{"id":"caf13304-31ce-43dd-a09b-56d2b8e198fa","arxiv_id":"2608.12465","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A symmetry-preserving neural-network variational state finds d-wave superconducting order in the 2D t-t' Hubbard model at 1/8 doping, with an extrapolated thermodynamic-limit order parameter of 0.022(1).","lead":"Researchers built a new symmetry-preserving neural-network wave function and computed the ground state of the 2D Hubbard model on lattices up to 24 by 24. They report robust d-wave superconducting order at 1/8 doping, supporting a contested result with an independent variational method.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Extrapolated d-wave order may reflect the ansatz's explicit Euclidean-distance decay and d-wave seed rather than the true ground state.","rationale":"The reader's weakest assumption matches the point I would defend as most load-bearing. The paper's abstract and conclusion make a thermodynamic-limit statement ('robust evidence', 'resolving a long-standing question') from four finite-cluster variational energies and pairing correlations. For that statement to be true, the finite-size Δ_SC plateau must extrapolate to a nonzero value because the true ground state has d-wave off-diagonal long-range order, not because the ansatz architecture and initialization pressure the calculation in that direction. The explicit exponential decay in Eq. (4) and the d-wave seed in Methods D are the clearest such pressures; they are trainable or removable, but the manuscript does not report a systematic study of their effect on the long-distance correlation function or on the extrapolated order parameter. The half-filling benchmark is real independent support (no SC is found), and the agreement with the AFQMC estimate of Ref. [6] is encouraging, but the AFQMC comparison is not a substitute for a controlled variational-bias check. Because no code or data are shipped, the extrapolation cannot be independently reproduced today. I therefore do not see a reason to reject or reclassify; the concern is a testable caveat, and the honest verdict remains conditional on the proposed check. If the check confirms stability, the central claim is much stronger; if not, the 'robust evidence' framing would need to be withdrawn.","tokens_in":14366,"tokens_out":10623,"duration_ms":113009,"concrete_test":"Re-optimize the SBP wave function for L = 16, 20, and 24 at t'/t = -0.2, U/t = 8, δ = 1/8 under two variants: (a) fix λ = 0 in Eq. (4) to remove the Euclidean-distance decay, and (b) omit the d-wave seed f_d described in Methods D. Evaluate Δ_SC from Cp(r) using the same Eq. (8) and extrapolate to 1/L → 0 for each variant. If the extrapolated value shifts by more than the quoted error bar of 0.001, or if the long-distance plateau disappears, the claimed robustness is not established. As a secondary check, optimize a translation-invariant backflow Slater determinant with no pairing term and measure Cp(r); a spurious plateau in that control would indicate the definition of Δ_SC itself is sensitive to the variational manifold.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim — a nonzero thermodynamic-limit d-wave order parameter Δ_SC = 0.022(1) — rests on the long-distance plateau of the pairing correlation Cp(r) being an unbiased property of the ground state. The SBP ansatz is not bias-free here: Eq. (4) multiplies every pairing-matrix element by a normalized exponential e^{-λ d(i,j)}, so the variational manifold is explicitly biased toward short-distance pairing amplitudes, and Methods D adds a configuration-independent d-wave seed f_d to accelerate convergence. The paper states that the seed only affects learning speed and that final results are independent of it, but no supporting comparison is shown. The finite-size order parameter in Eq. (8) averages Cp(r) only over the saturation shell |r| ≥ d_max/2, and the extrapolation uses just four square clusters (L = 12, 16, 20, 24). If the true ground state has pairing correlations with a longer tail or modulated by the stripe envelope visible in Fig. 5, the exponential cutoff plus the d-wave seed could produce a finite plateau at these system sizes even when the unbiased optimum has weaker or no superconducting order. The half-filling QMC benchmark (Fig. 6) is genuine supporting evidence that the ansatz does not blindly produce d-wave order, but at