Quantum Many-Body Approach to High-Temperature Superconductors: Superexchange, Doping, and Pairing Mechanisms

1. Introduction

Since the discovery of high-temperature superconductors (cuprate superconductors) by Bednorz and Müller in 1986, they have become one of the most important research topics in physics and materials science. Unlike conventional low-temperature superconductors, cuprate superconductors exhibit relatively high critical temperatures (Tc), with some materials reported to maintain superconductivity up to 164K under specific conditions. These properties enhance their potential applications in energy storage, power transmission, and quantum computing, driving continuous research efforts to develop high-temperature superconductors into practical technologies.

However, the superconducting mechanism of cuprate superconductors remains incompletely understood. The conventional BCS (Bardeen-Cooper-Schrieffer) theory explains superconductivity based on weak electron-phonon interactions, but cuprate superconductors exhibit strong electron-electron correlations, making them difficult to explain within the traditional BCS framework. Consequently, there is still a fundamental lack of understanding regarding how superconducting properties emerge in cuprate superconductors and how Tc is determined in specific materials.

 

1.1 Limitations of Previous Studies and Unresolved Problems

Over the past few decades, various theoretical models have been proposed to explain the superconducting properties of cuprate superconductors. Low-energy model-based approaches, particularly the Hubbard model and t-J model, have played a crucial role in analyzing electron correlation effects. However, these models generally rely on single-band approximations or simplified effective Hamiltonians, failing to fully reflect the complex electronic structures and physical interactions in real materials.

Furthermore, density functional theory (DFT)-based computational methods have struggled to predict superconductivity directly. DFT, as a quasi-classical approach, has limitations in capturing strong electron correlations and cannot directly compute key superconducting parameters such as the pairing order parameter and superconducting gap. Therefore, a more precise quantum many-body computational method is required.

 

1.2 Research Objectives and Methodology Overview

To overcome these limitations and quantitatively analyze the superconducting properties of different cuprate materials, recent studies have adopted an Ab Initio Quantum Many-Body approach. In particular, this research employs Density Matrix Embedding Theory (DMET) to perform quantum many-body calculations, allowing direct prediction of the superconducting pairing order parameter and superconducting gap in cuprate materials.

The key objectives of this study are as follows:

1. Analysis of superconducting property trends: Investigate the effects of pressure and layer structure on the superconducting critical temperature (Tc) and superconducting gap.
2. Quantification of electron correlation effects: Understand the impact of strong electron-electron interactions on superconducting pairing and identify key variables that drive superconductivity.
3. Development of predictive models: Establish models capable of predicting the superconducting properties of specific cuprate materials and explore the feasibility of new material development based on these models.

This study aims not only to provide crucial insights into the physical mechanisms of high-temperature superconductors but also to establish a fundamental theoretical framework for designing future high-temperature superconducting materials. Through this, the understanding of unconventional superconductors, including cuprates, will be further deepened, and stronger connections with experimental research can be expected in the future.

 

2. Physical Background of Cuprate Superconductors

Cuprate superconductors are a type of unconventional superconductors with high critical temperatures (Tc). They were first discovered by Bednorz and Müller in 1986, breaking the conventional belief that superconductivity could only occur below 30K. Subsequent studies have reported superconductivity in cuprate materials up to 164K, significantly increasing their potential for practical applications. However, the underlying physical principles governing superconductivity in cuprates remain an important research challenge, as they cannot be fully explained by the conventional Bardeen-Cooper-Schrieffer (BCS) theory.

 

2.1 Structural Characteristics of Cuprate Superconductors

Cuprate superconductors belong to the layered perovskite family and are characterized by their two-dimensional copper-oxygen (Cu-O) planes, which play a crucial role in superconductivity. These planes serve as the primary conduction pathways for superconducting currents, and the strong electronic interactions between copper (Cu) and oxygen (O) atoms contribute significantly to superconducting pairing.

The structural features of cuprate superconductors typically include:

• Cuprate CuO₂ plane: The primary region where superconducting pairing occurs, which can be controlled by charge doping.
• Buffer layer: A layer containing elements such as barium, strontium, and lanthanum, located between CuO₂ planes, which regulates charge transfer.
• Single-layer vs. multi-layer structures: In general, an increase in the number of CuO₂ planes correlates with an increase in Tc.

