{"id":"280ace56-543e-4348-98f4-c2e9510c0067","arxiv_id":"2608.03894","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"FFT-accelerated determinant quantum Monte Carlo reaches larger kagome lattices, yielding superfluid critical exponents at Dirac filling and evidence that the proposed triangle-rule CDW is a finite-size effect.","lead":"Researchers used fast Fourier transforms to speed up quantum Monte Carlo simulations on the kagome lattice, reaching system sizes twice as large as earlier studies. The bigger simulations reveal a superfluid quantum transition at a specific filling and suggest a previously proposed charge pattern disappears in the thermodynamic limit.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Projection length β=30 is likely underconverged for the largest lattices (L=18–24), making the reported superfluid critical exponents potentially size-biased; a β-dependence test is required.","rationale":"The paper's algorithmic contribution (FFT for composite lattices combined with delayed update) is internally consistent and the performance claims scale as expected; the physics claims, however, depend on ground-state convergence. The reader's weakest assumption — that β=30 is converged for all sizes — is the most load-bearing uncertainty. The absence of a reported Δτ and of any β-dependence test makes it impossible to rule out a size-dependent bias in the finite-size scaling of the superfluid transition. Our proposed test directly checks this by repeating a small number of L=24 simulations at larger β, which is feasible given the claimed speedup. If the test shows convergence, the paper's central conclusions stand; if not, the exponents and |U_c| must be revised. Thus the appropriate verdict remains CONDITIONAL (unchanged), pending this verification. We do not see a stronger internal inconsistency or a need to reject.","tokens_in":12243,"tokens_out":9290,"duration_ms":85990,"concrete_test":"Run the largest size L=24 at |U|=4.5, 4.79, and 5.2, with β = 30, 45, 60, and 80 (all with the same Δτ, e.g., 0.1 or 0.05, and enough sweeps that statistical error bars are smaller than the expected β-dependence). Compute the pairing structure factor SPair(Γ) and the correlation ratio RPair. If these observables vary beyond error bars between β=30 and β=80, or if the crossing between L=24 and smaller L shifts by more than the quoted 0.01 in |U| when β is increased, then β=30 is not converged and the FSS must be repeated with a proper β extrapolation. Also report the value of Δτ used, since both Trotter and projection errors depend on it.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the reliable extraction of the superfluid critical point at the Dirac filling (Sec. IV). All data are taken at a single projection length β=30, with Trotter step Δτ never specified. In ground-state projector DQMC, observables converge to ground-state values with corrections of order exp(−ΔE β). For a Dirac-like finite-size gap ΔE ~ v_F/L (v_F ~ t), L=24 gives ΔE β ~ 1.2, so excited-state contamination is ~30% and is much larger for L=24 than for L=12, where ΔE β ~ 2.5. This size-dependent projection error acts as an unwanted scaling correction in the finite-size scaling analysis, Eqs. (9)–(10), and can bias |U_c|, ν, ζ, η. No β-dependence is shown for any observable, and the FSS fits (Fig. 4) assume ground-state scaling forms. The claim that the exponents are 'reliable' therefore rests on an unverified assumption. If β=30 is not converged for the largest sizes, the quoted exponents could be systematically off.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript presents an FFT-based acceleration scheme for propagator multiplications in projector determinant quantum Monte Carlo (DQMC) on composite lattices, combining it with delay updates. The method is applied to the attractive Hubbard model on the kagome lattice at Dirac filling. The authors report a superfluid transition at |U_c|=4.79(1) with critical exponents ν=0.88(3), ζ=-0.68(2), η=0.55(7) from finite-size scaling on lattices up to L=24, and conclude that the previously proposed triangle-rule CDW order does not survive in the thermodynamic limit. They also characterize an effective computational scaling of N^2.49 for experimentally relevant system sizes.","tokens_in":12602,"tokens_out":4761,"duration_ms":47910,"significance":"The algorithmic claim is well motivated and, if