δ = 1/8 no such sign-problem-free control exists, so the robustness of the thermodynamic extrapolation is the single load-bearing assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces the Symmetry-Preserving Backflow Pairing (SBP) ansatz, a neural-network variational wave function for the square-lattice t-t' Hubbard model that enforces translational symmetry by construction. The authors show that an unconstrained backflow-pairing ansatz falls into multiple stripe-ordered local minima, while SBP converges reproducibly and reaches lower variational energies on L=12,...,24 at t'/t=-0.2, U/t=8, δ=1/8. Using the long-distance plateau of the d-wave pairing correlation function, they extract a finite order parameter Δ_SC and extrapolate it to Δ_SC=0.022(1), consistent with a constrained-path AFQMC estimate. A half-filling QMC benchmark shows that the same ansatz produces no spurious d-wave order in the undoped case.","tokens_in":14651,"tokens_out":5779,"duration_ms":54587,"significance":"The result, if correct, would settle a longstanding controversy for this parameter point and support d-wave superconductivity coexisting with stripe correlations in the 2D Hubbard model. The methodological contribution is substantial: a symmetry-preserving, scalable fermionic NQS that reaches 24x24 with 504 electrons, with transfer learning across sizes, and a clean half-filling control demonstrating that the ansatz can yield vanishing pairing correlations. The agreement with AFQMC for Δ_SC provides genuine external support. The main weakness is the underdocumented thermodynamic-limit extrapolation and the untested seed-independence claim; these are local enough to be fixable but directly bear on the headline number.","major_comments":[{"comment":"The central claim Δ_SC=0.022(1) is an extrapolation from four system sizes, but the paper does not state the fit function, the uncertainties on the per-size Δ_SC values, the goodness of fit, or the error bar on the intercept. Please report the fitting procedure, show the stability of the intercept under excluding L=12 or including a quadratic correction in 1/L, and state how the fit uncertainty is propagated. Without this, the 'robust evidence' claim is stronger than the displayed data support.","section":"Inset of Fig. 4 and Eq. (8)"},{"comment":"The text asserts that the d-wave seed f_d only accelerates learning and that final results are independent of it, but no comparison is shown. Because the measured observable is the d-wave pairing correlation in the same symmetry channel as the seed, this is load-bearing. Please provide a direct comparison of Cp(r), Δ_SC, and variational energy with and without seed (or with different Δ_d values), and report the optimized value of λ and the sensitivity of the plateau to λ. The half-filling benchmark is reassuring but does not control the doped case.","section":"Methods D and Eq. (4)"},{"comment":"The order parameter uses only one of the four spin-resolved singlet correlators and one disconnected contraction, with the justification that the omitted terms coincide asymptotically. The finite-size Δ_SC, however, averages over distances that are not necessarily asymptotic (|r|≥d_max/2). Please quantify the finite-distance error, for example by computing the full expression (10) on 12×12, and by showing that Cp(r) is flat within the averaging shell rather than still decaying. This is a finite-size systematic that should be controlled before the extrapolation is trusted.","section":"Eq. (11) and Fig. 4"}],"minor_comments":[{"comment":"There is a typo '1/8-doped model Hubbard model'; the duplicated word 'model' should be removed.","section":"Section I"},{"comment":"The 'effectively convex' landscape claim rests on only two random seeds for SBP and BP; reporting a few more independent initializations would considerably strengthen the claim.","section":"Fig. 2 and Section III"},{"comment":"For the 20×20 and 24×24 lattices, no previous variational energies are listed; the 'state-of-the-art' claim for those sizes should be qualified or contextualized with the closest available results in the literature.","section":"Table I"},{"comment":"The inset would be much easier to evaluate if the extrapolation curve and error bars on each Δ_SC point were drawn; currently only markers are shown.","section":"Inset of Fig. 4"},{"comment":"The quantity d_max is not defined precisely; on