{"id":"795285ca-52ad-43ae-a53c-a34df9b1fd28","arxiv_id":"2607.06460","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":6,"one_line_summary":"In the Emery model, the maximum superconducting critical temperature increases with copper-oxygen energy distance at fixed charge gap, correlating with increased oxygen hole content and deeper charge-transfer character.","lead":"Using a computational method called cellular dynamical mean-field theory, this paper finds that in the three-band Emery model of cuprate superconductors, the maximum superconducting critical temperature increases with the copper-oxygen energy distance when the charge gap is held fixed. This matters because it identifies a new parameter regime for optimizing superconductivity and connects model parameters to experimental trends in real cuprate materials.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The T_c^max variation across the five parameter sets is ~20% (0.020–0.024 in units of t_pp) with no reported error bars; statistical significance of the trend is unestablished.","rationale":"The reader correctly identified the narrow T_c range and absence of error bars as a concern (point 1 in their rationale) but chose the single-Δ_I generalization issue as the weakest assumption. I think the statistical significance issue is more load-bearing: if the trend at Δ_I = 0.6 is within noise, there is nothing to generalize. The paper's methodology (CDMFT + CT-HYB on a minimal plaquette) is sound and well-established, and the physical mechanism (oxygen hole content driving T_c) is plausible and connects to experimental trends. The Zaanen-Sawatzky-Allen framework provides a useful organizing principle. However, the quantitative claim rests on a ~20% variation in T_c^max across five points with no error bars and a discrete temperature grid whose resolution is not stated. The verdict remains CONDITIONAL — the paper is a legitimate contribution, but the central quantitative result needs statistical validation before it can be fully accepted. The companion article [28] addressing generality across Δ_I values is necessary but not sufficient; error bars on the existing data are the first priority.","tokens_in":12100,"tokens_out":2067,"duration_ms":134687,"concrete_test":"Re-run all five parameter sets (points 1–5 in Fig. 1) with: (a) a finer temperature grid (e.g., spacing ≤ 0.0005) around the transition, and (b) sufficient CT-HYB samples to estimate statistical errors on the superconducting order parameter at each temperature. Report error bars on each T_c^CDMFT value. If the 1σ error bars on adjacent points in Fig. 2(f) overlap (e.g., if point 2 and point 4 are statistically indistinguishable), the monotonic trend is not established and the central claim weakens to 'no significant variation detected.'","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is that T_c^max increases with Cu-O energy distance at fixed Δ_I = 0.6. Figure 2(f) shows five points spanning T_c^max from ~0.020 to ~0.024 — a relative variation of about 20%. No statistical uncertainties are reported on any of these values. The T_c^CDMFT values are extracted from CT-HYB quantum Monte Carlo simulations, which have inherent statistical errors from the stochastic sampling. Additionally, T_c^CDMFT is determined by locating where the superconducting order parameter transitions from zero to nonzero across a discrete temperature grid (T ∈ [1/50, 1/40]). If the temperature grid spacing is comparable to the ~0.004 spread in T_c^max, the precision of each T_c value is grid-limited. Typical CT-HYB statistical errors at these temperatures can be ~0.001–0.002, which is a substantial fraction of the observed variation. If error bars across the five points overlap, the monotonic trend in Fig. 2(f) — the paper's first main result — may not be statistically distinguishable from noise. This is more fundamental than the single-Δ_I concern (which the reader identified as weakest) because it questions whether the trend exists at all in the data presented, not merely whether it generalizes. The companion article [28] cannot rescue a trend that is within statistical noise at Δ_I = 0.6.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"This manuscript uses cellular dynamical mean-field theory (CDMFT) with a CT-HYB quantum Monte Carlo solver to study the Emery (three-band) model on a 2×2 CuO₂ plaquette. The central claim is that, at fixed parent charge-transfer gap size Δ_I = 0.6, the maximum superconducting critical temperature T_c^max increases with increasing bare Cu–O energy distance ε̃_p − ε_d (equivalently, decreases with increasing bare charge-transfer energy Δ). The authors connect this to increased oxygen hole content 2p_p, reproducing the experimental correlation of Refs. [6,7]. They identify a third condition for optimizing T_c: being deep in the charge-transfer regime (U_d ≫ Δ). The methodology is well-established and the superconducting domes in Figs. 2(a–e) and 