{"id":"5afb74a8-1eb0-41bf-b8a2-5c1629657e41","arxiv_id":"2412.21072","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"In the adiabatic Hubbard-Holstein model, increasing Hubbard repulsion accelerates charge density wave domain growth, restoring Allen-Cahn t^(1/2) coarsening via screening of the Holstein coupling.","lead":"This paper uses a machine-learning force field trained on approximate quantum solutions to simulate how charge density wave domains grow in a correlated electron model. It finds that electron-electron repulsion speeds up domain coarsening by screening the electron-phonon coupling, a counterintuitive result for non-equilibrium physics.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central claim rests entirely on Gutzwiller-approximation forces with no exact many-body benchmark; the U-dependent coarsening enhancement could be an artifact of GA.","rationale":"The reader's conditional verdict is well calibrated. The paper's central claim, that Hubbard U enhances CDW coarsening and restores the t^(1/2) Allen-Cahn law for U/tnn >= 0.9, is based entirely on simulated dynamics driven by Gutzwiller-approximation forces. The ML component is validated only against GA (Fig. 1 and Fig. 2), so it cannot remove any bias inherent in GA. The key physical quantity that determines the coarsening law is the effective force on CDW domain walls; these are non-equilibrium, inhomogeneous configurations where a variational ground-state ansatz such as GA is most likely to be quantitatively inaccurate. A concrete test against DQMC forces on small systems, including domain-wall configurations, would either confirm or refute the GA-based result. If the exact many-body forces reproduce the same U-trend in L(t), the claim stands; if not, the central result is an artifact. This concern is the same load-bearing weakness identified by the reader, so no change to the conditional verdict is needed. Secondary concerns about the unspecified computation of the density-based order parameter phi_i in the ML trajectories and an unpublished baseline are also worth addressing, but they are less fundamental than the missing exact benchmark for GA.","tokens_in":17810,"tokens_out":8085,"duration_ms":90576,"concrete_test":"Use determinant quantum Monte Carlo (DQMC) at half-filling on an 8x8 lattice with frozen lattice configurations to compute the adiabatic force F_i = -dF/dQ_i via Hellmann-Feynman or energy differences for representative configurations: uniform staggered Q, a straight domain wall, and random quenched states. Compare against the GA force for U/tnn = 0.3, 0.6, 0.9, 1.2, and 1.5. If domain-wall-site forces differ by more than about 10% or change sign, run Langevin dynamics with DQMC forces on a small system (e.g., 14x14) and measure L(t); if the U-trend in Fig. 4a is not reproduced, the central claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's physical conclusion that Hubbard U enhances CDW coarsening is established only within the Gutzwiller approximation (GA): the ML force field is trained to reproduce GA forces (Fig. 1), and the benchmark in Fig. 2 only verifies that ML reproduces GA on 14x14 systems. GA is a variational ground-state ansatz; its accuracy for the non-equilibrium, domain-wall-rich configurations sampled during coarsening is never tested against an exact method such as determinant quantum Monte Carlo (DQMC) at half-filling. The effective quasiparticle Hamiltonian in Eq. (12) contains site-dependent renormalization factors R_i,sigma that renormalize both hopping and the Holstein potential; whether this correctly captures the energy landscape of CDW domain walls (which may involve mid-gap soliton states) is unknown. If GA underestimates the barrier for domain-wall motion at small U or overestimates the screening effect at intermediate U, the claimed recovery of the Allen-Cahn t^(1/2) law at U/tnn >= 0.9 and the monotonic speed-up with U in Fig. 4a would be artifacts. The unpublished self-cited baseline [83] does not provide an independent check.