Hierarchical structure of primary and hybridization-induced superconducting correlations in bilayer nickelates

Using variational Monte Carlo simulations on a bilayer two-orbital Hubbard model, this study reveals that the high-temperature superconductivity in La$_3$Ni$_2$O$_7$ arises from a hierarchical pairing structure where the primary interaction stems from Ni $3d_{z^2}$ orbital splitting, while orbital hybridization redistributes these correlations to the $d_{x^2-y^2}$ channel, thereby reconciling competing theoretical scenarios and explaining the robustness of the $s_{\pm}$ state.

The Gist

The Big Picture: A New Kind of Superconductor

Imagine a material called La₃Ni₂O₇ (a type of nickelate) that has recently been discovered to conduct electricity with zero resistance (superconductivity) at very high temperatures—close to 80 Kelvin. This is a big deal because it rivals the famous copper-based superconductors (cuprates) but works under high pressure.

Scientists have been arguing about how this happens. Is it driven by the electrons in one specific layer? Or is it a mix? This paper solves that mystery by showing that the superconductivity has a hierarchical structure—like a building with a strong foundation and a decorated top floor.

The Cast of Characters: The "Orbitals"

To understand this, we need to meet the main characters: the electrons living in specific "rooms" called orbitals. In this material, there are two main types of rooms:

1. The $z^2$ Room: These electrons live in a vertical, dumbbell-shaped room that connects the top and bottom layers of the material. They are very good at jumping between layers.
2. The $x^2-y^2$ Room: These electrons live in a flat, horizontal room. They are great at moving sideways within a single layer but are shy about jumping to the other layer.

The Discovery: The "Primary Driver" vs. The "Redistributor"

The authors used a powerful computer simulation (Variational Monte Carlo) to figure out what's going on. Here is what they found, broken down into a story:

1. The Foundation: The $z^2$ Room is the Engine

Think of the $z^2$ electrons as the engine of a car.

• Because these electrons can easily jump between the top and bottom layers, they create a strong "bonding" and "antibonding" split (like two gears meshing perfectly).
• This splitting creates a powerful attractive force that pairs electrons up. This is the primary reason superconductivity starts. Without this engine, the car doesn't move.
• In the paper's language, this is the "primary pairing interaction."

2. The Confusion: Why is the $x^2-y^2$ Room also Superconducting?

Here is where it gets tricky. Other scientists noticed that the $x^2-y^2$ electrons (the flat room) also seemed to be pairing up strongly, even though they don't have a strong engine of their own.

• The Analogy: Imagine a dance floor. The $z^2$ electrons are the professional dancers who know the steps perfectly and are leading the dance. The $x^2-y^2$ electrons are the beginners who don't know the steps and have no music of their own.
• The Twist: Even though the beginners have no music, they end up dancing perfectly in sync with the pros. Why? Because the two groups are holding hands (this is called orbital hybridization).
• The strong "engine" of the $z^2$ electrons pulls the $x^2-y^2$ electrons along. The $z^2$ electrons do the heavy lifting, but the $x^2-y^2$ electrons get swept up in the rhythm and end up dancing just as well.

The "Hierarchical Structure" Explained

The paper calls this a hierarchical structure:

• Level 1 (The Origin): The superconductivity is born in the $z^2$ channel because of the strong connection between layers.
• Level 2 (The Result): Through the "hand-holding" (hybridization), the superconducting "dance" is redistributed to the $x^2-y^2$ channel.
• The Result: Even though the $x^2-y^2$ electrons didn't start the party, they end up with just as much "dance energy" (superconducting correlation) as the $z^2$ electrons.

Why This Matters: The "Ghost" in the Machine

One of the biggest debates in the field was about the shape of the "Fermi surface" (the map of where electrons live). Some scientists thought if a specific part of this map (the $\gamma$ sheet) disappeared, superconductivity would vanish.

• The Paper's Verdict: It doesn't matter if that specific part of the map disappears!
• The Analogy: Imagine you are driving a car. Some people argued that if you lose a specific tire (the $\gamma$ sheet), the car stops. But this paper shows that the car is actually driven by the engine (the $z^2$ bonding). Even if you lose a tire, as long as the engine is running and the other wheels are connected, the car keeps driving.
• This explains why the superconductivity is so robust (stable) and doesn't easily break when the material's structure changes slightly.

