Unconventional Superconductivity in $\mathrm{La_{3}Ni_{2}O_{7}}$ from the Perspective of Symmetry

By employing a phenomenological symmetry-based method combined with DFT+$U$ calculations, this study reveals that while both pressurized bulk and ambient-pressure thin-film $\mathrm{La_{3}Ni_{2}O_{7}}$ exhibit $s_{\pm}$-wave two-gap superconductivity, the significantly lower transition temperature in the thin film arises from a shift in dominant pairing from out-of-plane Ni-$d_{z^2}$ to in-plane Ni-$d_{x^2-y^2}$ orbitals driven by reduced inter-layer hopping.

The Gist

Imagine a new type of material, La₃Ni₂O₇, that has recently been discovered to conduct electricity with zero resistance (superconductivity) at surprisingly high temperatures. This is a big deal because usually, you need to cool things down to near absolute zero to get this super-power.

However, there's a mystery. When scientists squeeze this material with massive pressure (like a giant hydraulic press), it becomes a superconductor at about 80 Kelvin. But when they grow it as a very thin film (like a layer of paint on a wall) without any pressure, it still superconducts, but only at about 40 Kelvin—half as hot.

Why does the "squeezed" version work so much better than the "thin film" version? This paper tries to solve that puzzle by looking at the material's symmetry (its geometric shape and rules) rather than just its chemical ingredients.

Here is the breakdown of their findings using simple analogies:

1. The "Dance Floor" Analogy (Symmetry)

Think of the atoms in this material as dancers on a floor.

• The Pressurized Bulk: When you squeeze the material, the dancers are forced into a specific, tight formation (a high-symmetry space group).
• The Thin Film: When it's a thin film, the floor underneath it (the substrate) stretches the dancers out slightly differently.
• The Connection: Even though the floor looks different, the authors found that the rules of how the dancers can move relative to each other (the "layer group symmetry") are actually the same for both. This shared rulebook allowed the scientists to use one mathematical method to study both versions.

2. The "Handshake" Analogy (Pairing)

In a superconductor, electrons don't move alone; they pair up and dance together. This is called "pairing."

• The Problem: Scientists didn't know how these electrons were holding hands. Were they holding hands vertically (up and down)? Or horizontally (side to side)?
• The Method: The authors created a "symmetry filter." Instead of guessing the microscopic details, they asked: "Given the shape of the room and the temperature, what kind of handshake is physically possible?"

3. The Big Discovery: Two Different Handshakes

The paper reveals that while both versions of the material use the same type of handshake (called s±-wave, which is a specific, complex way electrons pair up), the dominant way they pair is different.

• In the Pressurized Bulk (The Squeezed Version):
  The electrons are mostly holding hands vertically (out-of-plane). Imagine two dancers in a double-decker building shaking hands through the floor between them. This vertical connection is very strong and allows the superconductivity to happen at the higher temperature (80 K).

  • Key Orbitals: The electrons involved are from the $d_{z^2}$ orbitals (think of these as the "vertical" dancers).

• In the Thin Film (The Flat Version):
  Because the film is stretched differently, the vertical connection gets weaker. The electrons switch to holding hands horizontally (in-plane). Now, the dancers are shaking hands with their neighbors on the same floor. This horizontal connection is weaker, which is why the superconductivity drops to the lower temperature (40 K).

  • Key Orbitals: The electrons involved are from the $d_{x^2-y^2}$ orbitals (think of these as the "horizontal" dancers).

4. Why the Temperature Drops

The authors explain the temperature drop like this:
Imagine you have a team of two people trying to lift a heavy box.

• Scenario A (Bulk): They are standing on a solid, compressed foundation. They can lift the box high (High $T_c$).
• Scenario B (Thin Film): The foundation shifts, and they have to change their grip. They are now lifting the box with a different, less efficient muscle group. They can still lift it, but not as high (Low $T_c$).

The paper argues that the thin film's lower temperature isn't because the material is "broken," but because the dominant pairing strategy changed from a strong vertical grip to a weaker horizontal one.

5. Checking the Work

To make sure their theory was right, the authors compared their calculated "dance moves" (energy gaps) with real-world experiments:

• ARPES (Angle-Resolved Photoemission Spectroscopy): Like taking a high-speed photo of the dancers' paths. The paper's predictions matched the photos perfectly.
• STM/STS (Scanning Tunneling Microscopy): Like listening to the rhythm of the dancers. The "sound" (density of states) predicted by the paper matched the experimental recordings, showing a "V-shape" pattern that confirms their theory.

Summary

The paper concludes that symmetry is the boss. By looking at the geometric rules of the material, they figured out that:

1. Both the squeezed bulk and the thin film are superconductors.
2. They both use the same general "handshake" style.
3. However, the squeezed version relies on vertical electron connections, while the thin film relies on horizontal ones.
4. This switch in strategy is exactly why the thin film is "cooler" (lower temperature) than the squeezed version.

This method of using symmetry to predict how electrons pair up could be a new tool for understanding other strange superconductors in the future.