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper presents a machine-learning force-field approach to simulate the coarsening dynamics of charge density waves in the adiabatic square-lattice Hubbard-Holstein model. The force on each lattice displacement is computed in the Gutzwiller approximation and learned by a six-layer neural network acting on D4-symmetric local descriptors, enabling 200x200 Langevin simulations. The authors report that increasing the Hubbard repulsion U accelerates CDW domain growth, with L(t) recovering the Allen-Cahn t^(1/2) law for U/tnn >= 0.9, and attribute this to screening of the Holstein potential by electron correlations. They also report dynamic scaling collapse of the correlation function at late times.","tokens_in":18036,"tokens_out":7291,"duration_ms":70612,"significance":"If the conclusions hold, the paper provides a novel and nontrivial result: electron repulsion, which competes with CDW order, accelerates rather than suppresses domain coarsening, and the proposed ML protocol demonstrates a promising route to linear-scaling simulations of correlated electron dynamics. The surrogate itself is carefully validated against the Gutzwiller reference (force standard deviations about 0.006 in the units used, and correlation-function benchmarks on 14x14 systems), which gives confidence in the ML step. The main limitation is that the reference method (Gutzwiller approximation) is not benchmarked against an exact many-body method for the non-equilibrium domain-wall states relevant to coarsening; the physical claim is therefore conditional on the accuracy of GA in this regime. The paper also leaves unspecified several simulation parameters (e.g., damping, cutoff radius) needed to reproduce and assess the results.","major_comments":[{"comment":"The benchmarks in Fig. 1 and Fig. 2 only establish that the ML model reproduces the Gutzwiller-approximation (GA) forces, not that GA is accurate for the domain-wall-rich configurations sampled during coarsening. Since the central physical claim is about the Hubbard-Holstein model, the approximation dependence is load-bearing. Please provide a direct benchmark of GA against an exact/near-exact method (e.g., determinant quantum Monte Carlo or exact diagonalization on small lattices) for representative static lattice configurations with domain walls, and state the expected accuracy of GA for this regime.","section":"Sec. II, ML force field and Fig. 2"},{"comment":"The cutoff radius Rc in Eq. (3) is never specified and no convergence test with respect to Rc is reported. The linear-scaling claim relies on this truncation; without a test that the learned forces (or the resulting L(t)) have converged with Rc, the surrogate may be missing longer-range correlation effects. Please report Rc, the corresponding number of neighbors, and a convergence study.","section":"Sec. II, Eq. (3) and locality paragraph"},{"comment":"The Langevin equation (2) contains the damping gamma and mass m, but neither is specified in the text, and the integration timestep and total simulation time are not given. The coarsening rate and the exponent in Fig. 4a depend on the dynamics being in the overdamped regime and on the thermostat implementation. Please provide gamma, m, the numerical integrator, and the dimensionless time convention; otherwise the quantitative Allen-Cahn result is not reproducible.","section":"Sec. II, Eq. (2)"},{"comment":"The disorder-screening mechanism is inferred from a DMFT result for static random disorder, Eq. (8), but the Holstein displacement Q_i is a dynamical degree of freedom and the simulations do not directly measure the screening of the potential or the domain-wall barrier. Please either test this mechanism directly (for example, by comparing the GA potential barrier for a domain wall as a function of U, or by computing the variance of the effective on-site potential along the trajectory) or clearly label it as a plausible but unverified interpretation.","section":"Sec. III, Eqs. (7)-(8)"}],"minor_comments":[{"comment":"The scaling collapse is shown for U/tnn = 0.6, 0.9, 1.2, 1.5 but not for U/tnn = 0.3; the text's statement that 'the late-stage coarsening dynamics ... obeys dynamic scaling invariance' is broader than the presented evidence. Please show the U=0.3 collapse (or explain its absence) and qualify the claim.","section":"Fig. 4"},{"comment":"Please clarify whether n_i is the Gutzwiller expectation value obtained simultaneously with the force computation, and how the phase exp(iQ·r_i) is treated for finite systems with periodic boundary conditions.","section":"Eq. (4)"},{"comment":"The training dataset description is incomplete. Please specify the system size and the range of lattice configurations used to generate the 300 snapshots, the number of training/validation splits, and the number of independent ML models trained for