Fermi liquid and isotropic superconductivity of Hund scenario for bilayer nickelates

Using a dynamic Schwinger boson approach, this study demonstrates that while the Hund's coupling scenario in bilayer nickelates predicts isotropic $s$-wave superconductivity and Fermi liquid normal states, it fails to fully account for experimental observations, suggesting that this mechanism alone is insufficient to explain the material's high-$T_c$ superconductivity.

The Gist

Imagine a superconductor as a busy dance floor where electrons usually bump into each other and lose energy (creating resistance). In a superconductor, these electrons pair up and glide across the floor without any friction. Scientists recently discovered a new type of dance floor made of "bilayer nickelates" (a specific material with two layers of nickel atoms) that allows this friction-free dancing at surprisingly high temperatures.

However, there is a debate about how the electrons find their partners. The paper you provided investigates two different theories on how this pairing happens, using a specific material called La₃Ni₂O₇ as the test case.

Here is the breakdown of the two theories and what the authors found, explained simply:

The Two Competing Theories

Think of the electrons in this material as living in two different neighborhoods:

1. The "dx²-y²" Neighborhood: These are the main dancers, moving freely around.
2. The "dz²" Neighborhood: These are the local residents, sitting more still in the background.

Theory A: The "Hybridization" Scenario (The Handshake)
In this view, the two neighborhoods are connected by a bridge. The "dz²" residents and "dx²-y²" dancers constantly shake hands and swap places (hybridize). This constant interaction creates a strong glue that helps the dancers pair up.

• What the paper says: This theory predicts that the dance floor should look a bit messy (non-Fermi liquid) with some dancers moving in a chaotic, linear pattern. It also predicts a very high maximum temperature for superconductivity.

Theory B: The "Hund's Coupling" Scenario (The Coach)
In this view, the "dz²" residents are very stubborn and stay in their own houses (localized). They don't shake hands with the dancers. Instead, they act like a strict coach (Hund's coupling) who yells instructions to the dancers, telling them how to pair up.

• What the paper says: The authors tested this "Coach" theory using advanced math (Schwinger boson approach).

What the "Coach" Theory (Hund Scenario) Actually Predicted

When the authors ran their calculations for the "Coach" theory, they found three major things that didn't quite match the real-world experiments:

1. The Dance is Too Perfect (Isotropic Gap):
   The "Coach" theory predicts that the electrons pair up in a perfectly uniform circle (isotropic s-wave). Imagine a dance where everyone holds hands in a perfect, smooth circle.

   • Reality Check: Experiments show the dance is actually a bit uneven (anisotropic), with some spots on the floor being "tighter" than others.

2. The Party Ends Sooner (Lower Tc):
   The "Coach" theory predicts the superconducting party (the high-temperature state) ends at a much lower temperature than the "Handshake" theory does.

   • Reality Check: The real material stays superconducting at higher temperatures than the "Coach" theory allows.

3. The Crowd is Too Calm (Fermi Liquid):
   The "Coach" theory predicts that before the dancing starts, the electrons behave like a calm, orderly crowd (Fermi liquid).

   • Reality Check: In the bulk (thick) version of the material, the electrons behave like a chaotic, rushing crowd (non-Fermi liquid) before they start dancing.

The "Hole" in the Theory

The authors also tested what happens if you add "holes" (empty spots) to the "dz²" neighborhood.

• In the "Handshake" theory, adding holes helps the pairing.
• In the "Coach" theory, adding holes actually kills the superconductivity. The more holes you add, the less likely the electrons are to pair up. This contradicts what is seen in some thin-film experiments.

The Final Verdict

The authors conclude that while the "Coach" (Hund's coupling) theory explains some things (like why thin films act calmly), it cannot explain the whole story on its own.

• It fails to explain the chaotic behavior seen in the thick bulk material.
• It fails to explain the uneven shape of the electron pairs seen in experiments.
• It predicts the superconductivity dies out too quickly.

The Bottom Line: The paper suggests that the "Coach" theory is not the whole answer. The "Handshake" (Hybridization) theory seems to fit the experimental data much better, explaining both the chaotic and calm behaviors, as well as the high temperatures. However, the authors suggest that in the real world, both the "Coach" and the "Handshake" might be working together to get the electrons to dance.

Technical Summary

Technical Summary: Fermi liquid and isotropic superconductivity of Hund scenario for bilayer nickelates

Problem Statement
Recent experimental discoveries of high-$T_c$ superconductivity in bulk bilayer nickelate $La_3Ni_2O_7$ (under pressure) and compressively strained thin films (at ambient pressure) have necessitated a clarification of the pairing mechanism. While both systems host dominant $3d_{z^2}$ and $3d_{x^2-y^2}$ bands near the Fermi level, they exhibit distinct normal state properties: bulk $La_3Ni_2O_7$ displays linear-in-$T$ resistivity (non-Fermi liquid, NFL), whereas thin films exhibit Fermi liquid (FL) transport. Theoretical debates persist regarding whether the primary driver of superconductivity is the inter-orbital hybridization between $d_{x^2-y^2}$ and $d_{z^2}$ orbitals or the Hund's coupling between them. Previous studies have largely treated these scenarios separately, often using different theoretical frameworks, making a direct comparison difficult.