correct, represents a practical advance: the block-diagonalization derivation in Eqs. (2)-(6) is clean and the reduction of propagator multiplication from O(N^3) to O(N^2 log N) is a genuine mathematical identity. Combining this with delay updates is a timely contribution. The physics claims—reliable critical exponents at the Dirac filling and the absence of triangle-rule CDW—are significant for the kagome Hubbard model, but the quantitative conclusions rest on a single, unverifiable projection length (β=30) and an unreported Trotter step. Without convergence evidence, the reported exponents and phase-diagram conclusions remain provisional. No code or data are released, which limits reproducibility but is not a flaw in the derivation itself.","major_comments":[{"comment":"All data used for the finite-size scaling are taken at projection length β=30, with no β-dependence shown for any observable. The FSS forms in Eqs. (9)-(10) assume ground-state scaling, but finite-β contamination is generally size-dependent: for L=24, a Dirac-like gap ΔE ~ v_F/L (with v_F ~ t) gives ΔE β ~ 1.2, so the largest lattice can have substantially larger excited-state contributions than L=12. Such size-dependent projection error would bias |U_c| and the exponents. A β-convergence study for representative L (e.g., L=12, 18, 24) and observables (R_Pair, P, D(Γ), D(K)) is required before the quoted error bars can be considered reliable.","section":"Sec. IV, Figs. 3-5"},{"comment":"The Trotter time step Δτ is never specified. Since Eq. (5) defines the discrete-time propagator and all reported results depend on this parameter, the manuscript should state the value used and demonstrate convergence (e.g., by comparing Δτ and Δτ/2 for a representative system). Without this, the possibility of a Trotter-discretization bias in the extracted critical parameters cannot be assessed.","section":"Eq. (5) and all simulations"},{"comment":"The χ2 minimization procedure is not fully documented. The manuscript reports the minimum of χ2 but does not give the reduced χ2 value, the number of data points, the fit ranges, or the sensitivity to the polynomial order. A fourth-order polynomial with five parameters can overfit, and the claim of a 'very high' collapse quality should be quantified. This is particularly important because the central conclusion of 'reliable' exponents depends on the FSS fits, not just on the algorithmic speedup.","section":"Sec. IV, Fig. 4 and Sec. V, Fig. 6"}],"minor_comments":[{"comment":"The text contains several typographical errors (e.g., 'attract ive' in the title, 'partical', 'hebavior', 'deffinition', 'acclerated'). A careful proofreading pass is needed.","section":"Throughout"},{"comment":"The range 12 ≤ L ≤ 27 is mentioned, but the actual values of L used in the runtime analysis are not listed. Please specify the data points and the number of cores used for each point.","section":"Fig. 2 caption"},{"comment":"The definition of δk is vague ('the smallest available neighboring wave vector'). Please specify it explicitly, e.g., in terms of the reciprocal lattice vectors, to reproduce the correlation ratio.","section":"Eq. (7)"},{"comment":"No information is given about the boundary conditions (presumably periodic), the number of independent samples used for error bars, or the autocorrelation times. These are standard details that should be included for reproducibility.","section":"Uncited material"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of cond-mat.str-el and the algorithmic part is solid. The main blocker is the missing convergence evidence for β and Δτ, which directly affects the validity of the reported physics results. The authors should be asked to supply these data or, if already available, to include them in the manuscript. The high rate of self-citations in the concluding outlook is not a problem by itself, but it does not substitute for external validation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nYou should know two things. The paper delivers a genuinely useful algorithmic step: FFT-accelerated propagator multiplication generalized to composite lattices and combined with delay updates. And it uses that to report a superfluid critical point at the Dirac filling on the kagome lattice, with exponents, and argues the earlier triangle-rule CDW is a finite-size effect. The algorithm part is solid; the physics part is plausible but under-verified.