a periodic square lattice the minimum-image convention for Euclidean distances should be stated explicitly.","section":"Eq. (8)"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely correct in its main conclusion, and the half-filling benchmark plus AFQMC agreement provide genuine support. However, the headline number rests on an underdocumented four-point extrapolation and an unsupported seed-independence claim. These are fixable within revision, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear X,\n\nThe short version: this paper deserves a serious referee. The SBP ansatz is a real methodological step, and the central d-wave superconductivity claim is plausible—but the \"robust evidence\" framing is ahead of what the extrapolation actually shows.\n\nWhat's genuinely new: enforcing translational symmetry in the pairing matrix by construction, with a transformer backflow and a trainable distance-decay kernel. That removes the broken-symmetry stripe minima that have plagued other NQS optimizations, and it lets the authors reach 24×24 with 504 electrons, beating previous variational energies. The half-filling QMC benchmark is the right control: the same ansatz produces vanishing d-wave pairing there, so it does not blindly output order. The quantitative agreement with the independent AFQMC estimate for Δ_SC is also real support.\n\nThe soft spots are in the extrapolation and documentation. The order parameter is extracted from four cluster sizes (12, 16, 20, 24) with no stated fit function, no alternative scaling checks, and no error estimate on the infinite-size limit beyond the point scatter. The paper asserts the d-wave seed only accelerates learning and does not affect final results, but shows no comparison. And the explicit exponential decay in Eq. (4) does bias the variational manifold toward short-distance pairing amplitudes; the half-filling control mitigates this, but it does not eliminate the worry that the finite-size plateau in C_p(r) could be inflated by the ansatz's locality bias and the seeded channel. None of these is a fatal flaw, but together they mean the thermodynamic-limit conclusion is less \"robust\" than the abstract claims. I'd also flag that no code or data is shipped, so the 24×24 energies cannot be independently checked.\n\nOn the citation pattern: the heavy reliance on Ref. [21] (their companion paper) is not a red flag in itself, but readers outside the group will want the present paper to stand a bit more on its own.\n\nBottom line: a genuinely useful method paper with a plausible physics claim, currently over-stated at the margins. A good referee can push for the missing fit details and seed-independence data. I would send it to review, and I'd probably cite it for the energy benchmarks and the symmetry-preserving construction.\n\nBest","headline":"A genuinely new symmetry-preserving neural-network ansatz with a plausible d-wave superconductivity claim, but the thermodynamic-limit evidence is thinner than the abstract suggests.","tokens_in":15158,"tokens_out":1929,"would_cite":true,"duration_ms":16407,"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":"A neural-network wave function that preserves translational symmetry finds d-wave superconducting order in the 1/8-doped t-t' Hubbard model, with a thermodynamic-limit order parameter of 0.022(1).","keywords":["Hubbard model","d-wave superconductivity","neural quantum states","backflow","translational symmetry","stripe order","variational Monte Carlo","strongly correlated electrons"],"falsifier":"Compute $C_p(|r|)$ for the same parameters on larger periodic clusters or with a pairing kernel that is not forced to decay with Euclidean distance; if the saturated plateau moves toward zero with system size instead of converging to about $\\Delta_\\mathrm{SC}\\approx 0.022$, the claimed thermodynamic-limit order is a variational artifact.","tokens_in":14180,"feed_emoji":"⚛️","tokens_out":14748,"duration_ms":116702,"temperature":0.7,"pith_summary":"The paper tries to settle a long-standing question: at 1/8 hole doping, with next-nearest-neighbor hopping $t'/t=-0.2$ and on-site repulsion $U/t=8.0$, does the two-dimensional Hubbard model harbor superconductivity? Its answer is yes. Using a neural-network wave function that keeps translational symmetry by construction, the authors reach lower variational energies than competing pure-stripe