Importance of effective Coulomb interactions for $T_c$ in cuprates

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.

• The paper establishes that oxygen p-orbital Coulomb interactions, captured via Wannierization and cRPA, significantly modulate T₍c₎ in cuprates.
• The authors employ ab initio downfolding into single-band Hubbard and three-band Emery models to quantitatively correlate bandstructure features with T₍c₎ variations.
• Statistical regression reveals that both next-nearest neighbor hopping and Holstein-type retardation effects are crucial predictors for superconducting behavior.

Summary of "Importance of effective Coulomb interactions for $T_c$ in cuprates" (2606.21323)

Motivation and Scope

The paper presents a systematic, ab initio investigation into the determinants of the superconducting critical temperature ($T_c$) in cuprate materials, aiming to quantitatively correlate experimental $T_c$ with low-energy lattice model parameters downfolded from first-principles calculations. The focus is on capturing the impact of effective Coulomb interactions, especially those involving oxygen $p$-orbitals, via Wannierization and cRPA, in both single-band Hubbard and three-band Emery models. Robust statistical analysis is employed to link material-specific parameters with maximal observed $T_c$ values, providing a basis for future theoretical and computational studies.

Dataset Construction and Bandstructure Classification

The dataset encompasses approximately 40 stoichiometric parent cuprate compounds with documented maximal $T_c$, spanning different families and structural variants. For each, DFT calculations yield bandstructures, further downfolded to minimal lattice models via MLWFs and cRPA for interaction parametrization. Bandstructure analysis exposes several categorical distinctions, notably the presence or absence of additional bands near the Fermi level arising from buffer-layer atoms or chain structures.

Figure 1: DFT and downfolded bandstructures for representative cuprate categories, indicating orbital contributions to bands near the Fermi level.

Maximal $T_c$ is strongly modulated by such bandstructure categories; highest $T_c$ values are associated with compounds displaying additional bands proximate to the Fermi energy and multiple CuO$_2$ layers per unit cell.

Figure 2: Distribution of maximal observed $T_c$ by compound category and number of CuO$T_c$0 layers.

Downfolding and Effective Interaction Parametrization

Wannierization (via Wannier90 and RESPACK) yields tight-binding parameters ($T_c$1, $T_c$2, $T_c$3 for Hubbard; $T_c$4, $T_c$5 for Emery) alongside orbital spreads, with cRPA supplying frequency-dependent effective interactions $T_c$6. Notably, significant frequency dependence is observed, and the quantitative amplitude of coupling constants varies between models and compounds. The spatial spread of Wannier orbitals anticorrelates with on-site interaction strength.

Figure 3: Hopping amplitude illustration for single-band and three-band models, central to tight-binding parametrization.

Figure 4: MLWF spreads versus corresponding on-site Coulomb interactions; anti-correlation indicates increased delocalization decreases bare interactions.

Figure 5: Representative cRPA results showing frequency dependence of effective Coulomb interactions for 1-band and 3-band models; occasional inversion of $T_c$7 and $T_c$8 observed.

Statistical Correlation Analysis

Regression analysis seeks single-parameter and low-dimensional multivariate predictors of $T_c$9 from computed model parameters. Across the full dataset, $T_c$0 and ratios such as $T_c$1 in the single-band model display moderate correlation, corroborating previous claims that next-nearest-neighbor hopping is significant for superconductivity. In three-band models, the strongest correlations consistently involve local Coulomb interactions on oxygen $T_c$2-orbitals ($T_c$3, $T_c$4), along with ratios to oxygen-oxygen hopping ($T_c$5).

Figure 6: Correlation plots of model parameters versus $T_c$6, highlighting significance of $T_c$7 in single-band and $T_c$8/$T_c$9 in three-band models.

Importantly, this correlation is highly robust across different subsets—compounds with additional bands, Cu-O chains, ternary $p$0-element compounds—suggesting a universal role for oxygen-site interactions.

