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Importance of effective Coulomb interactions for $T_c$ in cuprates

T0 review · 1 major / 2 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read In cuprates, Tc correlates most strongly with the effective Coulomb repulsion on oxygen p-orbitals in the three-band model.

desk verdict The paper's new angle is a dataset-wide correlation showing Tc tracks most strongly with U_p from the three-band model, but the automated downfolding leaves the result on shaky ground. read the letter →

arxiv 2606.21323 v1 pith:V4Y4ISZM submitted 2026-06-19 cond-mat.str-el

Jakša Vučičević , Upendra Kumar , Chia-Nan Yeh , Miguel A. Morales , Malte Rösner This is my paper · ORCID
classification cond-mat.str-el
keywords cupratesTcCoulombinteractionsthree-bandEmerymodelsingle-bandHubbardoxygenp-orbitalsDFTdownfoldingstatisticalcorrelation
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

The paper compiles data on about 40 cuprate compounds and their experimental Tc values, then uses an automated procedure to compute DFT band structures and downfold them to the single-band Hubbard model and the three-band Emery model via MLWFs and cRPA. Statistical analysis of the resulting parameters against Tc shows that the Coulomb coupling on the p-orbitals exhibits the strongest correlation across the full dataset. This points to the oxygen sites in the copper-oxide planes as central to Tc variations, while also confirming a correlation with t' in the single-band model where Tc appears to require a minimum t' value. A sympathetic reader would care because identifying the dominant interaction parameter offers a route to understanding why Tc differs between compounds and potentially to guiding higher-Tc material design.

What carries the argument

The fully automated MLWF + cRPA downfolding of DFT band structures to the single-band Hubbard and three-band Emery models, followed by statistical correlation analysis with experimental Tc.

What would settle it

A cuprate compound or set of compounds in which the computed p-orbital Coulomb coupling fails to correlate with measured Tc while some other parameter shows a stronger correlation when an alternative downfolding procedure is used.

Watch

Extended reading notes

Core claim

Through a large-scale systematic study that downfolds DFT band structures for dozens of cuprates to effective lattice models and applies unbiased statistical analysis, Tc scales well with simple functions of the model parameters; throughout the dataset Tc correlates the most with the Coulomb coupling on the p-orbitals in the three-band model, highlighting the importance of the oxygen sites in the copper-oxide planes, while also confirming the previously observed trend that t' in the single-band model correlates with experimental Tc and that Tc appears to vanish below a finite value of t'.

Load-bearing premise

The automated MLWF + cRPA downfolding procedure from DFT band structures produces effective model parameters whose variations across compounds faithfully reflect the microscopic interactions that control experimental Tc.

Editorial extensions

If this is right

  • Tc correlates with t' in the single-band model and appears to vanish below a finite value of t'.
  • The coupling strength plays a role in addition to the hopping parameters.
  • More sophisticated models might be needed to describe all cuprates on the same footing.
  • The oxygen p-orbitals are important for understanding variations in Tc.

Reading between the lines

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

  • If p-orbital Coulomb terms dominate, then structural modifications that change oxygen coordination or environment could tune Tc more directly than changes focused on copper sites.
  • The observed correlation implies that differences in screening of interactions at oxygen atoms vary enough across crystal structures to control Tc differences.
  • Three-band model simulations that vary only the p-orbital U could be compared directly to experimental Tc trends for specific compounds to test predictive power.
  • Extending the same downfolding and correlation procedure to pressure-dependent Tc data or additional compounds would provide a direct test of whether the p-orbital trend 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

1 major / 2 minor

Summary. The manuscript conducts a large-scale study of ~40 cuprate compounds by compiling experimental Tc and crystal structures, computing DFT band structures, and downfolding them via an automated MLWF+cRPA procedure to single-band Hubbard and three-band Emery models. Statistical analysis of the resulting effective parameters identifies the strongest correlation between experimental Tc and the Coulomb interaction U_p on oxygen p-orbitals in the three-band model, while also confirming a trend with t' in the single-band model and noting that Tc appears to vanish below a finite t' value.

