Beyond the conventional Emery model: crucial role of long-range hopping for cuprate superconductivity

Using the dynamical vertex approximation, this study demonstrates that while the conventional Emery model captures qualitative features of cuprate superconductivity, incorporating long-range hopping parameters beyond the standard three is essential for achieving a quantitatively accurate phase diagram and a proper d-wave order parameter.

The Gist

Imagine you are trying to bake the perfect cupcake, but instead of flour and sugar, you are trying to understand how certain materials (called cuprates) conduct electricity without any resistance at all. This phenomenon is called superconductivity.

For decades, scientists have used a specific "recipe" to model these materials, called the Emery model. Think of this model as a simplified map of a city. In this city, there are copper "houses" and oxygen "houses." Electrons (the people) hop from house to house.

The traditional recipe for this map only allowed people to hop to their immediate neighbors (the next-door copper or oxygen houses). It was like saying, "You can only walk to the house directly next to yours."

The Problem with the Old Map

The authors of this paper, led by Eric Jacob and Karsten Held, decided to test this old map using a very powerful computer simulation (a method called the "Dynamical Vertex Approximation"). They found that the old map was missing something crucial.

In the real world, people don't just walk to the next door; they can also walk to the house two doors down, or even skip a few houses if the path is clear. In physics terms, these are called long-range hoppings.

When the scientists used the old, limited map (only next-door hops), the simulation failed to produce the right kind of superconductivity, especially when the material was "doped" (mixed with extra electrons or holes) to a level where real cuprates usually work best. It was like trying to bake a cake with only half the ingredients; the result just didn't rise properly.

The New Discovery

The team realized that to get the "perfect cake" (the correct superconducting behavior), they needed to add long-range hopping to their map. They had to let the electrons jump to houses further away, not just the immediate neighbors.

Here is what happened when they added these extra jumps:

1. The "Dome" Appeared: In superconductivity research, scientists look for a "dome" shape on a graph. This dome shows the range of conditions where superconductivity works best. The old map produced a tiny, narrow dome that didn't match reality. The new map, with long-range jumps, produced a big, healthy dome that looked exactly like the superconducting behavior seen in real cuprate materials.
2. The "Order" Made Sense: The old map created a weird, bumpy pattern for how the electrons paired up (called the "order parameter"). It was like a dance where the partners were stepping on each other's toes in a strange rhythm. The new map created a smooth, classic "d-wave" dance pattern, which is what scientists expect to see in these materials.

Why It Matters (According to the Paper)

The paper argues that for a long time, scientists have been using a "simplified" version of the physics that works okay for rough guesses but fails when you need precise numbers.

• The Old Way: Like using a hand-drawn sketch of a city to plan a subway system. It gets the general idea, but the trains would crash because the map missed the long tunnels.
• The New Way: Like using a high-tech GPS that accounts for every possible route, including the long-distance ones. This allows the simulation to predict exactly where and when the superconductivity happens.

The Bottom Line

The authors conclude that if you want to accurately describe how these superconducting materials work, you must include the long-distance jumps of the electrons. Ignoring them leads to the wrong predictions about when the material becomes superconducting and how it behaves. They didn't invent a new superconductor or a new medical device; they simply fixed the mathematical "map" we use to understand the ones we already have, showing that the old map was missing essential roads.

Technical Summary

Technical Summary: Beyond the Conventional Emery Model

Problem Statement
The Emery model, a three-band model comprising copper $d_{x^2-y^2}$ and in-plane oxygen $p_x, p_y$ orbitals, is the standard framework for describing cuprate superconductors. However, conventional applications of this model typically restrict hopping parameters to nearest-neighbor copper-oxygen ($t_{pd}$) and nearest- and next-nearest-neighbor oxygen-oxygen ($t_{pp}, t'_{pp}$) interactions. Previous studies utilizing this "conventional" set of parameters have struggled to quantitatively reproduce the superconducting phase diagram of cuprates, often relying on small cluster sizes that suffer from finite-size effects or on single-band Hubbard models that may lack necessary material realism. A critical open question is whether the conventional hopping parameters are sufficient to describe the $d$-wave superconducting dome and the correct order parameter symmetry, particularly at higher hole dopings.