Methodology
The authors employ a unified theoretical framework to compare the "Hund scenario" against the previously studied "hybridization scenario."

• Model: They utilize a bilayer two-orbital $t$-$J_H$-$J$ model defined on a square lattice. The Hamiltonian includes in-plane $d_{x^2-y^2}$ hopping ($t$), inter-orbital Hund's coupling ($J_H$), and interlayer $d_{z^2}$ superexchange ($J$). The model assumes localized $d_{z^2}$ spins and ignores weaker intralayer superexchange and Coulomb repulsion in the quarter-filled $d_{x^2-y^2}$ orbital to focus on the interplay between $J_H$ and $J$.
• Approach: The study uses the dynamic Schwinger boson approach. The $d_{z^2}$ spin operators are represented by Schwinger bosons, and the superexchange term is decomposed into an interlayer valence bond field ($\Delta$). The Hund's coupling term is factorized by introducing a fermionic holon field ($\chi$).
• Calculation: The authors solve the self-consistent equations for mean-field variables ($\Delta$ and Lagrange multipliers $\lambda$) and Green's functions. To determine the superconducting instability, they calculate the pairing vertex $\Gamma$ for $d_{x^2-y^2}$ electrons using the Bethe-Salpeter equation, incorporating self-energies for electrons, spinons, and holons.

Key Contributions and Results
The study reveals fundamental differences between Hund-driven and hybridization-driven superconductivity, summarized as follows:

1. Gap Structure:

   • Hund Scenario: The pairing vertex for $d_{x^2-y^2}$ electrons is momentum-independent, leading to a fully isotropic $s$-wave gap.
   • Hybridization Scenario (Reference [39]): Predicts an anisotropic $s_{\pm}$-wave gap with nodes or minima along the zone diagonal on the $d_{x^2-y^2}$ Fermi surfaces ($\alpha$ and $\beta$ bands).

2. Critical Temperature ($T_c$):

   • Magnitude: The maximum transition temperature in the Hund scenario ($T_c^{max}/J \approx 0.03$ after accounting for phase fluctuations) is approximately 40% lower than that of the hybridization scenario ($T_c^{max}/J \approx 0.05$).
   • Dependence on $d_{z^2}$ Holes: In the Hund scenario, $T_c$ is monotonically suppressed by $d_{z^2}$ hole doping ($\delta_d$). This contrasts sharply with the hybridization scenario, where $d_{z^2}$ metallization promotes superconductivity.
   • Mechanism: A finite Hund's coupling is required to transmit interlayer pairing to itinerant $d_{x^2-y^2}$ electrons. However, the lack of intrinsic short-time dynamics in the Hund-induced holon field leads to a small quasiparticle weight, suppressing the pairing vertex compared to the hybridization mechanism.

3. Normal State Properties:

   • Hund Scenario: The normal state is consistently a Fermi liquid (FL) for all values of $J_H$. The quasiparticle weight $Z_k$ decreases monotonically but remains finite, and no pseudogap or strange metal behavior is observed.
   • Hybridization Scenario: Predicts a non-Fermi liquid (NFL) normal state with linear-in-$T$ scattering rates around optimal $J$.
   • Theoretical Origin: The authors attribute this distinction to renormalization group flow: inter-orbital hybridization (Kondo-type) is enhanced, leading to NFL behavior, whereas local Hund's coupling (ferromagnetic Kondo-type) flows to a weak coupling fixed point, preserving the FL ground state.

4. Phase Diagram:

   • Superconductivity emerges only when $J_H$ exceeds a critical value ($J_H^c$).
   • $T_c$ increases rapidly with $J_H$ before saturating, following a linear relation with interlayer superexchange $J$ at large $J_H$.
   • The phase boundary between the metallic and superconducting states shifts significantly with $d_{z^2}$ hole doping, showing an abrupt change near the optimal superexchange $J_{opt}$.

Significance and Claims
The paper concludes that the Hund scenario alone is insufficient to explain the superconductivity observed in both bulk and thin-film bilayer nickelates.

• Mismatch with Bulk: The Hund scenario fails to account for the NFL behavior and linear-in-$T$ resistivity observed in bulk $La_3Ni_2O_7$ under high pressure.
• Mismatch with Thin Films: While the Hund scenario predicts an FL normal state (consistent with thin films), it fails to explain the experimentally observed anisotropic $s_{\pm}$-wave gap structure (with gap minima on the $\beta$ band). The authors argue that the existence of $\alpha$ and $\beta$ Fermi surfaces inherently requires hybridization with $d_{z^2}$ quasiparticle bands, which the pure Hund scenario (assuming localized $d_{z^2}$) cannot fully reproduce.
• Unified View: The authors suggest that while the Hund scenario provides a mechanism for FL behavior, the hybridization scenario offers a more natural and unified explanation for the full range of experimental observations, including the gap anisotropy, the maximum $T_c$, and the distinct normal state properties of bulk versus thin films. They propose that the two mechanisms may cooperate, but the hybridization mechanism appears dominant in explaining the specific phenomenology of these materials. Future measurements of $J$ and true gap structures are suggested to further test these hypotheses.