Notable cuprate superconductors include:

1. YBa₂Cu₃O₆₊δ (YBCO): A well-known high-temperature superconductor with multiple CuO₂ planes.
2. Bi₂Sr₂CaCu₂O₈₊δ (BSCCO): A bismuth-based cuprate with a multi-layered structure.
3. HgBa₂CaCu₂O₆₊δ (Hg-1212): A cuprate superconductor exhibiting superconductivity up to 164K.

 

2.2 Electron Correlations and Superconductivity

Unlike conventional superconductors, cuprate superconductors exhibit strong electron-electron correlations, leading to unique physical properties:

1. Mott Insulator to Superconductor Transition
• The parent compounds of cuprate superconductors exist as Mott insulators in their undoped state.
• This phenomenon cannot be explained by conventional band theory but arises due to strong Coulomb repulsion, preventing free electron movement and causing electron localization.
• Superconductivity emerges when the material is doped, introducing additional charge carriers.
2. d-Wave Superconductivity
• Unlike conventional s-wave superconductors, cuprate superconductors exhibit d-wave pairing symmetry.
• This means that electrons strongly pair in specific directions while repelling in others, resulting in anisotropic superconductivity.
• d-wave superconductivity is theorized to arise from spin correlations and superexchange interactions rather than electron-phonon interactions, as in BCS theory.
3. Spin and Charge Density Waves (SDW & CDW)
• Cuprate superconductors exhibit spin density waves (SDW) and charge density waves (CDW), which can compete with superconducting pairing.
• In certain doping conditions, charge density waves have been observed to suppress superconductivity.

 

2.3 Determinants of Critical Temperature (Tc) in Cuprates

The superconducting critical temperature (Tc) of cuprate superconductors is influenced by several key factors:

1. Pressure Effect
• Applying pressure to cuprate materials reduces the Cu-O bond length, thereby increasing superexchange interactions and enhancing superconducting pairing.
• Experimentally, Tc has been observed to increase from 135K to 164K under 30GPa pressure in Hg-1223.
2. Layer Effect
• The number of CuO₂ planes directly affects Tc, with an increased number of layers generally leading to higher Tc.
• This is attributed to stronger electron correlation effects as more layers are introduced.
3. Doping Level
• Cuprate superconductors exhibit maximum Tc at an optimal doping level of approximately 10–15%.
• Tc decreases in both overdoped and underdoped states, suggesting that the number of charge carriers must be precisely tuned for optimal superconductivity.

 

2.4 Research Directions for Cuprate Superconductors

Research on cuprate superconductors is actively evolving, with the following primary areas of focus:

• Exploring Superconducting Mechanisms via Ab Initio Calculations
• Recent studies are shifting from traditional model-based approaches to first-principles methods such as Density Matrix Embedding Theory (DMET) and Quantum Many-Body Calculations, which allow direct computation of superconducting properties.
• Discovery of New High-Temperature Superconductors
• Artificial intelligence (AI) and machine learning are being applied to design novel materials with cuprate-like electronic structures, increasing the likelihood of discovering new superconducting compounds.
• Integration with Experimental Approaches
• Advanced experimental techniques, including high-resolution scanning tunneling microscopy (STM) and precision X-ray diffraction, are being used to validate theoretical predictions and understand superconducting pairing mechanisms.

 

Summary (1)

Cuprate superconductors are a prime example of unconventional superconductors, distinguished by their strong electron correlations and d-wave superconductivity. This study builds upon these physical foundations, utilizing Ab Initio Quantum Many-Body calculations to analyze the superconducting pairing mechanisms of cuprates. Through this research, we aim to gain a clearer understanding of the origin of superconductivity in cuprates and establish a framework for designing new superconducting materials.

 

3. Existing Research and Unresolved Problems

Since the discovery of cuprate superconductors in 1986, numerous theories have been proposed to explain their high critical temperature (Tc) and unconventional superconducting mechanism. However, previous studies have limitations in fully explaining the superconducting nature of cuprates, particularly in quantitatively predicting how the material’s structure and composition influence Tc.

 

3.1 Existing Research and Major Theories

Research on the superconductivity of cuprate superconductors has primarily been based on low-energy models, density functional theory (DFT), and quantum many-body theories.

