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Correlation of maximum superconducting critical temperature with copper-oxygen energy distance and oxygen hole content in the Emery model

T0 review · 2 major / 5 minor · reviewed 2026-07-08 · glm-5.2

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Pith's one-line read Wider Cu–O energy gap raises cuprate superconducting temperature

desk verdict 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 read the letter →

arxiv 2607.06460 v1 pith:GWACB2SX submitted 2026-07-07 cond-mat.str-el cond-mat.quant-gascond-mat.supr-con

Eleanor M. O'Callaghan , Nicolas Kowalski , A.-M. S. Tremblay , Giovanni Sordi This is my paper · ORCID
classification cond-mat.str-elcond-mat.quant-gascond-mat.supr-con PACS 74.72.-h74.20.Mn71.27.+a
keywords cupratesuperconductivityEmerymodelcharge-transferinsulatoroxygenholecontentCDMFTZaanen-Sawatzky-Allencriticaltemperatureoptimization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

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.

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 (2)
  1. 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
  2. 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.
minor comments (5)
  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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.

Simulated Author's Rebuttal

1 responses · 0 unresolved

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.

read point-by-point responses

  1. Referee: 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.

    Authors: 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: no

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity. The central result is computed from CDMFT simulations; self-citations provide supporting context only.

full rationale

The paper's main claim—that T_c^max increases with Cu-O energy distance (ε̃_p - ε_d) at fixed charge gap Δ_I = 0.6—is obtained directly from CDMFT + CT-HYB quantum Monte Carlo simulations (Figs. 2a-e). The Hamiltonian parameters (U_d, ε̃_p - ε_d, t_pd, t_pp) are inputs; T_c^CDMFT is an output of the self-consistency loop; the charge gap Δ_I is computed from the n_tot(μ) plateau and used as a selection constraint, not a fitted constant. The oxygen hole content 2p_p = 2(2 - n_p) is also a computed output, not an input. No step in the derivation chain reduces to its inputs by construction. The companion article [28] (shared authors) is cited for generalization beyond Δ_I = 0.6 and for methodological details (T_c extraction, constant-Δ_I contours), but the central result at Δ_I = 0.6 is self-contained in the present paper's Figures 2 and 3. Reference [44] (shared authors) is cited for a prior conjecture that the present paper confirms through independent calculation, not by re-derivation. The Zaanen-Sawatzky-Allen framework is from Ref. [55] (external authors). The ε̃_p renormalization follows Ref. [42] (external). No uniqueness theorem is invoked, no ansatz is smuggled, and no prediction is equivalent to a fit. The one minor self-citation to [28] for generalization is acknowledged as a limitation, not presented as proof. This is a standard computational physics paper with honest self-citation for companion results; the central derivation is independent. Score 1 reflects the minor load on [28] for the generalization claim, which is not the paper's primary result. The skeptic's concern about statistical significance of the ~20% variation in T_c^max is a correctness/precision issue, not a circularity issue.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities, particles, or forces. It works within the established Emery model framework and uses standard CDMFT methodology. The parameters are either inherited from prior literature (t_pd, t_pp, t'_pp) or are the independent variables of the study (ε̃_p - ε_d, U_d). The charge gap constraint Δ_I = 0.6 is a methodological choice, not a new entity. No new axioms beyond standard domain assumptions of the Emery model and CDMFT are introduced.

free parameters (6)
  • t_pd = 1.5
    Cu-O hopping amplitude, set to 1.5 following prior work [17,19,43,44]. Not fitted to data in this paper but chosen from literature.
  • t'_pp = 1.0
    Next-nearest-neighbor O-O hopping, set to 1.0. Used as energy/temperature unit.
  • t_pp = 1.0 (model unit); 0.4 eV (Kelvin conversion)
    Nearest-neighbor O-O hopping, set to 1 as unit of energy. The 0.4 eV conversion to Kelvin is a physical scale choice for plotting.
  • Δ_I (charge gap target) = 0.6
    Fixed charge gap size used to select the five parameter sets. Chosen because it is close to the metal-insulator boundary. This is a constraint, not a fitted parameter, but it is a specific choice that affects the result.
  • U_d (per data point) = 7.76, 9.41, 11.15, 12.94, 14.8
    Onsite Coulomb repulsion, varied to maintain Δ_I = 0.6 at each ε̃_p - ε_d value. These are determined by the self-consistent CDMFT computation, not independently fitted.
  • ε̃_p - ε_d (per data point) = 2, 4, 6, 8, 10
    Bare Cu-O energy distance, the independent variable scanned in this study. Chosen to span the charge-transfer regime.
assumptions (4)
  • domain assumption The Emery three-band model with the specified hopping parameters captures the relevant physics of cuprate superconductivity.
    Invoked throughout; the model (Eq. 1-2) is the starting point. Standard in the field but a simplification of real cuprates (no apical oxygen, no long-range hopping, etc.).
  • domain assumption CDMFT on a 2x2 Cu plaquette (4 Cu + 8 O sites) is sufficient to capture d-wave superconducting correlations.
    Stated in 'Model and Method': 'we use the minimal cluster that captures d-wave superconductivity.' This is a standard but uncontrolled approximation.
  • domain assumption T_c^CDMFT, the mean-field transition temperature within the cluster, is a meaningful proxy for the true superconducting T_c.
    The paper acknowledges this neglects Kosterlitz-Thouless physics (citing Mermin-Wagner [56]) but uses T_c^CDMFT throughout as the figure of merit.
  • ad hoc to paper The charge gap Δ_I = 0.6 is representative enough to draw general conclusions about T_c^max trends.
    The paper studies only Δ_I = 0.6 and defers other values to the companion article [28]. The generality of the result depends on this assumption.

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Pith. "Pith review of Correlation of maximum superconducting critical temperature with copper-oxygen energy distance and oxygen hole content in the Emery model." pith.science (2026). https://pith.science/paper/GWACB2SX

@misc{pith2026260706460,
  author       = {Pith},
  title        = {Pith review of: Correlation of maximum superconducting critical temperature with copper-oxygen energy distance and oxygen hole content in the Emery model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GWACB2SX}},
  note         = {Machine review of arXiv:2607.06460}
}
abstract

Identifying microscopic parameters that optimize the maximum superconducting critical temperature $T_c^{\rm max}$ in the canonical model of the copper-oxygen plane of cuprates, the Emery model, remains challenging. Using cellular dynamical mean-field theory at finite temperature, we find that for a fixed charge gap size in the parent charge-transfer insulating state, $T_c^{\rm max}$ unexpectedly increases with increasing the copper-oxygen energy distance, as this favors the transfer of electrons from oxygen to copper orbitals. We show that these findings emerge naturally in the Zaanen-Sawatzky-Allen scheme and capture observed trends in hole-doped cuprates. Overall, our study uncovers that $T_c^{\rm max}$ is optimized in the Emery model under three conditions: upon doping a charge-transfer insulator, close to the charge-transfer insulator to metal boundary, and deep into the charge-transfer regime. This finding indicates new paths for optimizing $T_c^{\rm max}$.

Figures

Figures reproduced from arXiv: 2607.06460 by the authors.

Figure 1
Figure 1. FIG. 1. Zaanen-Sawatzky-Allen diagram [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a)-(e) Superconducting temperature [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a)-(e) Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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