Squeezing dynamical singlets in bilayer nickelates

This paper presents realistic calculations demonstrating that interlayer "dynamical singlets" formed between $3z^{2}-r^{2}$ and $x^{2}-y^{2}$ orbitals govern the physics of bilayer nickelates, successfully explaining experimental discrepancies between bulk crystals and thin films through their distinct responses to hydrostatic pressure and epitaxial strain.

The Gist

Imagine a microscopic world made of layers of atoms, specifically a material called a bilayer nickelate. Think of this material not as a solid block, but as a sandwich made of two thin slices of bread (the layers) with a filling in between. Inside this sandwich, electrons are the busy workers zipping around, and they have different "jobs" or "personalities" based on the shape of their orbitals (the paths they take).

In this specific sandwich, there are two main types of electron workers:

1. The "Planar" Workers ($x^2-y^2$): These are like commuters who love to run around on the flat surface of the bread, moving freely and quickly.
2. The "Vertical" Workers ($3z^2-r^2$): These are the workers who prefer to stand up and connect the two slices of bread together, bridging the gap between the layers.

The Big Discovery: The "Dynamic Handshake"

The paper argues that the secret to how this material behaves isn't just about how fast the electrons move, but about a special relationship between the two "Vertical" workers on opposite layers.

When the material is squeezed in a specific way (using compressive strain, like pressing down on the sides of the sandwich), these two vertical workers lock hands and form a tight, inseparable pair called a "dynamical singlet."

Think of it like two dancers who, when the music gets a certain way, stop dancing individually and lock into a perfect, synchronized embrace. They become so tightly bound to each other that they effectively stop interacting with the rest of the crowd. They form a "singlet" (a pair with no net spin), creating a quiet, stable island in the middle of a busy dance floor.

The Two Ways to Squeeze the Sandwich

The researchers found that you can squeeze this material in two different ways, and the electrons react very differently to each:

1. The "Squeeze from the Sides" (Compressive Strain):
Imagine pressing your hands against the sides of the sandwich, making it wider and flatter.

• What happens: The two vertical dancers (the $z$-orbitals) get pushed closer together. They lock hands tightly and form that "dynamical singlet."
• The Result: Because they are so busy holding each other, they stop helping the horizontal commuters. The material behaves like a "strange metal" where the usual rules of electricity don't quite apply in the same way. The vertical workers become "Mott localized," meaning they are stuck in their spot, holding hands, while the horizontal workers keep running around.

2. The "Squeeze from Top and Bottom" (Hydrostatic Pressure):
Imagine putting the whole sandwich in a press that pushes down from the top and up from the bottom, squeezing it evenly from all sides.

• What happens: The vertical dancers don't lock hands as tightly. Instead, the whole sandwich gets denser, and the horizontal commuters (the $x$-orbitals) get more space to run around.
• The Result: The material starts acting more like a normal metal where electrons flow freely. The "lock" between the vertical dancers is weaker, and they interact more with the rest of the system.

Why This Matters (According to the Paper)

The paper explains a mystery that scientists have been puzzling over: Why does this material act one way when you make thin films (strained) and a completely different way when you have a big chunk of it (pressurized)?

• The Thin Film (Strained): The "dynamical singlets" are strong. The vertical workers are locked in a pair, creating a specific type of electronic behavior that matches what scientists see in experiments on thin films.
• The Bulk Crystal (Pressurized): The "dynamical singlets" are weaker. The vertical workers are more free to interact with the horizontal ones, leading to a different kind of behavior that matches experiments on large crystals.

The Bottom Line

The authors used powerful computer simulations to show that the key to understanding this material is realizing that the electrons aren't just independent runners. Under certain conditions, the electrons on the top and bottom layers pair up into "dynamical singlets."

• Strain makes these pairs tight and strong, isolating them from the rest of the system.
• Pressure keeps them looser, allowing them to mix with the free-flowing electrons.

This "pairing" mechanism is the missing piece of the puzzle that explains why the material's electrical properties change so drastically depending on how you squeeze it. It suggests that the material is a unique playground where some electrons get stuck in a tight embrace while others run free, a state the authors call an "orbital-selective" regime. This specific arrangement of electrons is likely the foundation for the material's ability to conduct electricity without resistance (superconductivity) under high pressure, though the paper focuses on explaining the normal state before superconductivity kicks in.

Technical Summary

Problem Statement
The bilayer Ruddlesden-Popper (RP) nickelates, specifically $La_3Ni_2O_7$, have emerged as a system of significant interest due to their correlated electron behavior and the observation of high-temperature superconductivity under pressure. However, a central challenge remains in developing a unified electronic structure framework that consistently describes the normal state of these materials across different structural perturbations. Experimentally, the material exhibits a dichotomy: bulk single crystals under hydrostatic pressure and epitaxial thin films under compressive strain show systematically different evolutions of their properties. While previous theoretical work has established that the low-energy physics involves $x^2-y^2$ and $3z^2-r^2$ orbitals with weak inter-bilayer coupling, the specific role of non-local interlayer correlations and how they respond differently to pressure versus strain remains to be fully elucidated.