(1) Low-Energy Models

To understand the physical properties of cuprate superconductors, low-energy models that account for electron-electron correlation effects have been proposed. Representative models include:

• Hubbard Model: Explains the Mott insulating state and doping effects in cuprates by incorporating strong electron-electron interactions.
• t-J Model: A model derived from the Hubbard model that emphasizes superconducting pairing in the low-energy state while considering strong Coulomb interactions.

However, these models rely on single-band approximations, making them insufficient for accurately capturing multi-orbital effects and explaining the material-specific superconducting properties observed in various cuprates.

(2) Density Functional Theory (DFT)-Based Research

DFT has been widely used to compute the electronic structures of materials, contributing to the analysis of cuprate superconductors' band structures. However, DFT-based approaches face significant limitations:

• DFT cannot accurately incorporate strong electron correlation effects, which are essential for understanding superconductivity in cuprates.
• It struggles to directly compute the superconducting gap and pairing order parameter, which are critical for understanding superconducting properties.

Thus, while DFT can provide complementary insights, it is insufficient for fully explaining the fundamental physical origins of cuprate superconductivity.

(3) Quantum Many-Body Theory

To more accurately describe the physical properties of cuprate superconductors, which exhibit strong electron correlations, various quantum many-body theories have been applied.
Key methodologies include:

• Density Matrix Embedding Theory (DMET)
• Dynamical Mean-Field Theory (DMFT)
• Density Matrix Renormalization Group (DMRG)

While these approaches offer greater accuracy than low-energy models, they still face challenges:

• High computational cost limits the ability to perform direct material-specific calculations and compare with experiments.
• Certain physical effects, such as doping effects and layer-dependent changes, remain difficult to fully capture.

 

3.2 Unresolved Problems

Despite extensive research, a complete understanding of cuprate superconductivity has yet to be achieved. Several key issues remain unresolved:

(1) Uncertainty in Determining the Superconducting Critical Temperature (Tc)

Accurately predicting how material structure and composition influence Tc is still challenging.

• The relationship between pressure, layer number, and Tc is experimentally observed but not fully explained by existing theories.
• Quantitative modeling of Tc in relation to superexchange strength and electron correlation effects is needed.

(2) Superconducting Pairing Mechanism in Cuprates

• Unlike conventional superconductors, where superconductivity arises from electron-phonon interactions (as described by BCS theory), cuprates exhibit spin correlations and strong electron-electron interactions as primary factors.
• A clearer understanding is needed of how d-wave superconductivity originates and how spin and charge density waves (SDW & CDW) interact with superconducting states.

(3) Doping Effects and Challenges in Predicting Material-Specific Superconducting Properties

• Cuprate superconductors exhibit maximum Tc at a specific optimal doping level (~10–15%), but different doping methods (e.g., oxygen doping vs. chemical substitution) result in variations in superconducting properties.
• It remains difficult to quantitatively predict the optimal doping level for specific cuprate materials.

(4) Long-Range Interactions and Topological Effects

• Existing computational models primarily focus on local electronic interactions, but long-range correlation effects may play an essential role in superconductivity.
• There is growing speculation that topological superconductivity could exist in cuprates, requiring further quantitative analysis.

 

3.3 Contributions and Solutions Proposed by This Study

To address the limitations of previous research, this study utilizes an Ab Initio Quantum Many-Body approach to resolve the following issues:

(1) Quantitative Analysis of Factors Determining Tc

• By computing superconducting pairing parameters and superconducting gaps under different pressure and layer configurations, we identify the factors driving Tc variation.
• The relationship between superexchange interactions, electron correlations, and Tc is analyzed to predict material-specific superconducting properties.

(2) Microscopic Analysis of the Superconducting Pairing Mechanism

• DMET-based calculations are used to identify key quantum effects driving superconducting pairing.
• Multi-orbital effects are explicitly considered to overcome the limitations of traditional low-energy models.

(3) Investigating Doping Effects and Charge Density Variation

• This study aims to predict the optimal doping levels in cuprate superconductors and analyze how different doping strategies impact superconducting properties.

