On the theory of supermodulation of the superconducting order parameter created by structural supermodulation of apex distance in optimally doped Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$

This theoretical study using a five-orbital LCAO approximation and Kondo double electron exchange successfully reproduces the experimentally observed supermodulation of the superconducting order parameter in Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$, providing strong evidence that super-exchange interactions mediated by Cu4s orbital hybridization constitute the electron pairing mechanism in optimally doped cuprates.

The Gist

The Big Picture: A Mystery in the Superconducting World

Imagine you have a magical material (a high-temperature superconductor) that conducts electricity with zero resistance. Scientists have been trying to figure out how the electrons in this material pair up to do this for decades. It's like trying to figure out the secret handshake that allows a crowd of people to move through a hallway without bumping into each other.

Recently, a team of experimentalists (led by Seamus Davis) used a super-powerful microscope to look at this material. They found something strange: when they slightly changed the distance between a specific atom (an "apex" oxygen) and the copper atoms below it, the strength of the superconductivity changed in a very predictable way.

The Problem: The experimentalists said, "We think this proves our theory about how the electrons pair up." But the scientific community is skeptical. They said, "Show us the math. Prove it."

The Solution (This Paper): Two Bulgarian physicists, Varonov and Mishonov, stepped in. They said, "We can explain exactly why that distance change matters, using standard physics rules." They built a mathematical model that matches the experimental data perfectly, without needing to "fake" any numbers.

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The Creative Analogy: The Trampoline and the Bouncer

To understand their theory, let's use an analogy.

1. The Players

• The Electrons: Imagine them as dancers trying to pair up to waltz across a floor.
• The Copper Floor (CuO2 Plane): This is the dance floor where the magic happens.
• The "Apex" Oxygen: This is a bouncer standing on a ladder above the dance floor.
• The Pairing Mechanism: This is the secret rule that tells the dancers how to hold hands.

2. The Experiment

The experimentalists noticed that if the bouncer (the apex oxygen) moves closer to the dance floor, the dancers pair up better. If the bouncer moves farther away, the pairing gets worse.

The experimentalists concluded: "The bouncer is controlling the dance!" But they didn't explain how.

3. The Authors' Theory: The "Hidden Elevator"

The authors say: "We know exactly how the bouncer controls the dance."

In their model, the copper atoms have a secret "elevator" (a specific orbital called Cu4s) that usually stays hidden or inactive.

• The Bouncer's Job: The apex oxygen acts like a lever. When it moves closer, it pushes down on the copper atom.
• The Effect: This push activates the "elevator" (the Cu4s orbital).
• The Result: Once the elevator is active, it connects with the main dance floor (the Cu3d orbital). This connection creates a super-strong "handshake" (called Kondo-Zener exchange) that forces the electron dancers to pair up efficiently.

The Metaphor:
Think of the electrons as two people trying to shake hands across a wide river.

• Without the bouncer moving, the river is too wide; they can't reach each other.
• When the bouncer moves closer, it drops a bridge (activates the Cu4s orbital).
• Now, the handshake happens easily.
• The authors calculated exactly how much the bridge drops based on the bouncer's height, and it matched the experiment perfectly.

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The "Fitting" Controversy: Did They Cheat?

This is the most dramatic part of the paper.

When the authors submitted their explanation to a top physics journal (Physical Review B), the editors rejected it.

• The Editor's Accusation: "You just tweaked your numbers (fitted parameters) until your line looked like the experimental line. Anyone can do that. You didn't really predict anything."
• The Authors' Defense: "No! We didn't tweak anything! We used the exact same numbers that nature gave us (from other reliable experiments). We didn't change a single number to make it fit. We just did the math, and the math naturally landed on the experimental result."

The Analogy:
Imagine you are trying to predict how far a ball will roll down a hill.

• The Editor says: "You just guessed the speed until it matched the video. That's cheating."
• The Authors say: "We used the exact weight of the ball, the exact angle of the hill, and the exact friction of the grass. We didn't guess. The fact that our calculation matched the video proves our physics is right."