3(a–e) are internally consistent across panels.","tokens_in":12724,"tokens_out":1142,"duration_ms":152217,"significance":"The paper addresses a well-motivated question—the dependence of T_c on Cu–O energy distance at fixed charge gap—that is inaccessible to single-band models and directly relevant to cuprate experiments. The CDMFT methodology and CT-HYB solver are standard and well-deployed. The finding that T_c^max increases with ε̃_p − ε_d at fixed Δ_I is a falsifiable, parameter-free prediction (no fitting to the target result). The connection to oxygen hole content and the Zaanen–Sawatzky–Allen framework provides physical intuition. The prediction for ultracold-atom implementations adds timeliness. However, the quantitative claim rests on five data points with a ~20% spread in T_c^max and no reported error bars, which limits the strength of the conclusion as presented.","major_comments":[{"comment":"Figure 2(f) and the associated text present the paper's first main result: T_c^max decreases with increasing Δ (increases with ε̃_p − ε_d) at fixed Δ_I = 0.6. The five data points span T_c^max ≈ 0.020–0.024 in units of t_pp, a relative variation of about 20%. No statistical uncertainties are reported on any of these values. Since T_c^CDMFT is extracted from CT-HYB quantum Monte Carlo simulations (which have inherent stochastic errors) and is determined by locating where the superconducting order parameter transitions from zero to nonzero across a discrete temperature grid (T ∈ [1/50, 1/40], i.e., spacing ~0.005), the precision of each T_c value may be grid-limited and/or comparable to the observed spread. The authors should report error bars (or at minimum discuss the statistical and grid-related uncertainties) and demonstrate that the monotonic trend in Fig. 2(f) is distinguishable from","section":null},{"comment":"Section 'Setting the model in the charge-transfer regime' and Fig. 1: The central quantitative result is established only at a single value of the charge gap, Δ_I = 0.6. While the companion article [28] is cited as showing the trend generalizes, this manuscript itself does not provide evidence at other gap sizes. The authors should either (a) include at least one additional Δ_I value in this paper to demonstrate robustness, or (b) explicitly state in the main text that the claim is presently limited to Δ_I = 0.6 and that generalization depends on the companion paper.","section":null}],"minor_comments":[{"comment":"The temperature grid T ∈ [1/50, 1/40] gives a spacing of ~0.005 in units of t_pp, which is comparable to the ~0.004 spread in T_c^max across the five parameter sets. The authors should clarify how T_c^CDMFT is extracted within this grid (interpolation? nearest grid point?) and discuss the resulting precision of T_c extraction.","section":null},{"comment":"Fig. 2(f): The secondary x-axis (ε̃_p − ε_d) decreases left-to-right while the primary x-axis (Δ) increases left-to-right. This is potentially confusing; a brief note in the caption would help.","section":null},{"comment":"The conversion to Kelvin using t_pp = 0.4 eV is mentioned in captions but not in the main text. Stating this once in the body would improve readability.","section":null},{"comment":"Reference [28] is cited extensively for methodological details (T_c extraction, constant-Δ_I contours, generalization beyond Δ_I = 0.6). Since several key aspects of the argument depend on [28], a brief summary of what is established there would help the reader assess this paper's standalone contribution.","section":null},{"comment":"In the Discussion, the phrase 'the mixed d-p character of the doped holes is enhanced' could benefit from quantitative support—e.g., reporting η values across the five parameter sets.","section":null}],"recommendation":"major_revision","confidential_remarks":"The paper is well-written and the question is important. The main concern is whether the central quantitative claim survives scrutiny once error bars are included—the ~20% variation in T_c^max across five points, combined with CT-HYB statistical errors and grid spacing, could undermine the trend. If the authors can show the trend is statistically robust (or strengthen the presentation with additional data points), this could become a solid contribution. The heavy reliance on the companion [28] for generalization beyond Δ_I = 0.6 is also worth noting for scope assessment."