Significance. If the downfolded parameters accurately reflect compound-to-compound variations in microscopic interactions, the result would indicate that oxygen-site Coulomb couplings are the dominant factor controlling Tc variations across cuprates, with implications for materials design. The alignment with recent Hubbard-model ground-state calculations on the t' threshold adds supporting context. The automated pipeline and statistical approach enable the scale of the study, but the significance hinges on validation of the downfolding procedure.

major comments (1)
  1. [Methods] Methods paragraph (and any associated supplementary details on the pipeline): The central claim that Tc correlates most strongly with U_p relies on the automated MLWF+cRPA downfolding producing parameters whose variations across compounds faithfully track the interactions controlling experimental Tc. No benchmarks against manual cRPA results, no sensitivity tests to DFT functional choice or energy windows on multiple cuprates, and no assessment of compound-dependent self-interaction errors are reported, despite known sensitivities that can shift screened interactions by several eV.
minor comments (2)
  1. [Abstract] Abstract: The phrase 'more sophisticated models might be needed to describe all cuprates on the same footing' is stated without specifying which discrepancies in the data support this conclusion.
  2. [Abstract] Abstract: The 'previously observed trend' for t' should include a citation to the relevant prior work.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful review and constructive feedback on our manuscript. We address the major comment on the methods and validation of the automated downfolding procedure below.

read point-by-point responses

  1. Referee: [Methods] Methods paragraph (and any associated supplementary details on the pipeline): The central claim that Tc correlates most strongly with U_p relies on the automated MLWF+cRPA downfolding producing parameters whose variations across compounds faithfully track the interactions controlling experimental Tc. No benchmarks against manual cRPA results, no sensitivity tests to DFT functional choice or energy windows on multiple cuprates, and no assessment of compound-dependent self-interaction errors are reported, despite known sensitivities that can shift screened interactions by several eV.

    Authors: We agree that the absence of explicit benchmarks and sensitivity analyses represents a limitation, as the referee notes. A comprehensive set of manual cRPA validations or tests across all ~40 compounds would be computationally prohibitive for this large-scale study. We will revise the methods section and supplementary information to include (i) a discussion of known cRPA sensitivities to DFT functionals, energy windows, and self-interaction errors with references to the literature, and (ii) results from sensitivity tests performed on a representative subset of compounds. This will clarify that while absolute parameter values may carry systematic uncertainties, the automated and consistent treatment across the dataset supports the reliability of relative trends, including the observed correlation with U_p. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: parameters obtained from independent DFT+MLWF+cRPA downfolding; correlations with experimental Tc are statistical observations, not reductions by construction.

full rationale

The paper computes effective model parameters (U_p, t', etc.) via an automated first-principles pipeline applied to DFT band structures of each compound, then performs statistical correlation analysis against separately measured experimental Tc values. No step defines a parameter in terms of Tc, fits a model parameter to Tc data and then calls the result a prediction, or reduces the central correlation claim to a self-citation chain. The confirmation of the prior t' trend is presented as an external observation rather than a load-bearing uniqueness theorem. The derivation chain therefore remains self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Based on abstract only; the paper invokes standard DFT and downfolding methods without introducing new free parameters or entities. No explicit ad-hoc assumptions are stated beyond the validity of the chosen low-energy models.

assumptions (1)
  • domain assumption DFT band structures plus MLWF+cRPA downfolding yield physically meaningful effective Hubbard and Emery parameters for cuprates
    Invoked when the paper states it computes these parameters for each compound and correlates them with Tc.