each U.","section":"Methods, 'Neural network and training'"},{"comment":"There are several typographical errors: 'U/nn = 0.3' should be 'U/tnn'; 'Gutswiller' in the Fig. 2 caption should be 'Gutzwiller'; 'indicting' in Sec. II should be 'indicating'; and 'two eigenvalues problems' in Methods should be 'two eigenvalue problems'.","section":"Throughout"},{"comment":"The notation rho_{ij,sigma} in Eq. (11) and Delta_{i,sigma} in Eq. (13) should be explicitly defined with indices, and the sentence about the two eigenvalue problems should be rewritten for clarity.","section":"Methods, 'Gutzwiller approximation'"},{"comment":"Reference [83] is cited as 'unpublished'; it is used both for the slow-coarsening U=0.3 behavior and for the t^(1/2) weak-coupling limit. Please clarify whether this reference is under review or replace it with a publicly available version.","section":"Reference [83]"},{"comment":"The data points for L(t) are shown as symbols without error bars. Since the correlation function is averaged over 70 runs, please indicate the statistical uncertainty (e.g., standard error of the mean) for L(t).","section":"Fig. 4a"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on self-citations, including the unpublished baseline [83], which weakens the independent verification of the central comparison. The editor may wish to ask the authors to provide the unpublished work or cite a public version. The manuscript is technically sound in its ML-to-GA validation, but the absence of an exact benchmark for GA in the non-equilibrium context is a substantive gap that should be addressed before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the result—Hubbard U speeds up CDW coarsening in the adiabatic Hubbard-Holstein model—is new and counterintuitive, and the ML force-field machinery is solid. But the physics rests entirely on the Gutzwiller approximation, so the claim is 'true within GA' until someone checks GA against an exact method on the relevant non-equilibrium configurations.\n\nWhat the paper does well: the linear-scaling ML force field trained on GA forces is a clean extension of the authors' prior Holstein and itinerant-magnet papers. Force errors are small (σ ≈ 0.006 and 0.004), and the 14×14 Langevin benchmark shows the ML model reproduces GA correlation functions over time, which is the right kind of validation. The scaling collapse in Fig. 4 is a decent check of dynamic scaling. The screening mechanism—Hubbard U renormalizes the effective on-site Holstein potential—is plausible and consistent with known disorder-screening results.\n\nWhere it's soft: the whole story is inside GA. Because the ML model is a surrogate for GA, the benchmark only proves the fit is good; it says nothing about whether GA is reliable for domain-wall-rich configurations during coarsening. The stress-test concern about mid-gap soliton states is legitimate. An exact DQMC check on a small system—even just forces or a short trajectory—would help a lot. The L(t) curves have no error bars despite 70 runs, and no code or data is provided. Ref [83], an unpublished self-cited baseline, is used to interpret the Holstein strong-coupling limit; that's not fatal but should be cleaned up.\n\nNone of these are fatal. The paper is honest about using GA, and within GA the result is exactly what the simulations produce. The open question is GA's reliability out of equilibrium. That is a referee question, not a desk-reject reason.\n\nMy recommendation: send to peer review. The right referees know GA and ML force fields. I'd ask for a small-system exact benchmark or, failing that, a sharp statement of when GA could break down for this problem. If the authors can provide that, the paper becomes a solid contribution.","headline":"Hubbard U accelerates CDW coarsening in the adiabatic Hubbard-Holstein model—a new result that holds within the Gutzwiller approximation, with the GA itself as the main soft spot.","tokens_in":18605,"tokens_out":3471,"would_cite":true,"duration_ms":33473,"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":"Electron repulsion speeds up charge-density-wave coarsening in a Hubbard-Holstein model.","keywords":["charge density wave","coarsening dynamics","Hubbard-Holstein model","machine learning force field","Gutzwiller approximation","disorder screening","Langevin dynamics","phase ordering kinetics"],"falsifier":"Run the same thermal-quench coarsening protocol on small systems using an unbiased