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful and constructive report. The referee correctly identifies that the quantitative claim in Fig. 2(f) rests on a modest spread of T_c values and that the result is established at a single charge gap. We address both points below and will revise the manuscript accordingly.","responses":[{"response":"The referee raises a legitimate concern that we must address. We agree that error bars and a discussion of uncertainties are necessary, and we will add them in the revised manuscript. We provide here the substance of what will be added. First, regarding the temperature grid: the simulations are performed at a discrete set of temperatures. The values of T_c^CDMFT shown in Figs. 2(a–e) are extracted by identifying the lowest temperature at which the superconducting order parameter is zero and the highest temperature at which it is nonzero, so T_c is bracketed between two adjacent grid points. The grid spacing in the relevant temperature range is 0.002 in units of t_pp (not 0.005 as the referee estimated from the overall range [1/50, 1/40]). The spread in T_c^max across the five data points is approximately 0.004 (from ~0.020 to ~0.024), i.e., about two grid spacings. Second, regarding statistical errors: each CT-HYB simulation has a stochastic uncertainty on the superconducting order parameter. We will report the statistical error on the order parameter at each temperature and propagate it to the T_c extraction. In practice, the order parameter values near the transition are well separated from zero (or clearly zero) at the temperatures adjacent to T_c, so the bracketing is unambiguous within statistical noise. Third, we will add a discussion of the grid-limited precision: each T_c^max value is known to within ±0.002 (one grid spacing). The monotonic trend in Fig. 2(f) spans ~0.004, which is twice the grid spacing. While this is a modest effect, the trend is consistent across all five points with no reversals, and it is corroborated by the companion analysis in Ref. [28] at other gap sizes. We will state these limitations transparently and temper the quantitative strength","revision_made":"no","referee_comment":"Figure 2(f): No error bars reported; ~20% spread in T_c^max may be comparable to stochastic and grid-related uncertainties. Authors should report error bars and demonstrate the monotonic trend is distinguishable from noise."}],"tokens_in":11676,"tokens_out":1360,"duration_ms":80098,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"The main thing to know: this paper reports a genuinely new computational result — that at fixed charge-transfer gap, T_c^max in the Emery model increases with the bare Cu-O energy distance, equivalently decreasing with bare charge-transfer energy Δ. This is the first time this parameter axis has been isolated in CDMFT, and it connects to the experimental correlation between T_c and oxygen hole content (Refs. 6, 7). The finding that high-T_c systems sit deep in the charge-transfer regime (U_d >> Δ) is a clean, physically interpretable result that the Zaanen-Sawatzky-Allen diagram makes intuitive sense of. The methodology is standard and well-established: CDMFT with CT-HYB on a 2×2 plaquette, same code and approach as the group's prior work. The superconducting domes in Figs. 2 and 3 are internally consistent across panels, and the orbital-occupancy analysis connecting to oxygen hole content is a real strength — it grounds the parameter-space scan in a measurable quantity. The paper is honest about T_c^CDMFT being a mean-field quantity that neglects KT physics, and about the minimal cluster size. Now the soft spots. The stress-test concern about error bars is the one that matters most. Fig. 2(f) shows five points spanning T_c^max from ~0.020 to ~0.024 in units of t_pp — a 20% relative variation. These come from CT-HYB, which has statistical errors, and T_c is extracted on a discrete temperature grid (T ∈ [1/50, 1/40], spacing ~0.005). If the QMC error bars are ~0.001-0.002, that's a substantial fraction of the observed spread. Without reported uncertainties, the monotonic trend could be within noise. This is more pressing than the single-Δ_I concern, though that one also matters: the entire quantitative claim rests on one gap value, with generalization deferred to the companion article [28] which I cannot verify here. The reader's assessment is fair and roughly on target. I'd push the soundness down slightly below their 6.0 given the error-bar gap, and I agree the circularity burden is low — the parameters are inputs, not fitted to the target. This paper is for researchers working on cuprate superconductivity modeling, particularly those using multi-band approaches. It deserves a serious referee who should require: (1) error bars on all five T_c^max points, (2) confirmation that the trend survives at least one other Δ_I value, and (3) discussion of grid-spacing effects on T_c precision. The core idea is worth pursuing; the evidence just needs to be shored up.","headline":"New CDMFT result: T_c^max increases with Cu-O energy distance at fixed charge gap in the Emery model, but the trend rests on five points with no error bars","tokens_in":13170,"tokens_out":648,"would_cite":false,"duration_ms":118085,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["74.72.-h","74.20.Mn","71.27.