Technical Summary

Technical Summary: Unconventional Superconductivity in La$_3$Ni$_2$O$_7$ from the Perspective of Symmetry

Problem Statement
The discovery of high-temperature superconductivity ($T_c \approx 80$ K) in pressurized bulk La$_3$Ni$_2$O$_7$ and the subsequent observation of ambient-pressure superconductivity in thin-film variants have sparked intense interest. However, a significant discrepancy exists: the $T_c$ in thin films is approximately half that of the pressurized bulk. While previous studies have suggested $s_{\pm}$-wave or $d$-wave pairing driven by strong coupling or orbital hybridization, the specific microscopic mechanisms governing the pairing symmetry and the origin of the $T_c$ reduction in thin films remain unclear. The central challenge is to determine how the distinct structural symmetries of the bulk (under high pressure) and the thin film (under strain) influence the superconducting gap structure and dominant pairing configurations.

Methodology
The authors develop a phenomenological, symmetry-based framework to investigate the superconducting pairing in La$_3$Ni$_2$O$_7$ without imposing a priori assumptions about the microscopic pairing configuration. The approach integrates:

1. Symmetry Analysis: The study utilizes the eight one-dimensional irreducible representations of the $4/mmm$($D_{4h}$) point group to classify time-reversal symmetry (TRS) preserving superconducting states. Both the pressurized bulk (transitioning to $I4/mmm$ space group) and the thin film ($P4/mmm$ space group) share the same bilayer layer group symmetry ($p4/mmm$), providing a unified framework.
2. Electronic Structure: A two-orbital (Ni-$d_{z^2}$ and $d_{x^2-y^2}$) tight-binding model is constructed via Wannier downfolding from Density Functional Theory with Hubbard $U$ (DFT+U) calculations. This model respects the symmetries of the type-II magnetic layer group.
3. Mean-Field Theory: The system is modeled using a Hamiltonian with strong on-site repulsion and short-range attraction. The pairing potential is derived via Hubbard–Stratonovich decoupling within the Bogoliubov–de Gennes (BdG) formalism.
4. Selection Criteria: To identify the physically realized pairing state, the authors calculate the BCS condensation energy for all symmetry-allowed pairing channels. Crucially, they impose a physical constraint: the pairing interaction strength ($V_i$) required to reproduce the experimental $T_c$ must be physically realizable (specifically, $V_i$ must be comparable to or less than the corresponding hopping integrals $|t|$). The leading pairing type is identified by minimizing the free energy among feasible configurations.

Key Results
The study reveals that while both pressurized bulk and thin-film La$_3$Ni$_2$O$_7$ exhibit an overall $s_{\pm}$-wave pairing symmetry with two-gap superconductivity, the dominant microscopic pairing configurations differ fundamentally:

• Pressurized Bulk: Superconductivity is dominated by out-of-plane pairing of the Ni-$d_{z^2}$ orbitals ($\Delta^z_{\perp}$, $s_{z^2}$-wave symmetry). The in-plane pairing of $d_{x^2-y^2}$ orbitals ($\Delta^x_{\parallel}$) is a secondary, coexisting component. The transition temperature is primarily determined by the inter-layer coupling strength.
• Thin Film: The dominant pairing shifts to in-plane pairing of the Ni-$d_{x^2-y^2}$ orbitals ($\Delta^x_{\parallel}$, $s_{x^2+y^2}$-wave symmetry). The out-of-plane $d_{z^2}$ pairing becomes sub-dominant.
• Origin of $T_c$ Reduction: The reduction in $T_c$ in the thin film is attributed to this transition in the dominant pairing type. The thin film exhibits a decreased ratio of inter-layer to intra-layer hopping ($|t^z_{1,\perp}/t^x_{1,\parallel}|$). This structural change weakens the dominant inter-layer $d_{z^2}$ channel and enhances the intra-layer $d_{x^2-y^2}$ channel. Since the intra-layer attractive interaction in the $d_{x^2-y^2}$ orbitals is weaker than the inter-layer interaction in the $d_{z^2}$ orbitals, the overall $T_c$ is reduced.
• Spectral Agreement: The calculated energy spectra and density of states (DOS) show closed agreement with experimental Angle-Resolved Photoemission Spectroscopy (ARPES) and Scanning Tunneling Microscopy/Spectroscopy (STM/STS) data. Specifically, the model reproduces the "V-shape" nodal gap signature and the two-gap features observed in experiments.
• Exclusion of $d$-wave: The study systematically excludes $d$-wave pairing ($B_{1g}$ and $B_{2g}$) as the leading instability. Strong inter-band scattering processes, particularly within the $\beta$ Fermi surface sheet, suppress $d$-wave channels in favor of $s_{\pm}$-wave symmetry.

Significance and Claims
The paper claims to provide a unified symmetry-guided explanation for the superconductivity in both bulk and thin-film La$_3$Ni$_2$O$_7$. Its primary contributions are:

1. Mechanism Identification: It identifies the specific orbital and layer configurations driving superconductivity, resolving the ambiguity regarding the dominant pairing channel in different structural environments.
2. Explanation of $T_c$ Discrepancy: It attributes the lower $T_c$ in thin films not merely to disorder or volume fraction, but to a fundamental transition in the dominant pairing mechanism driven by symmetry-constrained hopping ratios.
3. Methodological Generalization: The authors propose that their symmetry-based phenomenological method, which relies on experimental $T_c$ and structural symmetry to "select" the pairing configuration, can be generalized to a broader range of unconventional superconductors.
4. Validation: The results are validated by their consistency with RPA calculations and experimental spectroscopic data, reinforcing the role of symmetry in determining the gap structure of nickelate superconductors.

The authors conclude that their work highlights the close relationship between crystal symmetry and superconductivity, offering a rationalization for why superconductivity often emerges in high-symmetry crystalline environments.