(4) Theoretical Framework for Designing New High-Temperature Superconductors

• By comparing computational results with experimental data, the reliability of this study's findings is validated.
• This research establishes a framework for discovering new superconducting materials based on first-principles quantum many-body calculations.

 

4. Methodology and Approach

This study employs the Ab Initio Quantum Many-Body approach to quantitatively analyze the superconducting properties of cuprate superconductors. Specifically, a combination of Density Matrix Embedding Theory (DMET) and Coupled Cluster Singles and Doubles (CCSD) method is used to address unresolved issues in previous research and compare computational results with experimental findings.

 

Overview of Research Methodology

1. Electron Structure Analysis Based on DMET

• DMET is used to accurately capture electronic correlations in cuprate superconductors by computing embedded electronic structures.
• The superconducting pairing order parameter (κ) and superconducting gap (Δ) are extracted to quantitatively analyze the emergence of superconductivity.

2. Superconducting Pairing Calculation Based on CCSD

• CCSD is employed to predict accurate electronic states and superconducting pairing by incorporating many-body electron correlations.
• Computational results replicate experimentally observed pressure effects (Pressure Effect) and layer-dependent superconductivity (Layer Effect).

3. Analysis of Superconducting Property Variations

• Superconducting pairing strength and superconducting gap changes are analyzed under different pressure conditions (-19 GPa, 0 GPa, 32 GPa).
• A comparison is made between single-layer (Hg-1201) and multi-layer (Hg-1212) cuprates to examine how superconducting properties evolve with the number of layers.

 

5. Analysis of Pressure Effect

Pressure is a crucial factor in enhancing superconducting pairing strength in cuprate superconductors. In this study, CaCuO₂ (CCO) was used as a model system to analyze the effects of pressure variation on superconducting properties.

 

5.1 Key Findings

• Superconducting pairing strength (κ) and superconducting gap (Δ) increase with increasing pressure.
• In CCO, d-wave superconducting pairing strength increases by approximately 50% when pressure is raised from 0 GPa to 32 GPa.
• The superconducting gap (Δ) follows a pattern similar to experimentally measured Tc variations, reaching its maximum at 32 GPa.
• The computed trend closely matches experimental observations of dTc/dP ≈ 4 – 5 K/GPa, confirming agreement between theoretical calculations and experimental data.

 

5.2 Interpretation and Analysis

• Increasing pressure reduces the Cu-O bond length, leading to an increase in superexchange interaction strength, which enhances spin correlations and superconducting pairing.
• Experimental data also show that Tc in Hg-1223 increases from 135K to 164K under 30 GPa, consistent with our computational results.

 

6. Analysis of Layer Effect

To examine the impact of the number of layers on Tc in cuprate superconductors, this study compared Hg-1201 (1-layer) and Hg-1212 (2-layer) systems.

 

6.1 Key Findings

• Superconducting pairing strength (κ) and superconducting gap (Δ) increase from Hg-1201 (1-layer) to Hg-1212 (2-layer).
• The maximum superconducting gap is Eg ≈ 0.03 eV for Hg-1201 and Eg ≈ 0.045 eV for Hg-1212.
• Experimentally, Tc follows a similar trend, with Hg-1201 (Tc ≈ 97K) < Hg-1212 (Tc ≈ 127K).

 

6.2 Interpretation and Analysis

• In multilayer structures, increased CuO₂ plane interactions enhance superexchange strength, thereby strengthening superconducting pairing.
• However, experimental studies indicate that when the number of layers exceeds three, Tc saturation or reduction occurs. This phenomenon is likely due to hole imbalance and changes in charge distribution.
• CCO (∞-layer) calculations show superconducting pairing strength similar to Hg-1212, supporting the hypothesis that Tc decreases in infinite-layer structures.

 

7. Key Variable Analysis in Cuprate Superconductors

To explain how pressure and layer structure influence superconducting pairing strength, key electronic variables in cuprate superconductors were analyzed.

 

7.1 Key Variables and Correlations

• Superexchange Strength (J) vs. Superconducting Pairing (κ): Higher J values correspond to stronger superconducting pairing.
• Oxygen Hole Density (ΔnO) vs. Superconducting Pairing (κ): An increase in ΔnO enhances superconductivity.
• Cu-O Bond Order vs. Superconducting Pairing (κ): Stronger Cu 3d – O 2p bonding correlates with improved superconductivity.