\n\nWhat's actually new: the block-diagonalization of the kinetic matrix for multi-sublattice lattices, spelled out for kagome, triangular, and honeycomb. The complexity reduction from O(N^3) to O(N^2 log N) for the propagator multiplication is real, and the runtime benchmark showing N^2.49 total scaling in the experimentally relevant size window is credible. The physical result—reaching L=24 at the Dirac filling—is a genuine step beyond Ref [18], which stalled at L=12. The FSS analysis is standard and the data collapse is reported as high quality. If the extrapolation of the triangle-rule CDW to zero holds, it resolves a concrete discrepancy with the earlier DQMC study.\n\nThe soft spots are real but not fatal to the method. The paper never reports the Trotter time step Δτ, and more importantly all production data are taken at a single projection length β=30 with no β-convergence test. The stress-test concern is legitimate: for Dirac spectra the finite-size single-particle gap scales as 1/L, so for L=24, ΔEβ ~ 1.2, meaning excited-state contamination is size-dependent and could bias the FSS and the quoted exponents. The crossing plot shows small drift, which is reassuring but not a quantitative check. The \"reliable\" characterization of |U_c|, ν, ζ, η is therefore premature. I would not call this a fatal flaw—the exponents could survive a β=50 check—but authors should be asked to show it, and to release at least the binned data and a statement of Δτ. Lack of code is a minor issue given the algorithm is simple, but raw data tables would help.\n\nThere is no circularity or internal contradiction. The self-citations are to the same SU(2N)/SU(3) framework and are relevant. The work is clearly written and honest about what it claims.\n\nWho should read this: anyone doing DQMC on frustrated or composite lattices, and anyone interested in the attractive Hubbard model on kagome. It deserves a serious referee, and I would send it out. I'd ask for a β-dependence test and the missing simulation parameters before accepting the physics claims; the algorithmic contribution can stand on its own.","headline":"A clean algorithmic advance for DQMC on composite lattices, with physics results that need a convergence check before the exponents are quoted.","tokens_in":12999,"tokens_out":3534,"would_cite":true,"duration_ms":33512,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A fast-Fourier-transform acceleration doubles the reach of determinant quantum Monte Carlo on kagome lattices, revealing a superfluid quantum critical point at |U_c|=4.79(1) with exponents ν=0.88(3), ζ=−0.68(2), η=0.55(7), and showing the p","keywords":["kagome lattice","attractive Hubbard model","determinant quantum Monte Carlo","FFT acceleration","superfluid quantum criticality","charge-density-wave order","finite-size scaling","Dirac filling"],"falsifier":"Measure the pairing correlation ratio R_Pair at β=40, 60, and 80 for the smallest (L=12) and largest (L=24) systems at |U| around 4.8; if the crossing point |U_c| shifts by more than the quoted statistical error, the β=30 convergence assumption is violated. Alternatively, compute D(Γ) at |U|=8 for L=18 and L=24 with β=60; if D(Γ) stops decreasing with L, the triangle-rule CDW is not a pure finite-size effect.","tokens_in":12196,"feed_emoji":"⚛️","tokens_out":11114,"duration_ms":83428,"temperature":0.7,"pith_summary":"This paper claims that a fast-Fourier-transform (FFT) acceleration of propagator multiplications, combined with the delay-update algorithm, makes determinant quantum Monte Carlo (DQMC) simulations of the kagome-lattice Hubbard model feasible on lattices twice as large as previously accessible (up to L=24). On these sizes, the attractive Hubbard model at Dirac filling ρ=2/3 exhibits a zero-temperature superfluid quantum critical point at |U_c|=4.79(1) with critical exponents ν=0.88(3), ζ=−0.68(2), and η=0.55(7), extracted from correlation-ratio crossings and data collapses. The paper also reports that the previously proposed triangle-rule charge-density-wave order at Γ extrapolates to zero in the thermodynamic limit and is likely a finite-size effect, and that the √3×√3 CDW at the K point also vanishes. If correct, the results resolve a longstanding ambiguity about competing orders in the attractive kagome Hubbard model and demonstrate a general acceleration strategy for DQMC on composite