states on square lattices up to $24\\times24$ (504 electrons), and their $d$-wave pairing correlation function saturates at long distances. Extrapolating the resulting order parameter to the thermodynamic limit gives $\\Delta_\\mathrm{SC}=0.022(1)$, consistent with an earlier constrained-path auxiliary-field quantum Monte Carlo estimate and with the picture of superconductivity coexisting with stripe correlations. The methodological point is that broken-symmetry variational states fall into stripe minima that suppress pairing, so a symmetry-preserving ansatz is needed to expose the superconducting ground state.","feed_headline":"D-wave superconductivity wins in the doped Hubbard model","feed_subtitle":"A translation-symmetric wave function beats stripe states and extrapolates to a finite pairing order.","key_machinery":"The load-bearing object is the Symmetry-Preserving Backflow Pairing (SBP) ansatz: a neural-network pairing matrix $f_{ij}(n)$ whose weights depend only on the relative displacement $i-j$ and on configuration-dependent backflow vectors $y^\\sigma_i(n)$ produced by a translationally equivariant transformer. The matrix element is multiplied by a normalized exponential distance factor $e^{-\\lambda d(i,j)}/\\sum_{i'j'}e^{-\\lambda d(i',j')}$, suppressing pairing between distant sites. Because only up-down pairs are kept, the wave function is a determinant, $\\Psi_\\theta(n)=\\det[n_\\uparrow \\star f(n)\\star n_\\downarrow]$, rather than a Pfaffian. Enforcing translational symmetry in the parametrization removes pure-stripe broken-symmetry states from the variational manifold, making the optimization landscape effectively convex so different random initializations converge to the same physical state. A configuration-independent $d_{x^2-y^2}$ nearest-neighbor seed accelerates the optimization but is checked not to alter the final result.","core_discovery":"The central claim is that the ground state of the $t$-$t'$ Hubbard model at $t'/t=-0.2$, $U/t=8.0$, and doping $\\delta=1/8$ has genuine $d_{x^2-y^2}$ superconducting order in the thermodynamic limit. The evidence is the long-distance behavior of the $d$-wave pairing correlation function $C_p(|r|)$ on periodic clusters of linear size $L=12,16,20,24$: after a fast short-distance decay, $C_p(|r|)$ saturates to a finite plateau. Averaging the plateau over $|r|\\ge d_\\mathrm{max}/2$ defines the order parameter $\\Delta_\\mathrm{SC}$, and its extrapolation $1/L\\to0$ gives $\\Delta_\\mathrm{SC}=0.022(1)$, matching a constrained-path auxiliary-field quantum Monte Carlo value obtained on cylinders with a pinning field. No pinning field is needed in the present calculation, and the order parameter is read directly from correlation functions on periodic square clusters. In the same state the spin correlations show an antiferromagnetic pattern modulated by a longer-wavelength stripe envelope, so superconductivity and stripe correlations coexist at the level of fluctuations while the one-body density remains uniform.","pith_inferences":["Our inference, not the paper's claim: a decisive test would be the same SBP extrapolation at $t'=0$; if the order parameter vanished there, it would confirm that negative $t'$ is what stabilizes pairing in this regime.","Our inference: the built-in Euclidean-distance decay of the pairing kernel is the main hidden modeling choice; repeating the calculation with algebraic or learnable long-range kernels would show whether the saturated $C_p(|r|)$ is robust or an artifact of the kernel.","Our inference: the 'effectively convex' landscape suggests a general design principle for competing orders: imposing the Hamiltonian's exact symmetries on the variational form may be more reliable than optimizing a broken-symmetry state and projecting afterward.","Our inference: the same symmetry-preserving construction should transfer to multi-orbital or frustrated fermion models, where stripe-type local minima are also expected; the paper mentions this as future work but does not test it."],"forward_implications":["The contradiction between DMRG studies that find no superconductivity for $t'<0$ and neural-network studies that do is resolved, for this parameter point, in favor of $d$-wave superconductivity in the thermodynamic limit.","Stripe order and superconductivity are not mutually exclusive: the SBP ground state has stripe-modulated antiferromagnetic