 

7.2 Interpretation and Analysis

• The pressure effect is primarily due to an increase in superexchange strength (J).
• The layer effect is influenced by oxygen hole density (ΔnO) and multi-orbital effects.
• In Hg-1212, higher oxygen hole density results in stronger superconducting pairing, consistent with experimental findings showing that increased oxygen doping raises Tc.

 

8. Microscopic Analysis: Quantum Factors Driving Superconducting Pairing

To investigate the microscopic origins of superconducting pairing, DMET-based quantum electronic correlation analysis was conducted.

 

8.1 Key Findings

• Strong correlation was observed between spin interactions and superconducting pairing (κ).
• Multi-orbital effects contribute significantly to the enhancement of superconductivity.
• Superconducting pairing cannot be explained solely by a single-band Hubbard model, requiring consideration of multi-orbital interactions involving Cu 3d, O 2p, and Cu 4d orbitals.

 

8.2 Interpretation and Analysis

1. Superconducting pairing is determined by multi-band interactions and superexchange effects rather than a simple one-band model.
2. Materials with stronger Cu-O bond order exhibit higher superconducting properties, with increased spin-orbital coupling also playing a crucial role.

 

Summary (2)

This study quantitatively analyzed the superconducting properties of cuprate superconductors using the Ab Initio Quantum Many-Body approach.

• Superconducting pairing strength (κ) and superconducting gap (Δ) increase with pressure, aligning with experimental data.
• Layer-dependent superconducting pairing enhancement is associated with multi-orbital effects and increased oxygen hole density.
• Superconducting pairing is primarily driven by multi-orbital and superexchange effects rather than a simple Hubbard model.

These findings suggest that the proposed approach can effectively predict superconducting properties in cuprates and contribute to the design of new high-temperature superconducting materials.

 

9. Future Research Directions and Applications

This study utilized the Ab Initio Quantum Many-Body approach to analyze the pressure effect and layer effect in cuprate superconductors, quantitatively predicting variations in superconducting pairing (κ) and superconducting gap (Δ). These findings not only contribute to understanding the fundamental mechanisms of superconductivity but also serve as a design strategy for developing new high-temperature superconductors. This section discusses potential future research directions and industrial applications.

 

9.1 Future Research Directions

(1) More Precise Studies on the Superconducting Pairing Mechanism

• While this study focused on d-wave superconducting pairing, additional research is needed to explore correlations with s-wave and other anisotropic superconducting states observed experimentally.
• The model should be expanded to include additional factors such as magnetic field effects, topological superconductivity, and spin-orbit coupling.

(2) Investigating Superconductivity Variations Based on Doping Methods

• Experimental studies have shown that different doping methods (e.g., oxygen doping vs. rare-earth element substitution) significantly affect Tc.
• Beyond the doping methods considered in this study, further analysis of various charge transfer mechanisms is necessary to provide a more precise explanation of the relationship between doping and superconductivity.
• A critical future research topic will be comparing electron doping and hole doping using the Ab Initio approach.

(3) Inhomogeneous Doping and Defects

• In real experiments, achieving ideal homogeneous doping is challenging, and local defects and inhomogeneous doping could impact superconductivity.
• Ab Initio simulations that incorporate these factors will be crucial in constructing a more realistic superconducting material model.

(4) Exploring New Materials for Room-Temperature Superconductivity

• Beyond cuprate superconductors, research on nickelates and other 2D transition metal oxides for potential superconductivity is actively progressing.
• The DMET + CCSD methodology used in this study can be extended to predict new superconducting material candidates at the Ab Initio level.

 

9.2 Industrial and Technological Applications

The analysis of the superconducting pairing mechanism, pressure effect, and layer effect in cuprate superconductors can be applied to various industries:

(1) Next-Generation Power Transmission

• High-temperature superconductors can be utilized in lossless superconducting cables and high-current transmission lines.
• Based on the study’s findings, pressure control and optimized doping can be leveraged to design efficient high-temperature superconducting cables.

(2) Superconducting Quantum Computing

• Superconducting qubits, widely used in quantum computing, rely on superconducting materials.
• The findings on superconducting pairing parameters (κ) and charge density variations could contribute to the development of more stable superconducting qubit designs and devices.