Methodology
The authors employ the Dynamical Vertex Approximation (D$\Gamma$A) to study the Emery model. This diagrammatic extension of Dynamical Mean-Field Theory (DMFT) allows for the treatment of non-local correlations on fine momentum grids (up to $130 \times 130$), effectively simulating an infinite system and avoiding the sign problems associated with Quantum Monte Carlo simulations on large clusters.

The study utilizes a specific material realization: $\text{CaCuO}_2$. The non-interacting Hamiltonian is derived from Density Functional Theory (DFT) calculations using the Generalized Gradient Approximation (GGA) and projected onto maximally localized Wannier functions. The interaction strength $U = 6.5$ eV is calculated via the constrained Random Phase Approximation (cRPA). The authors compare three distinct models:

1. Conventional Emery Model: Includes only $t_{pd}$, $t_{pp}$, and $t'_{pp}$.
2. Covalent Emery Model: A conventional parameter set derived for $\text{Bi}_2\text{Sr}_2\text{CaCu}_2\text{O}_8$ (Bi-2212) in previous literature.
3. Full d-p Model: Includes all hopping elements derived from the DFT-Wannier projection, extending beyond nearest and next-nearest neighbors.

Superconducting instabilities are identified by solving the linearized Eliashberg equation within the ladder formulation of D$\Gamma$A, extracting the superconducting eigenvalue $\lambda_{SC}$. A value of $\lambda_{SC} \to 1$ indicates the transition temperature $T_c$.

Key Contributions and Results

• Insufficiency of Conventional Hopping: The authors demonstrate that the conventional Emery model significantly underestimates effective hopping parameters when downfolded to an effective one-band Hubbard model. Specifically, the next-nearest-neighbor hopping $t'$ in the downfolded model is too small (e.g., $|t'/t| \approx 0.07$ for $\text{CaCuO}_2$) compared to realistic values ($|t'/t| \approx 0.15 - 0.25$). This deficiency arises because the conventional model neglects long-range hopping paths that are physically present in the full d-p Hamiltonian.
• Band Structure Corrections: The inclusion of long-range hopping parameters in the full d-p model correctly shifts the van Hove singularity to lower energies, aligning the band structure with DFT-Wannier results. In contrast, the conventional model places the van Hove singularity too close to the Fermi energy.
• Superconducting Phase Diagram:
  • Conventional Model: Fails to show a superconducting instability for hole dopings $\delta \geq 0.15$ (the optimal doping range). It yields a very narrow superconducting dome (9%–14% doping) and predicts a strongly modulated $d$-wave order parameter with suppressed spectral weight at the antinodes. This modulation is attributed to a pseudogap developing prematurely due to the misplaced van Hove singularity.
  • Full d-p Model: Successfully reproduces a broad superconducting dome consistent with cuprate phenomenology, spanning approximately 7% to 22% hole doping. At $\delta = 0.15$, a clear superconducting instability is observed. The resulting $d$-wave order parameter is conventional, without the spurious modulation seen in the restricted models.
• Role of Long-Range Hopping: The study establishes that long-range hopping parameters (beyond the three conventional ones) are quantitatively necessary to obtain the correct phase diagram and order parameter. These parameters are of the same range as the $t'$ term in the Hubbard model, which is known to be crucial for superconductivity.

Significance
The paper claims that while the conventional Emery model is qualitatively useful, it is quantitatively insufficient for describing cuprate superconductivity beyond low dopings. The authors argue that restricting calculations to the three conventional hopping parameters leads to incorrect predictions regarding the extent of the superconducting dome and the nature of the order parameter. By utilizing the full d-p model derived from first principles and the D$\Gamma$A method, the work provides a more material-realistic description of cuprates that naturally includes oxygen orbitals and long-range hopping effects. This approach resolves discrepancies regarding the van Hove singularity position and the resulting pseudogap behavior, offering a path toward a more accurate theoretical understanding of high-$T_c$ superconductivity without resorting to empirical parameter fitting.