many-body method such as determinant quantum Monte Carlo or exact diagonalization for U/tnn = 0.6 and 0.9, measure L(t), and check whether the domain growth is still faster than at U/tnn = 0.3 and approaches t^(1/2); if the enhancement disappears or the exponent differs, the reported acceleration is an artifact of the Gutzwiller force training.","tokens_in":17575,"feed_emoji":"⚛️","tokens_out":3573,"duration_ms":37251,"temperature":0.7,"pith_summary":"The paper tries to establish that in the adiabatic half-filled square-lattice Hubbard-Holstein model, increasing the Hubbard repulsion U accelerates rather than suppresses the coarsening of checkerboard charge-density-wave domains after a thermal quench. It further claims this acceleration is a disorder-screening effect: electron correlations renormalize the effective random on-site potential produced by the Holstein phonons, weakening the pinning that normally slows domain growth. To test this in large systems, the authors build a machine-learned force field trained on Gutzwiller-approximation forces, achieving linear-scaling Langevin dynamics on 200x200 lattices. If true, the result shows that standard universal coarsening classes can be reached in correlated electron systems once correlation-induced screening is accounted for, and it validates a route to multiscale dynamical simulations.","feed_headline":"Hubbard repulsion accelerates CDW coarsening","feed_subtitle":"Machine-learned force fields show electron correlation screens phonon pinning, restoring t^(1/2) domain growth.","key_machinery":"The carrying mechanism is a machine-learned force field that predicts the local electronic force on each lattice site from a symmetry-invariant description of its neighborhood, trained on forces computed by the Gutzwiller/slave-boson approximation. The local descriptors are built from irreducible representations of the D4 point group, and a deep neural network approximates the force as a function of those descriptors, giving O(1) per-site cost and linear overall scaling. The physical explanation of the enhanced coarsening rests on a self-energy renormalization formula epsilon'_i = epsilon_i + Sigma_i(0), which shows that the variance of the effective disorder potential is reduced by a factor 1/(1+U chi_ii)^2, where chi_ii is the local compressibility.","core_discovery":"The central claim is that electron-electron repulsion enhances the growth of charge-density-wave domains in the adiabatic Hubbard-Holstein model, opposite to the naive expectation that Hubbard U suppresses Holstein-driven charge order. After a quench to low temperature, the characteristic domain length L(t) grows faster for larger U, and for U/tnn >= 0.9 it recovers the Allen-Cahn law L(t) ~ t^(1/2), while at small U growth is slow and nearly arrested. The late-stage correlation functions collapse onto a single curve when rescaled by L(t), confirming dynamic scaling invariance. The paper attributes the enhancement to a screening mechanism: the Hubbard interaction reduces the effective random on-site potential generated by the Holstein phonons, analogous to disorder screening in correlated systems, and this weaker pinning allows domains to grow more freely.","pith_inferences":["If the screening mechanism generalizes beyond the specific model, moderately correlated materials could show faster CDW ordering kinetics than simple electron-phonon models predict, a difference that may be observable in ultrafast pump-probe experiments.","The same machine-learning pipeline could be applied to other emergent orders such as spin density waves or orbital order, where the feedback between quasiparticles and the order parameter may produce different coarsening universality classes.","A direct test of the approximation dependence would be to train the identical ML architecture on forces from exact diagonalization or determinant quantum Monte Carlo on small systems; if those forces reproduce the same enhancement, the result is not an artifact of the Gutzwiller approximation.","The paper's mechanism implies that the coarsening rate is controlled by the local compressibility chi_ii, so tuning U or the filling to maximize chi_ii could systematically accelerate or decelerate domain growth in experiments."],"forward_implications":["For Hubbard U/tnn of at least 0.9, CDW domain growth obeys the Allen-Cahn t^(1/2) law, meaning electron correlations restore the standard curvature-driven coarsening universality