The authors argue that the editor didn't read their paper carefully and accused them of "fitting" when they actually did a "first-principles" calculation (solving it from the ground up).

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Why Does This Matter?

If the authors are right, they have solved a 40-year-old mystery.

1. The "Smoking Gun": They proved that the Cu4s orbital (the hidden elevator) is essential for high-temperature superconductivity. Most other theories ignore this part of the atom.
2. The Mechanism: They identified the specific force (Kondo-Zener exchange) that pairs the electrons. It's not a new, weird magic; it's a known force in physics that was just overlooked in this context.
3. The Future: If this is true, we can design better superconductors by engineering the distance between these atoms, potentially leading to lossless power grids or super-fast computers.

The "Open Letter" Drama

Because the journal rejected them based on what they felt were unfair and incorrect accusations, the authors wrote a very long, passionate "Open Letter" (included in the text you provided).

• They call the editor's decision "unprofessional" and "offensive."
• They accuse the editor of not understanding the science.
• They say, "We are just trying to solve the biggest puzzle in physics, and you are blocking us because you didn't read the fine print."

Summary in One Sentence

Two physicists claim they have finally cracked the code on how high-temperature superconductors work by showing that a specific atom's distance controls a hidden "bridge" between electrons, but they are currently fighting a bitter battle with journal editors who they believe rejected their proof unfairly.

Technical Summary

1. Problem Statement

The paper addresses the long-standing mystery of the electron pairing mechanism in high-temperature cuprate superconductors (specifically hole-doped $\text{Bi}_2\text{Sr}_2\text{CaCu}_2\text{O}_{8+x}$).

• Context: Recent experiments by O'Mahony et al. (PNAS 2022) using Scanning Josephson Tunneling Microscopy (SJTM) observed a "super-modulation" of the superconducting order parameter. This modulation is spatially correlated with the structural super-modulation of the distance ($\delta$) between the planar Copper (Cu) ions and the apical Oxygen (O) ions.
• The Gap: While the experiment suggested a strong correlation between the electron-pair density ($n_p$) and the apical distance $\delta$, a rigorous theoretical explanation linking this observation to a specific microscopic pairing mechanism was missing. The authors aim to provide a first-principles theoretical derivation of this dependence to identify the pairing interaction.

2. Methodology

The authors employ a Linear Combination of Atomic Orbitals (LCAO) approach combined with BCS theory and Kondo-Zener exchange interactions.

• Electronic Structure Model:

  • They construct a 5-orbital Hamiltonian per unit cell: $\text{Cu}4s$, $\text{Cu}3d_{x^2-y^2}$, $\text{O}2p_x$, $\text{O}2p_y$, and $\text{O}2p_z$.
  • The Hamiltonian includes on-site energies ($\epsilon_s, \epsilon_d, \epsilon_p$) and hopping integrals ($t_{sp}, t_{pd}, t_{pp}, t_{as}$).
  • Crucially, the hopping integral between the apical oxygen and the Cu $4s$ orbital ($t_{as}$) is treated as dependent on the apical distance $\delta$ (scaling as $\delta^{-2}$).
  • The model utilizes a re-normalization factor ($Z_\epsilon$) to align the energy scale with experimental ARPES data and effective mass measurements, ensuring the Fermi contour shape matches observations without altering the physics of the pairing mechanism.

• Pairing Mechanism:

  • The authors reject the standard super-exchange ($J_{dd}$) between Cu $3d$ orbitals as the primary pairing glue for superconductivity, arguing it explains antiferromagnetism but not necessarily superconductivity.
  • Instead, they propose the Kondo-Zener $s-d$ double exchange interaction ($J_{sd}$) between the $\text{Cu}4s$ and $\text{Cu}3d_{x^2-y^2}$ orbitals as the dominant pairing mechanism.
  • This interaction is mediated by the in-plane oxygen ligands. The authors argue that $J_{sd}$ is significantly larger than $J_{dd}$ and possesses an antiferromagnetic sign, leading to singlet pairing.