lattices.","feed_headline":"Doubled simulations pin superfluid point, rule out kagome CDW","feed_subtitle":"FFT acceleration doubles quantum Monte Carlo reach, pinning the superfluid critical point and dismissing a charge order.","key_machinery":"The FFT block-diagonalization of the kinetic matrix on composite lattices: for a lattice with n sublattices and L^d unit cells, the spinless kinetic matrix K is conjugated into a block-diagonal matrix Λ^{n×n} with L^d independent n×n blocks (Eq. 4). Propagator multiplication then becomes forward FFT on each sublattice component, block-diagonal multiply, and inverse FFT, lowering complexity from O(N^3) to O(N^2 log N) for the matrix-matrix product. The delay-update algorithm accelerates the auxiliary-field update part, and their combination shifts the practical bottleneck so that the total cost in the experimentally relevant size range (12≤L≤27) scales as N^2.49.","core_discovery":"The central discovery is that the kinetic-energy propagator e^{-Δτ K} on a composite lattice can be applied in O(N log N) per column by Fourier-transforming each sublattice component separately, multiplying by the block-diagonal momentum-space matrix Λ^{n×n}, and transforming back. Combined with delay updates for the auxiliary-field sweeps, this reduces the practical cost of projector DQMC and enables ground-state simulations of the kagome lattice with L=24. Using these larger sizes, the authors find a clean crossing of the pairing correlation ratio R_Pair at |U_c|=4.79(1) with critical exponents ν=0.88(3), ζ=−0.68(2), η=0.55(7). They further extrapolate the CDW order parameters D(Γ) and D(K","pith_inferences":["The same FFT acceleration should carry over to finite-temperature DQMC and to repulsive Hubbard models, since it relies only on the kinetic propagator structure; testing this would broaden the method's reach beyond the ground-state attractive case.","If the reported exponents are confirmed, the transition at the Dirac filling may belong to a known universality class; comparing ν≈0.88 and η≈0.55 with analytic or large-scale results for chiral XY or Gross-Neveu-Yukawa fixed points would be a natural next step.","The vanishing of D(K) in the attractive Hubbard model while a √3×√3 CDW appears in the Holstein model suggests that the phonon-mediated interaction in the Holstein model, rather than the pure Hubbard attraction, is what stabilizes that order; checking this by adding phonon degrees of freedom to the Hubbard model could isolate the mechanism.","The N^2.49 scaling is a crossover artifact: for lattices beyond L≈27 the delay update dominates and the cost returns to O(N^3), so the method's advantage is strongest in the experimentally relevant size regime."],"forward_implications":["DQMC on kagome lattices can now reach L=24, doubling the previous limit and enabling reliable finite-size scaling of correlation ratios at the Dirac filling.","The attractive Hubbard model at ρ=2/3 has a superfluid quantum critical point at |U_c|=4.79(1) with exponents ν=0.88(3), ζ=−0.68(2), η=0.55(7), giving a benchmark for future studies.","The triangle-rule CDW order proposed in earlier DQMC work does not survive in the thermodynamic limit; the √3×√3 CDW seen in the Holstein model is also absent in the attractive Hubbard model.","The FFT block-diagonalization applies to other composite lattices (triangular, honeycomb) and to models beyond the attractive Hubbard model, such as SU(2N) and three-component Hubbard models."],"supporting_citations":[{"why":"Supplies the previous DQMC study up to L=12 and the triangle-rule CDW proposal that this paper re-examines and refutes.","marker":"[18]"},{"why":"Provides the delay-update scheme for determinant quantum Monte Carlo that this work combines with FFT acceleration.","marker":"[23]"},{"why":"Extends delay-update to ground-state auxiliary-field QMC, the version used for the projector simulations here.","marker":"[25]"},{"why":"Gives the FFT method for propagator multiplications on square and cubic lattices that this paper generalizes to composite lattices.","marker":"[31]"},{"why":"Recent demonstration of FFT-based acceleration for the 3D Hubbard model, establishing the baseline the kagome extension builds on.","marker":"[32]"},{"why":"Establishes the finite-size scaling formalism for fermionic quantum criticality from quantum Monte Carlo, used here for R_Pair and P