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Cite this review

Pith. "Pith review of Importance of effective Coulomb interactions for $T_c$ in cuprates." pith.science (2026). https://pith.science/paper/V4Y4ISZM

@misc{pith2026260621323,
  author       = {Pith},
  title        = {Pith review of: Importance of effective Coulomb interactions for $T_c$ in cuprates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V4Y4ISZM}},
  note         = {Machine review of arXiv:2606.21323}
}
abstract

Cuprate superconductors exhibit the highest observed superconducting $T_c$ at atmospheric pressure. However, the magnitude of $T_c$ varies significantly between different cuprates. At present, it is unclear what properties of the crystal structure affect $T_c$ most strongly, yet such an understanding must underpin any efforts toward high-$T_c$ materials design. To address this issue, we perform a large scale systematic study, employing a combination of data collection, state-of-the-art numerical methods, and statistical analysis. We identify about 40 different cuprate compounds, and we compile detailed data about their $T_c$'s and crystal structures from literature and the available databases. Using a fully automated procedure, for each compound we compute the DFT bandstructure and downfold it to two of the most commonly studied low-energy lattice models, namely the single-band Hubbard and the three-band Emery models. The downfolding is based on the approach of MLWFs and cRPA. Finally, we apply a thorough and unbiased statistical analysis to investigate the correlations between the experimentally measured $T_c$'s and the computed parameters of our theoretical models. Our data indicates that more sophisticated models might be needed to describe all cuprates on the same footing. Nevertheless, we find that $T_c$ scales well with simple functions of model parameters. We confirm a previously observed trend that $t'$ in the single-band model correlates with the experimental $T_c$, and we find that $T_c$ appears to vanish below a finite value of $t'$, in agreement with recent ground-state calculations for the Hubbard model. However, we find that the coupling strength also plays a role: throughout our entire dataset, $T_c$ correlates the most with the Coulomb coupling on the $p$-orbitals in the 3-band model, highlighting the importance of the oxygen sites in the copper-oxide planes.

Figures

Figures reproduced from arXiv: 2606.21323 by the authors.

Figure 1
Figure 1. Examples of our DFT and downfolded bandstructures. Crystals are oriented so that copper-oxide layers are parallel to the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Maximal observed Tc (regardless of doping) in our dataset of parent compounds. Symbol denotes the category of the compound (based on the presence and the orbital character of additional bands near the Fermi level in the DFT band structure; the categories are illustrated in different columns of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Illustration of the hopping amplitudes in the copper-oxide [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: Correlation between individual model-parameters (as defined in the main text) and the experimentally measured [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 7
Figure 7. Figure 7: Analysis restricted to compounds featuring copper-oxide chains and parameters of the 3-band model. All the plotting conventions 0.490 0.485 0.480 0.475 0.470 0.465 0 0 1%5% 52 [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: Analysis restricted to ternary f-element compounds and parameters of the 3-band model. All the plotting conventions are as in previous figures. We find that, regardless of the subset of the data, the interactions on and the hopping between the oxygen-site orbitals in t…
Figure 9
Figure 9. Figure 9: Analysis restricted to individual parameters of the 3-band Emery-Holstein model, covering the entire dataset. All the plotting [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: Conventional unit cells of cuprate compounds considered in this work, illustrated using VESTA[ [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: Example of the partial density of states computed for the [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]
Figure 12
Figure 12. Figure 12: Spreads of the MLWF vs. the corresponding on [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]
Figure 13
Figure 13. Figure 13: Examples of cRPA results. Both axes are given in units of eV. Square, dot and the horizontal dashed line denote the values used [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]
Figure 14
Figure 14. Figure 14: Region of the 1-band model parameter-space populated by our compounds. On the right panel, the axes are given in units of eV. [PITH_FULL_IMAGE:figures/full_fig_p024_14.png]
Figure 15
Figure 15. Figure 15: Region of the 3-band model parameter-space populated by our compounds. Axes are given in units of eV. Similarly to the 1-band [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]
Figure 16
Figure 16. Figure 16: Correlations between effective (cRPA) interactions at dif [PITH_FULL_IMAGE:figures/full_fig_p026_16.png]

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