• Theoretical Framework:

  • The pairing interaction is treated within the BCS formalism. The gap equation is solved for a separable potential $V_{p,q} = -2J_{sd}\chi_p\chi_q$, where $\chi_p$ represents the gap anisotropy function derived from the orbital mixing.
  • The superconducting order parameter $\Xi(T)$ is calculated, and the zero-temperature order parameter $\Xi_0$ is used to define the electron-pair density $n_p \propto \Xi_0^2$.

3. Key Contributions

• First-Principles Derivation: The paper claims to calculate the dependence of the electron-pair density on the apical distance ($n_p(\delta)$) without any fitting parameters related to the specific experimental data being explained. All parameters are derived from ab initio LDA calculations and standard tight-binding values.
• Identification of the Pairing Mechanism: The study explicitly links the modulation of the apical distance to the modulation of the $\text{Cu}4s$ energy level. It concludes that the $\text{Cu}4s$ orbital is not a passive spectator but an active participant in the pairing mechanism via the Kondo $s-d$ exchange.
• Definition of a Dimensionless Test Parameter: The authors introduce the logarithmic derivative $Q_{np} \equiv (\delta/n_p) (dn_p/d\delta)$ as the critical dimensionless parameter to test pairing theories. They argue this parameter is the "crucial experiment" metric for cuprates, analogous to the isotope effect in phonon-mediated superconductors.

4. Results

• Quantitative Agreement: The theoretical calculation of $n_p(\delta)$ yields a linear dependence that matches the slope of the experimental data from O'Mahony et al. (Fig. 5C of the PNAS paper) with acceptable accuracy (within ~10%).
• Prediction of Non-Linearity: The theory predicts a small negative curvature in the $n_p$ vs. $\delta$ relationship, which is not yet resolved in the current experimental data but is a specific testable prediction of the model.
• Correlation with $T_c$: The model successfully explains the correlation between the Fermi contour shape and the critical temperature ($T_c$) across different cuprate families, reinforcing the idea that the $\text{Cu}4s$ hybridization strength controls $T_c$.
• Validation of the "Crucial Experiment": The authors conclude that the SJTM experiment confirms the hypothesis that the pairing mechanism involves the exchange of electrons between $\text{Cu}4s$ and $\text{Cu}3d_{x^2-y^2}$ orbitals.

5. Significance and Controversy

• Scientific Impact: If validated, this work provides a definitive solution to the pairing mechanism in cuprates, shifting the paradigm from pure $d$-wave super-exchange to a Kondo-like $s-d$ exchange involving the $4s$ orbital. It suggests that the apical oxygen distance acts as a "tuning knob" for the pairing strength by modulating the $\text{Cu}4s$ level.
• Publication Context (The Appeal): The paper is accompanied by a detailed account of its rejection by Physical Review B (PRB) and npj Quantum Materials.
  • Rejection Reasons: Reviewers claimed the work was "phenomenological," relied on "fitting parameters," and lacked testable predictions. They also criticized the writing style and the novelty of the insight that $T_c$ depends on apical distance.
  • Authors' Defense: The authors argue the rejection was based on misunderstandings (e.g., they used ab initio parameters, not fits) and that the requirement to test "other Hamiltonians" is scientifically invalid for a "crucial experiment" that has a unique solution. They assert that the paper represents a fundamental ab initio calculation, not a phenomenological fit.
• Conclusion: The paper presents a robust theoretical framework that quantitatively explains a key experimental anomaly in high-$T_c$ superconductivity, proposing the Kondo $s-d$ exchange as the missing link in the pairing mechanism, while simultaneously highlighting a conflict between the authors and the editorial policies of major physics journals regarding the evaluation of "crucial" theoretical breakthroughs.