collapses.","marker":"[36]"},{"why":"Provides the symmetric pairing structure factor and scaling forms for attractive SU(3) Dirac fermions, adapted for the kagome analysis.","marker":"[40]"},{"why":"Supplies the automatic finite-size scaling analysis program used in the χ2 minimization for critical parameters.","marker":"[47]"},{"why":"Gives the specific χ2 minimization procedure applied to locate the optimal critical parameters and exponents.","marker":"[48]"},{"why":"Identifies the √3×√3 CDW in the kagome Holstein model, the competing order that this paper tests via D(K) and finds absent.","marker":"[49]"}],"fun_headline_variants":["FFT trick doubles QMC reach, pins superfluid, kills CDW","Accelerated QMC: superfluid point found, CDW ruled out","Kagome QMC: FFT speedup reveals superfluid, kills CDW","Double-size QMC: superfluid criticality, no CDW","FFT boost: larger QMC shows superfluid, no CDW"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The simulations assume that a projection length of β=30 is converged to the ground state for all system sizes and couplings, and the Trotter time step is never reported; if finite-β or finite-Δτ effects differ across sizes, the critical-point and CDW extrapolations could be biased.","fun_headline_variants_meta":{"raw":{"variants":["FFT trick doubles QMC reach, pins superfluid, kills CDW","Accelerated QMC: superfluid point found, CDW ruled out","Kagome QMC: FFT speedup reveals superfluid, kills CDW","Double-size QMC: superfluid criticality, no CDW","FFT boost: larger QMC shows superfluid, no CDW"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000163,"raw_usage":{"total_tokens":1113,"prompt_tokens":809,"completion_tokens":304,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":553,"completion_tokens_details":{"reasoning_tokens":205}},"tokens_in":553,"tokens_out":304,"duration_ms":3184,"temperature":1.0,"reasoning_tokens":205,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T10:13:04.649616+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the pairing correlation ratio R_Pair at β=40, 60, and 80 for the smallest (L=12) and largest (L=24) systems at |U| around 4.8; if the crossing point |U_c| shifts by more than the quoted statistical error, the β=30 convergence assumption is violated. Alternatively, compute D(Γ) at |U|=8 for L=18 and L=24 with β=60; if D(Γ) stops decreasing with L, the triangle-rule CDW is not a pure finite-size effect.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the previous DQMC study up to L=12 and the triangle-rule CDW proposal that this paper re-examines and refutes."},{"cited_title":"Sun and X","cited_arxiv_id":null,"evidence_quote":"Provides the delay-update scheme for determinant quantum Monte Carlo that this work combines with FFT acceleration."},{"cited_title":"Blankenbecler, D","cited_arxiv_id":null,"evidence_quote":"Extends delay-update to ground-state auxiliary-field QMC, the version used for the projector simulations here."},{"cited_title":"Du and Y .- Y","cited_arxiv_id":null,"evidence_quote":"Gives the FFT method for propagator multiplications on square and cubic lattices that this paper generalizes to composite lattices."},{"cited_title":"Chuang, Y .-S","cited_arxiv_id":null,"evidence_quote":"Recent demonstration of FFT-based acceleration for the 3D Hubbard model, establishing the baseline the kagome extension builds on."},{"cited_title":"Celledoni and A","cited_arxiv_id":null,"evidence_quote":"Establishes the finite-size scaling formalism for fermionic quantum criticality from quantum Monte Carlo, used here for R_Pair and P collapses."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the symmetric pairing structure factor and scaling forms for attractive SU(3) Dirac fermions, adapted for the kagome analysis."},{"cited_title":"Janke, Monte Carlo methods in classical statistical physics, in Computational Many-Particle Physics, edited by H","cited_arxiv_id":null,"evidence_quote":"Supplies the automatic finite-size scaling analysis program used in the χ2 minimization for critical parameters."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the specific χ2 minimization procedure applied to locate the optimal critical parameters and exponents."},{"cited_title":"Senthil, A","cited_arxiv_id":null,"evidence_quote":"Identifies the √3×√3 CDW in the kagome Holstein model, the competing order that this paper tests via D(K) and finds absent."}],"review_version":1}