spin correlations, yet its one-body density is translationally uniform and it carries a finite $d$-wave order parameter.","Variational approaches that first break translational symmetry and later restore it may systematically underestimate pairing because they get trapped in pure-stripe local minima; enforcing the symmetry in the ansatz itself bypasses this bias.","Because the optimized SBP wave function is translationally invariant, it can be transferred from one lattice size to the next, which is what makes the 24x24, 504-electron calculation and the four-point $1/L$ extrapolation practical."],"supporting_citations":[{"why":"Supplies the AFQMC reference $d$-wave order parameter on cylinders with twisted boundary conditions and a pinning field; the paper's extrapolated value agrees with it and with the prior coexistence claim.","marker":"[6]"},{"why":"Reports a Pfaffian-based neural wave function that finds superconductivity for $t'<0$; this is the direct neural-network precursor the SBP ansatz corroborates.","marker":"[7]"},{"why":"Provides variational energies of pure stripe states from a backflow neural ansatz, the competing broken-symmetry solutions that the SBP energies are measured against.","marker":"[13]"},{"why":"Reports DMRG results finding no superconductivity for $t'<0$; this is the conflicting conclusion the paper seeks to resolve.","marker":"[17]"},{"why":"Documents the variational-bias mechanism by which broken-symmetry stripe solutions display weak or no superconductivity, motivating the symmetry-preserving construction.","marker":"[21]"},{"why":"Supplies numerically exact auxiliary-field QMC spin correlations at half filling, used to benchmark the SBP ansatz's magnetic correlations.","marker":"[10]"},{"why":"Defines the $d$-wave order parameter as the average of the pairing correlation function over the saturation region, used for the thermodynamic-limit extrapolation.","marker":"[31]"},{"why":"Establishes the absence of superconductivity in the pure Hubbard model at $t'=0$, providing the contrast that highlights the role of negative $t'$.","marker":"[53]"}],"fun_headline_variants":["Symmetry-preserving neural net finds d-wave order in Hubbard model","Translation-symmetric wavefunction beats stripes, shows d-wave SC","Neural-network ansatz: robust d-wave pairing in doped Hubbard model","No pinning fields needed: neural net evidences Hubbard d-wave SC","State-of-the-art neural wavefunction supports Hubbard d-wave superconductivity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"That the finite-size extrapolation from four periodic clusters, together with the variational bias from the built-in distance decay and the seeded $d$-wave channel, does not itself manufacture the long-range pairing order it reports.","fun_headline_variants_meta":{"raw":{"variants":["Symmetry-preserving neural net finds d-wave order in Hubbard model","Translation-symmetric wavefunction beats stripes, shows d-wave SC","Neural-network ansatz: robust d-wave pairing in doped Hubbard model","No pinning fields needed: neural net evidences Hubbard d-wave SC","State-of-the-art neural wavefunction supports Hubbard d-wave superconductivity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000569,"raw_usage":{"total_tokens":2755,"prompt_tokens":1069,"completion_tokens":1686,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":685,"completion_tokens_details":{"reasoning_tokens":1595}},"tokens_in":685,"tokens_out":1686,"duration_ms":13257,"temperature":1.0,"reasoning_tokens":1595,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:08:01.585572+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $C_p(|r|)$ for the same parameters on larger periodic clusters or with a pairing kernel that is not forced to decay with Euclidean distance; if the saturated plateau moves toward zero with system size instead of converging to about $\\Delta_\\mathrm{SC}\\approx 0.022$, the claimed thermodynamic-limit order is a variational artifact.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies numerically exact auxiliary-field QMC spin correlations at half filling, used to benchmark the SBP ansatz's magnetic correlations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the $d$-wave order parameter as the average of the pairing correlation function over the saturation region, used for the thermodynamic-limit extrapolation."}],"review_version":1}