Summary in One Sentence

The superconductivity in this nickelate is started by the strong vertical connections of the $z^2$ electrons, but spread out to the flat $x^2-y^2$ electrons through a "hand-holding" effect, creating a stable, high-temperature superconductor that works even if the material's shape changes.

The Takeaway for the Future

This discovery tells us that in complex materials, you can't just look at the "visible" parts (like the Fermi surface map). You have to look at the hidden connections (hybridization) between different types of electrons. It's like realizing that a successful team isn't just about the star player, but about how the whole team passes the ball to each other.

Technical Summary

1. Problem Statement

The discovery of high-temperature superconductivity ($T_c \approx 80$ K) in the bilayer nickelate $\text{La}_3\text{Ni}_2\text{O}_7$ under high pressure has sparked intense theoretical debate regarding its microscopic origin. While an $s_{\pm}$ gap symmetry is widely accepted, the specific electronic degrees of freedom driving the pairing remain unresolved. Key ambiguities include:

• Orbital Origin: Is superconductivity driven primarily by the $3d_{z^2}$ orbitals (via interlayer bonding-antibonding splitting) or the $3d_{x^2-y^2}$ orbitals (often associated with cuprate-like physics)?
• Role of Hybridization: How does strong inter-orbital hybridization between $z^2$ and $x^2-y^2$ orbitals reshape the pairing mechanism?
• Fermi Surface Sensitivity: Some theories suggest the pairing is sensitive to the topology of Fermi surface sheets (specifically the $\gamma$ sheet derived from $z^2$), while others argue for robustness against such changes.
• Contradictory Observations: Different theoretical approaches (e.g., CDMFT, ladder models) yield conflicting results regarding which orbital channel exhibits the larger superconducting order parameter or gap.

2. Methodology

The authors employ a Variational Monte Carlo (VMC) method to investigate a bilayer two-orbital Hubbard model. This non-perturbative approach allows for a unified analysis of the superconducting gap structure and long-range correlations.

• Model: The Hamiltonian includes $z^2$ and $x^2-y^2$ orbitals on a bilayer lattice.
  • Kinetic Energy: Parameters are derived from first-principles calculations for $\text{La}_3\text{Ni}_2\text{O}_7$ at 20 GPa. The model features interlayer hopping ($t_\perp$) for $z^2$ orbitals and inter-orbital hybridization.
  • Tuning Parameter: The ratio $\Delta E / t_\perp$ (where $\Delta E$ is the energy offset between orbitals) is treated as a tunable parameter to mimic correlation-induced renormalization of orbital hierarchy.
  • Interactions: Includes intra- and inter-orbital Coulomb repulsion ($U, U'$), Hund's coupling ($J$), and pair-hopping ($J'$).
• Wave Function: A Gutzwiller-Jastrow type trial wave function $|\Psi\rangle = P_G^{(2)} P_{Jc} P_{Js} |\Phi\rangle$ is used.
  • $|\Phi\rangle$ is the ground state of an effective Bogoliubov–de Gennes (BdG) Hamiltonian containing variational pairing fields ($\tilde{\Delta}_{zz}, \tilde{\Delta}_{xx}$).
  • $P_G^{(2)}$ accounts for local charge/spin correlations in the two-orbital system.
• Observables:
  • Variational Gap Parameters ($\tilde{\Delta}$): Represent the strength of the local pairing field.
  • Long-range Correlation Functions ($P_{mm}$): Measure the spatial propagation of superconducting order in specific orbital channels ($z^2$ and $x^2-y^2$).
  • Orbital-Resolved Density of States (DOS): Analyzes the spectral weight distribution near the Fermi level.