Methodology
The authors employ a cluster dynamical mean-field theory (CDMFT) approach to model the normal-state electronic structure of bilayer RP nickelates.

• Low-Energy Models: They construct low-energy Wannier Hamiltonians from Density Functional Theory (DFT) band structures of $La_3Ni_2O_7$ under various conditions (ambient pressure, hydrostatic pressure up to 30 GPa, and compressive epitaxial strain up to -3%). The models retain the near-Fermi-level Ni-$e_g$ states: the in-plane $d_{x^2-y^2}$ (denoted $x$) and out-of-plane $d_{3z^2-r^2}$ (denoted $z$) orbitals.
• Cluster Definition: Following previous studies, they define a four-orbital cluster comprising the two Ni sites and their frontier $e_g$ orbitals within a bilayer. This cluster serves as the quantum impurity problem.
• Interactions: The Hamiltonian includes local Hubbard-Kanamori interactions parameterized by $U = 4$ eV (intraorbital), $U' = 2.4$ eV (interorbital), and $J = 0.8$ eV (Hund's coupling).
• Solver: The quantum impurity problem is solved using the continuous-time quantum Monte Carlo (CTHYB) method in the hybridization expansion, implemented via the TRIQS software library. Calculations are performed in the bonding-antibonding (BA) basis to mitigate the Monte Carlo sign problem, with an electronic temperature fixed at $T \approx 290$ K (above the spin-density wave transition).

Key Contributions and Results
The study reveals that the physics of bilayer nickelates is significantly controlled by interlayer "dynamical singlets" formed from the singly occupied $3z^2-r^2$ orbitals, which hybridize with itinerant planar $x^2-y^2$ orbitals.

1. Dichotomy between Pressure and Strain: The calculations demonstrate that hydrostatic pressure and compressive epitaxial strain drive the system along distinct evolutionary paths in the low-energy phase space:

   • Hydrostatic Pressure: Primarily enhances in-plane itinerancy ($\tilde{t}_{\parallel}$) by broadening the in-plane bandwidth and weakening correlations in the $x$ orbitals. It moderately modifies the interlayer coupling, keeping the $z$-derived $\gamma$ band close to the Fermi level with sharp quasiparticle weight. This leads to a linear-in-$T$ scattering rate (strange metal behavior).
   • Compressive Strain: Predominantly strengthens interlayer correlations ($\tilde{t}_{\perp}$) via the intersite self-energy $\Sigma_{12}$. This drives the system toward an orbital-selective singlet-paired Mott regime. Under strain, the $z$-derived $\gamma$ band shifts to higher binding energy and broadens, indicating reduced coherence. This results in a $T^2$ (Fermi liquid) scattering rate.

2. Dynamical Singlets: The authors identify a microscopic origin for these structural dependencies. Under compressive strain, the probability of the two Ni sites in a bilayer forming a spin-singlet configuration (one electron on each $z$ orbital with opposite spins) becomes dominant. This state resembles an orbital-selective Mott state where the $z$ orbitals become localized into dynamical singlets, decoupling from the itinerant $x$ bands. The hybridization between these singlets and the itinerant $x$ orbitals weakens as compressive strain increases.

3. Reproduction of Experimental Observations: The interacting theory qualitatively reproduces several experimental findings:

   • ARPES: It captures the orbital-selective mass renormalizations, estimating $m^*/m_{band} \approx 5-7$ for the $\gamma$ band and $2-3$ for the $\alpha/\beta$ bands, consistent with bulk single-crystal data.
   • Transport: The theory explains the contrasting transport signatures: the linear-in-$T$ resistivity observed in pressurized bulk crystals versus the $T^2$ behavior in strained thin films.
   • Superconductivity: The results offer a rationale for the enhancement of $T_c$ under combined strain and pressure, suggesting that while pressure promotes superconductivity via increased in-plane exchange, strain drives the system toward a strong singlet regime that shifts spectral weight into dynamical singlets.

Significance
The paper claims to provide a different paradigm for understanding correlated electron physics in bilayer nickelates. By extending the concept of the "orbital-selective Mott insulator," the authors propose a normal state where localized orbitals ($z$) form dynamical singlets that decouple from itinerant bands ($x$). This framework successfully accounts for the contrasting photoemission and transport signatures of strained thin films and bulk single crystals. The authors posit that this picture provides a natural starting point for future theoretical investigations into electronic instabilities, such as spin-density waves and superconductivity, and suggests that pairing via virtual scattering into excited triplet states of the $z$ dimers is a possibility worthy of further study. The work highlights the crucial role of non-local interlayer correlations in reshaping the low-energy electronic structure, a factor that must be considered to understand the emergence of superconductivity in these materials.