+a"],"model":"glm-5.2","headline":"Wider Cu–O energy gap raises cuprate superconducting temperature","keywords":["cuprate superconductivity","Emery model","charge-transfer insulator","oxygen hole content","CDMFT","Zaanen-Sawatzky-Allen","critical temperature optimization"],"falsifier":"If repeating the calculation at other charge-gap values (Δ_I ≠ 0.6) reveals that T_c^max no longer increases with Cu–O energy distance, or if larger clusters reverse the ordering, the central claim weakens.","tokens_in":12102,"feed_emoji":"🔬","tokens_out":916,"duration_ms":166167,"temperature":0.7,"pith_summary":"This paper uses cellular dynamical mean-field theory on the three-band Emery model to show that, for a fixed charge gap in the parent insulating state, the maximum superconducting critical temperature T_c^max increases as the bare energy separation between copper and oxygen orbitals grows. The mechanism is that a larger Cu–O energy distance drives more electron transfer from oxygen to copper, increasing the oxygen hole content, which experiments have correlated with higher T_c. The authors identify three conditions for optimizing T_c: doping a charge-transfer insulator, sitting near the metal–insulator boundary, and being deep in the charge-transfer regime where the on-site copper repulsion U_d greatly exceeds the charge-transfer energy Δ.","feed_headline":"Wider Cu–O energy gap raises cuprate Tc","feed_subtitle":"Simulations show maximum superconducting temperature grows with copper–oxygen orbital separation, identifying a third knob for optimization.","key_machinery":"Emery model, Zaanen–Sawatzky–Allen diagram, cellular dynamical mean-field theory (CDMFT), charge-transfer energy Δ = ε_d + U_d − ε_p, oxygen hole content 2p_p","core_discovery":"For five parameter sets sharing the same parent-state charge gap (Δ_I = 0.6), the authors find that T_c^max rises monotonically as the bare Cu–O energy distance ε_p − ε_d increases from 2 to 10 (equivalently, as the bare charge-transfer energy Δ decreases). This trend is explained by the corresponding increase in oxygen hole content 2p_p: deeper charge-transfer character redistributes electrons from oxygen to copper orbitals, enhancing the mixed d–p character of doped holes and raising T_c^max from roughly 95 K to 115 K.","pith_inferences":["If the trend holds across other charge-gap values (as the companion article reportedly shows), materials engineering could target large Cu–O energy separation as an independent knob for raising T_c, distinct from chemical pressure or doping.","The result suggests that single-band Hubbard models, which lack the Cu–O energy distance parameter, may systematically miss a third axis of T_c optimization that the three-band Emery model captures.","A natural experimental test would be to correlate measured oxygen hole content and charge-transfer energy Δ across cuprate families while controlling for charge-gap size, checking whether the monotonic T_c^max trend predicted here persists."],"forward_implications":["Cuprate families with larger Cu–O energy separation and higher oxygen hole content should exhibit higher maximum T_c, offering a materials-design criterion beyond doping and gap size.","Ultracold-atom implementations of the Emery model could test the predicted T_c^max vs. Cu–O energy distance trend by tuning orbital energy offsets in optical lattices.","The Zaanen–Sawatzky–Allen diagram becomes a navigational tool for optimizing superconductivity: the sweet spot is deep in the charge-transfer regime (large U_d, small Δ) but close to the metal–insulator boundary."],"fun_headline_variants":["Cu–O orbital separation raises cuprate Tc in Emery model","Deeper charge-transfer character increases cuprate Tc","Larger Cu–O energy distance enhances cuprate Tc","Oxygen hole content and Cu–O gap tune cuprate Tc","Charge-transfer regime elevates maximum cuprate Tc"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The central trend is established at a single fixed charge-gap value (Δ_I = 0.6), and the companion article is needed to confirm it generalizes to other gap sizes; the cluster is also small (4 Cu + 8 O sites), and the reported T_c is a mean-field quantity that omits Kosterlitz–Thouless physics.","fun_headline_variants_meta":{"raw":{"variants":["Cu–O orbital separation raises cuprate Tc in Emery model","Deeper charge-transfer character increases cuprate Tc","Larger Cu–O energy distance enhances cuprate Tc","Oxygen hole content and Cu–O gap tune cuprate Tc","Charge-transfer regime elevates maximum cuprate Tc"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":1057,"prompt_tokens":496,"completion_tokens":561,"prompt_tokens_details":null},"tokens_in":496,"tokens_out":561,"duration_ms":57751,"temperature":1.0,"reasoning_tokens":482,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T04:54:31.242366+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If repeating the calculation at other charge-gap values (Δ_I ≠ 0.6) reveals that T_c^max no longer increases with Cu–O energy distance, or if larger clusters reverse the ordering, the central claim weakens.","supporting_citations":[],"review_version":1}