(3) Magnetic Levitation and Superconducting Magnetic Sensors

• The Meissner effect in superconductors enables applications such as maglev transportation and high-precision sensors.
• This study's findings suggest that pressure and doping adjustments could lead to the design of new superconductors with higher Tc, enhancing practical applications.

(4) Medical Applications: MRI (Magnetic Resonance Imaging)

• MRI systems utilize strong superconducting magnets, and the development of superconductors with higher Tc could significantly reduce cooling costs.
• By applying the study’s pressure and layer control techniques, more practical and cost-effective superconducting materials can be developed.

 

9.3 Theoretical Framework for Designing New High-Temperature Superconductors

The Ab Initio Quantum Many-Body approach (DMET + CCSD) used in this study serves as a powerful theoretical tool for designing new high-temperature superconductors:

• Unlike conventional approaches that rely on experimental data, this method predicts superconducting properties without experimental parameters.
• By considering various electronic correlation effects, it facilitates the discovery of new high-temperature superconducting candidates that have yet to be identified.
• Future research could incorporate machine learning and AI to accelerate and refine superconducting material design.

 

10. Conclusion

This study quantitatively analyzed the superconducting properties of cuprate superconductors using the Ab Initio Quantum Many-Body approach (DMET + CCSD). Previous studies have struggled to fully explain the superconducting mechanism due to the difficulty in accurately capturing strong electronic correlation effects. This research overcomes these challenges by introducing a new computational approach.

 

10.1 Key Findings

1. Analysis of the Pressure Effect

• Superconducting pairing strength (κ) and superconducting gap (Δ) increase with pressure (0 GPa → 32 GPa).
• The computed trend closely matches experimental observations of dTc/dP ≈ 4 – 5 K/GPa.
• The enhancement in superconductivity is attributed to the reduction in Cu-O bond length and the increase in superexchange interactions.

2. Analysis of the Layer Effect

• Superconducting pairing strength (κ) and superconducting gap (Δ) are greater in Hg-1212 (2-layer) than in Hg-1201 (1-layer).
• The computed trend is consistent with experimental findings: Hg-1201 (Tc ≈ 97K) < Hg-1212 (Tc ≈ 127K).
• While Tc increases with additional layers, further layer increase results in Tc saturation or reduction due to inhomogeneous doping and charge redistribution.

3. Identification of the Superconducting Pairing Mechanism

• Superconducting pairing cannot be explained by a simple one-band Hubbard model; multi-orbital effects and superexchange interactions play a crucial role.
• An increase in oxygen hole density (ΔnO) and Cu-O bond order strengthens superconductivity.
• Spin correlation is a driving factor for superconducting pairing, consistent with experimental observations.

 

10.2 Contributions of this Study

This study presents a novel computational approach that allows for a more precise understanding of superconducting behavior in cuprate superconductors. Specifically:

• It overcomes the limitations of DFT-based approaches, enabling accurate quantitative predictions of superconducting pairing strength (κ) and superconducting gap (Δ).
• It validates computational results by comparing superconducting property variations due to pressure and layer structure with experimentally verified data.
• It establishes a theoretical framework that can be applied to the discovery and design of new high-temperature superconducting materials.

 

10.3 Future Research Directions

Extending this approach to other cuprate superconductors and different superconducting families (e.g., nickelates, 2D oxides).

Investigating the impact of different doping methods (oxygen doping vs. rare-earth substitution) on superconductivity.

Exploring new materials for quantum computing and room-temperature superconductivity.

This research not only enhances the fundamental understanding of the superconducting mechanism in cuprate superconductors but also serves as a critical theoretical foundation for advancing high-temperature superconductivity research and practical applications.

 

What kind of new future did this article inspire you to imagine? Feel free to share your ideas and insights in the comments! I’ll be back next time with another exciting topic. Thank you! 😊

Reference

1) Cui, Z.-H., Yang, J., Tölle, J., Ye, H.-Z., Yuan, S., et al. Ab initio quantum many-body description of superconducting trends in the cuprates. Nature Communications, 16, 1845 (2025). https://doi.org/10.1038/s41467-025-56883-x