3. Key Contributions

The paper establishes a hierarchical structure of superconductivity that reconciles competing theoretical scenarios:

1. Decoupling of Pairing Origin and Correlation Strength: The study demonstrates that the origin of the pairing interaction (the "driver") is distinct from the distribution of superconducting correlations (the "result").
2. Mechanism Clarification: It identifies the bonding-antibonding splitting of the $z^2$ orbitals as the primary driver of pairing, while orbital hybridization acts as a redistribution mechanism that induces correlations in the $x^2-y^2$ channel.
3. Robustness: The superconducting state is shown to be robust against changes in Fermi surface topology (e.g., the disappearance of the $\gamma$ sheet), challenging theories that rely solely on Fermi surface nesting.

4. Key Results

A. Interaction-Driven Electronic Reorganization

• As $\Delta E / t_\perp$ increases, the $z^2$ orbital occupancy ($n_{z^2}$) becomes pinned near half-filling due to strong electron correlations, indicating a redistribution of charge between orbitals.
• The $\gamma$ Fermi surface sheet (purely $z^2$-derived) shrinks and eventually disappears for large $\Delta E / t_\perp$, while the $\beta$ sheet is strongly renormalized.

B. Hierarchical Pairing Structure

• Primary Driver ($z^2$ Channel): The variational gap parameter $\tilde{\Delta}_{zz}$ becomes finite and grows with $\Delta E / t_\perp$, indicating that the primary attractive interaction arises from the interlayer bonding-antibonding splitting of the $z^2$ orbitals.
• Secondary Channel ($x^2-y^2$): The variational gap parameter $\tilde{\Delta}_{xx}$ remains negligible (effectively zero). This implies there is no intrinsic attractive pairing interaction in the $x^2-y^2$ channel.
• Redistribution of Correlations: Despite $\tilde{\Delta}_{xx} \approx 0$, the long-range superconducting correlation function $P_{xx}$ becomes comparable to, and in some regimes larger than, $P_{zz}$.
  • Explanation: Strong orbital hybridization redistributes the superconducting correlations generated in the $z^2$ channel to the $x^2-y^2$ channel. The magnitude of correlations in a channel is governed by the orbital character of the low-energy density of states, not just the pairing interaction strength.

C. Gap Structure and Symmetry

• The resulting state is an $s_{\pm}$ superconductor.
• Band Representation:
  • $\gamma$-$\delta$ Bands ($z^2$-dominated): Exhibit large, nearly momentum-independent gaps. This confirms the primary pairing occurs here.
  • $\alpha$-$\beta$ Bands (Hybridized): Exhibit strong momentum dependence. The gap vanishes along $k_x = \pm k_y$ lines where hybridization is symmetry-forbidden. This confirms the gap in these bands is induced by hybridization, not intrinsic pairing.
• Robustness: The gap structure and superconducting state remain stable even when the $\gamma$ Fermi surface sheet disappears, proving the mechanism is not dependent on specific Fermi surface nesting conditions.

D. Orbital-Resolved Density of States

• The total $z^2$ DOS remains finite even after the $\gamma$ sheet vanishes because $z^2$ character persists on the hybridized $\alpha$ and $\beta$ bands.
• The $x^2-y^2$ DOS increases with $\Delta E / t_\perp$, correlating with the growth of $P_{xx}$. This confirms that the strength of correlations in the $x^2-y^2$ channel is dictated by the availability of low-energy states with $x^2-y^2$ character, which are populated via hybridization.

5. Significance

• Resolution of Theoretical Conflict: The findings explain why different studies report conflicting dominant orbital characters. The "driving force" is $z^2$-based, but the "observable order" is shared between $z^2$ and $x^2-y^2$ due to hybridization.
• Beyond Cuprate Paradigms: The mechanism highlights the unique role of interlayer physics and orbital hybridization in bilayer systems, distinguishing them from single-layer cuprates where $d$-wave pairing is often linked to specific Fermi surface nesting.
• Stability of High-$T_c$: The insensitivity of the pairing state to Fermi surface topology (e.g., the presence/absence of the $\gamma$ sheet) provides a microscopic basis for the observed stability of superconductivity in $\text{La}_3\text{Ni}_2\text{O}_7$ under varying pressure conditions.
• General Implications: This hierarchical framework offers a new perspective for understanding multilayer correlated superconductors, emphasizing that orbital hybridization can decouple the origin of pairing from the spatial distribution of the superconducting order parameter.