Figure 7: Restriction to CuO-chain-containing compounds, further evidencing the preeminence of oxygen-site interactions in predicting $p$1.

Figure 8: Restriction to ternary $p$2-element compounds; $p$3 remains dominant predictor for $p$4.

Interaction Retardation and Holstein Modeling

Analysis of model-parameter correlations at different Matsubara frequencies suggests that instantaneous interaction assignments are insufficient; strong predictors often involve couplings evaluated at finite frequency, motivating a Hubbard-Holstein or Emery-Holstein model description. Coupling constants for Holstein bosons derived from cRPA (especially $p$5 for oxygen sites) yield the best fits to $p$6 across the entire dataset, implicating retardation effects as crucial for accurate modeling.

Figure 9: Regression by Holstein parameters ($p$7, $p$8) in Emery-Holstein model, surpassing instantaneous interaction predictors in statistical robustness.

Parameter Space, Phase Diagram, and Trends

Visualization of model parameter-space population reveals clustering by structural category and confirms that high and low $p$9 compounds often occupy overlapping regions—a counterintuitive result suggesting that subtle changes in interaction parameters, especially on oxygen sites, critically modulate $T_c$0.

Figure 10: Single-band model parameter-space, showing overlap of high/low $T_c$1 compounds; $T_c$2 threshold for superconductivity evident.

Figure 11: Three-band model parameter-space; strong correlation between bare Coulomb interactions on Cu and O sites, but maximal variation in charge-transfer gap.

Practical and Theoretical Implications

The findings emphasize the necessity of incorporating oxygen-site interactions and their retardation in minimal models for cuprate superconductivity. The regime of $T_c$3 required for nonzero $T_c$4 in square-lattice Hubbard models is confirmed quantitatively, consistent with recent ground-state studies. The statistical dominance of Holstein-type couplings for oxygens motivates direct many-body solutions of augmented Emery-Holstein models, combined with cRPA-derived parameter sets, to assess their predictive power for material-specific $T_c$5.

The methodology enables future materials design efforts to focus on maximizing oxygen-site coupling strengths and optimizing parameter regimes identified as robust across compound families. The evidence against $T_c$6 being determined solely by 1-2 model parameters underscores the complexity of the pairing mechanism and the important role of crystallographic and electronic structure nuances.

Conclusion

The paper delivers a comprehensive mapping from first-principles electronic structure to effective lattice models, robustly correlating oxygen-site Coulomb interactions (including their retardation) with experimental superconducting critical temperatures in cuprates. This work motivates further exploration of Holstein-type models and reinforces the indispensability of oxygen orbitals in accurate theoretical treatments of high-$T_c$7 superconductivity.

Figures

Figure 1: DFT and model bandstructure examples for key compound categories.

Figure 2: Maximal $T_c$8 per compound category and CuO$T_c$9 layer count.

Figure 3: Hopping matrices for single-band and three-band models.

Figure 6: Parameter-$T_c$0 correlations in full dataset for Hubbard and Emery models.

Figure 7: Parameter-$T_c$1 correlations restricted to CuO chain compounds.

Figure 8: Parameter-$T_c$2 correlations for ternary $T_c$3-element compounds.

Figure 9: Holstein parameter regression in Emery-Holstein model.

Figure 4: Wannier spread vs. Coulomb interaction (anti-correlation).

Figure 5: cRPA frequency dependence; comparison of $T_c$4 and $T_c$5.

Figure 10: Parameter-space region population for single-band model.

Figure 11: Parameter-space region population for three-band model.

Conclusion

The robust correlation between oxygen-site Coulomb interaction parameters and superconducting $T_c$6 demands theoretical models for cuprates that honor this physics. The work provides practical parameter sets for future theoretical exploration and establishes statistical foundations for materials optimization. Incorporating interaction retardation, particularly via Holstein coupling on oxygen orbitals, is posited as the most promising avenue for accurate and predictive modeling of high-$T_c$7 superconductivity in cuprates.