\sigma=i$ and $\\sigma=2i$ critical points of the large-$N$ Hubbard-Stratonovich effective action. Establishing this identification between the instanton calculus and the large-$N$ effective action is the paper's main claim.","pith_inferences":["A direct evaluation of the dropped $\\sigma^2/(2g)$ term at the constant saddles would test whether the numerical match survives beyond leading order; the paper's cited justification for dropping it is not demonstrated here.","The discrete ladder $\\sigma=ik$ suggests an infinite series of saddles, and whether all of them contribute to the path integral will depend on the steepest-descent contour, a question the paper does not settle.","The same strategy of mapping instantons to constant saddles may transfer to other fermionic vector models in fractional dimensions, offering a generic way to compute instanton actions at Wilson-Fisher fixed points.","If the identification is exact rather than approximate, it gives a practical generator of nonperturbative data: evaluate the Hubbard-Stratonovich free energy at imaginary quantized $\\sigma$ to read off instanton actions in any dimension $2\\le d\\le 4$."],"forward_implications":["A semiclassical expansion around the fermionic instantons becomes a well-defined starting point for computing nonperturbative corrections to anomalous dimensions and correlation functions in both models.","Because the on-shell action is independent of the moduli $C$ and $x_0$, each instanton carries exact zero modes, so collective-coordinate quantization will be needed and is expected to give the leading exponential nonperturbative factor.","The same action values are recovered from the Hubbard-Stratonovich saddles, so instanton effects in the microscopic fermion theory and in the large-$N$ effective theory describe the same nonperturbative sector.","The tower of constant solutions labeled by integers, through $\\sigma=ik$ with $k=n+1$ in $d=2$ and $k=n+2$ in $d=4$, predicts a discrete family of saddle actions whose lowest member reproduces the flat-space instanton."],"supporting_citations":[{"why":"Supplies the Hubbard-Stratonovich large-$N$ instanton method and the cited justification for dropping the $\\sigma^2/(2g)$ term; the paper's central comparison extends this approach to fermionic models.","marker":"[3]"},{"why":"Defines the Gross-Neveu model whose fermionic instantons are the first central object of the paper.","marker":"[4]"},{"why":"Provides the renormalization-group fixed-point couplings $g_* = 2\\pi\\epsilon/N$ and $g_1^{*2}, g_2^*$ used to convert the instanton actions into fixed-point values.","marker":"[6]"},{"why":"Supplies the reflection-Hermiticity convention that lets $\\psi$ and $\\bar\\psi$ be treated as independent fields in Euclidean space.","marker":"[7]"},{"why":"Gives the bosonic instanton of the Gross-Neveu-Yukawa model and the dimensional-regularization approach used to derive and evaluate the scalar solution.","marker":"[8]"},{"why":"Provides the conformally coupled scalar instanton on which the bosonic solution and its sphere mapping rely.","marker":"[9]"}],"fun_headline_variants":["Fermionic instantons match large-N saddles in GN and GNY","Instantons equal large-N actions at fixed points","GN/GNY instantons mirror Hubbard-Stratonovich saddles","Sphere mapping reveals exact instanton–large-N match"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the quadratic term $\\sigma^2/(2g)$ in the Hubbard-Stratonovich effective action may be dropped at the fixed point; the paper cites prior work for this rather than proving it here, and at the constant saddle $\\sigma=i$ on $S^2$ this term contributes at order $N/\\epsilon$, the same order as the fermion determinant, so if it is not truly irrelevant the exact matching fails.","fun_headline_variants_meta":{"raw":{"variants":["Fermionic instantons match large-N saddles in GN and GNY","Instantons equal large-N actions at fixed points","GN/GNY instantons mirror Hubbard-Stratonovich saddles","Sphere mapping reveals exact instanton–large-N match"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000216,"raw_usage":{"total_tokens":1383,"prompt_tokens":847,"completion_tokens":536,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":463,"completion_tokens_details":{"reasoning_tokens":466}},"tokens_in":463,"tokens_out":536,"duration_ms":6117,"temperature":1.0,"reasoning_tokens":466,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:27:16.133193+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the full Hubbard-Stratonovich effective action on $S^d$ keeping the term $\\sigma^2/(2g)$, at the constant saddles $\\sigma=i$ in $d=2+\\epsilon$ with $g_* = 2\\pi\\epsilon/N$ and $\\sigma=2i$ in $d=4-\\epsilon$ with the fixed-point couplings of (22); if either quadratic term contributes at order $N/\\epsilon$ rather than being suppressed, the claimed exact action equalities fail.","supporting_citations":[{"cited_title":"Gross and A","cited_arxiv_id":null,"evidence_quote":"Defines the Gross-Neveu model whose fermionic instantons are the first central object of the paper."},{"cited_title":"Zinn-Justin,Quantum field theory and critical phenomena, Fifth edition, Int","cited_arxiv_id":null,"evidence_quote":"Supplies the reflection-Hermiticity convention that lets $\\psi$ and $\\bar\\psi$ be treated as independent fields in Euclidean space."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the bosonic instanton of the Gross-Neveu-Yukawa model and the dimensional-regularization approach used to derive and evaluate the scalar solution."}],"review_version":1}