class.","The late-stage coarsening satisfies dynamic scaling invariance: equal-time correlation functions collapse onto a single time-independent curve when lengths are rescaled by L(t).","The Hubbard interaction screens the Holstein interaction in the adiabatic limit, effectively reducing the random on-site potential that pins CDW domains and thereby accelerating their growth.","Machine-learned force fields with locality-based descriptors provide a linear-scaling algorithm suitable for large-scale dynamical simulations of correlated electron systems, going beyond the limits of direct many-body methods.","This is presented as the first systematic study of electron-correlation effects on phase-ordering dynamics, opening a new direction in non-equilibrium physics of emergent orders."],"supporting_citations":[{"why":"Supplies the disorder-screening formula used to explain how Hubbard interactions renormalize the random on-site potential and thereby accelerate coarsening.","marker":"[84]"},{"why":"Provides the Gutzwiller/slave-boson method that generates the training forces for the ML model and captures correlation effects such as bandwidth renormalization and disorder screening.","marker":"[74-82]"},{"why":"Establishes the ML force-field approach for CDW phase-ordering dynamics and the descriptor construction that the present work adapts to the Hubbard-Holstein model.","marker":"[14]"},{"why":"Provides the weak-coupling Holstein-model coarsening result with the t^(1/2) law against which the recovered Allen-Cahn growth at large U is compared.","marker":"[83]"},{"why":"Justifies the semiclassical treatment of phonons by showing that CDW phases from hybrid Monte Carlo based on exact diagonalization agree with determinant QMC results.","marker":"[62]"},{"why":"Provides the nearsightedness principle that underlies the local cutoff in the ML force field and the linear-scaling design.","marker":"[17,18]"}],"fun_headline_variants":["Electron repulsion accelerates charge density wave growth","Hubbard U boosts CDW coarsening via screening","Repulsion enhances domain growth in charge density waves","ML simulations reveal correlation-driven CDW coarsening","Correlation accelerates CDW ordering in Hubbard-Holstein"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central physical result rests on the Gutzwiller approximation being accurate for the disordered, out-of-equilibrium lattice configurations sampled during domain growth, since the machine-learned forces are trained to reproduce Gutzwiller forces.","fun_headline_variants_meta":{"raw":{"variants":["Electron repulsion accelerates charge density wave growth","Hubbard U boosts CDW coarsening via screening","Repulsion enhances domain growth in charge density waves","ML simulations reveal correlation-driven CDW coarsening","Correlation accelerates CDW ordering in Hubbard-Holstein"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000479,"raw_usage":{"total_tokens":2345,"prompt_tokens":893,"completion_tokens":1452,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":509,"completion_tokens_details":{"reasoning_tokens":1389}},"tokens_in":509,"tokens_out":1452,"duration_ms":10931,"temperature":1.0,"reasoning_tokens":1389,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:03:04.069619+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same thermal-quench coarsening protocol on small systems using an unbiased many-body method such as determinant quantum Monte Carlo or exact diagonalization for U/tnn = 0.6 and 0.9, measure L(t), and check whether the domain growth is still faster than at U/tnn = 0.3 and approaches t^(1/2); if the enhancement disappears or the exponent differs, the reported acceleration is an artifact of the Gutzwiller force training.","supporting_citations":[{"cited_title":"Tanaskovi´ c, V","cited_arxiv_id":null,"evidence_quote":"Supplies the disorder-screening formula used to explain how Hubbard interactions renormalize the random on-site potential and thereby accelerate coarsening."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the weak-coupling Holstein-model coarsening result with the t^(1/2) law against which the recovered Allen-Cahn growth at large U is compared."},{"cited_title":"Esterlis, S","cited_arxiv_id":null,"evidence_quote":"Justifies the semiclassical treatment of phonons by showing that CDW phases from hybrid Monte Carlo based on exact diagonalization agree with determinant QMC results."}],"review_version":1}