Exact downfolding and its perturbative approximation

This paper presents a rigorous formulation of the downfolding procedure to derive exact effective models for arbitrary target spaces by integrating out high-energy degrees of freedom, establishes conditions for perturbative truncation, formally derives the constrained random phase approximation (cRPA) with identified corrections, and validates the approach using material examples like fcc Nickel and SrCuO$_2$.

The Gist

Imagine you are trying to understand how a massive, chaotic city functions. You want to know why traffic jams happen, how people get from A to B, and what makes the economy tick. But the city has millions of people, thousands of cars, billions of bricks, and complex weather patterns. If you try to simulate every single atom in every brick and every thought in every person's head, your computer would explode before you even started.

In physics, this is the "Many-Electron Problem." Electrons are the "people" of the material world. They are everywhere, interacting constantly, and trying to solve their behavior exactly is impossible for anything but the tiniest systems.

This paper, titled "Exact Downfolding and its Perturbative Approximation," is about a brilliant new way to simplify this problem without losing the truth. The authors are proposing a rigorous "recipe" to shrink a complex city down to a manageable neighborhood model that still tells you exactly how traffic flows.

Here is the breakdown using everyday analogies:

1. The Problem: The "Too Big to Handle" City

In materials like metals or superconductors, electrons interact in a hierarchy.

• The "Target Space" (The Neighborhood): These are the electrons that matter most for the property you care about (like magnetism or superconductivity). They are the "active citizens" causing the traffic jams.
• The "Rest Space" (The Rest of the City): These are the high-energy electrons far away from the action. They are the background noise, the distant suburbs, the people who aren't directly involved in the main event.

Traditionally, physicists have tried to ignore the "Rest Space" or approximate it roughly. But this paper asks: What if we could mathematically "integrate out" the rest of the city perfectly, leaving us with a perfect map of just the neighborhood?

2. The Solution: "Downfolding" (The Magic Compression)

The authors call their process Downfolding. Think of it like a high-tech photo compression algorithm.

• The Old Way: You take a photo of the whole city, crop it to the neighborhood, and hope the background blur doesn't change the colors of the houses in the foreground. Sometimes it works; sometimes the colors look wrong.
• The New Way (Exact Downfolding): The authors developed a mathematical "lens" that takes the entire city, calculates exactly how the background influences the foreground, and then compresses that influence into the neighborhood model itself.

They prove that if you do this correctly, the "neighborhood model" you get is exactly equivalent to the full city. If you run a simulation on this small model, the results are identical to running it on the whole universe.

3. The Catch: The "Perturbative Approximation"

Here is the tricky part. While the "Exact Downfolding" is perfect, the math is so complex it's like trying to write down the instructions for every single interaction in the universe. It's too heavy to carry.

So, the authors ask: Can we simplify this perfect recipe without ruining the cake?

They introduce a set of rules of thumb (approximations) to decide when it's safe to throw away the most complex parts of the math.

• The Analogy: Imagine you are baking a cake. The "Exact" recipe requires measuring the humidity of the air, the exact age of the eggs, and the vibration of the floor.
• The Approximation: The authors say, "If the eggs are fresh and the floor is stable, you can ignore the humidity and the vibration. You only need to worry about the flour and sugar."
• The Result: They show that for many materials, the "vibration" (complex interactions) is so small that you can safely ignore it. This allows them to use standard, fast computer methods to solve the problem.

4. The "cRPA" Connection: Fixing a Popular Tool

There is a very popular tool in physics called cRPA (constrained Random Phase Approximation). It's like a well-known, slightly old-fashioned map of the city. It's been used for decades and works well in many cases, but no one was 100% sure why it worked or where it failed.

The authors used their new "Exact Downfolding" lens to look at cRPA.

• The Discovery: They found that cRPA is actually a specific, simplified version of their exact theory.
• The Insight: They identified exactly what cRPA was ignoring (certain subtle interactions between the neighborhood and the suburbs) and showed when those ignored parts actually matter. It's like realizing your old map was missing a few side streets that only become important during rush hour.

5. Testing the Theory: Nickel and Cuprates

To prove their recipe works, they tested it on two real-world materials:

1. Nickel (a metal): They showed that for Nickel, the "simplified" version of their recipe works perfectly. The "background noise" is quiet enough to ignore.
2. SrCuO2 (a superconductor candidate): This is a more complex material. They found that while the simplified recipe works, you have to be very careful about how you define the "neighborhood." If you pick the wrong atoms to include, the math breaks down. This highlights that choosing the right starting point is crucial.

Why This Matters

This paper is a "user manual" for the future of material science.

• Before: Physicists were guessing which approximations were safe to use. It was like driving blindfolded, hoping the road didn't change.
• Now: They have a rigorous checklist. Before you start your simulation, you can check the "rules" to see if your simplified model will be accurate.

In summary: The authors built a perfect mathematical machine to shrink complex materials into simple models. They then figured out how to make that machine run faster by safely ignoring tiny details, and they used it to fix and improve the most popular tools scientists currently use. This means we can design better batteries, superconductors, and electronics with much higher confidence that our computer models are telling the truth.

Technical Summary

Here is a detailed technical summary of the paper "Exact downfolding and its perturbative approximation" by Profe et al.

1. Problem Statement

The construction of effective low-energy models for strongly correlated materials is a central challenge in condensed matter physics. The standard workflow involves:

1. Ab initio calculation (e.g., DFT) to characterize the full electronic structure.
2. Downfolding: Deriving an effective Hamiltonian for a specific "target space" (e.g., low-energy $d$-orbitals) by integrating out "rest space" degrees of freedom (high-energy orbitals).
3. Many-body solution: Solving the effective model using techniques like DMFT or QMC.

While significant progress has been made in step 3 (solving the models), the downfolding procedure (step 2) often relies on ad-hoc approximations or lacks a rigorous error control mechanism. Existing methods like the Constrained Random Phase Approximation (cRPA) are widely used but introduce uncontrolled approximations, particularly regarding the treatment of hybridization between target and rest spaces and the dependence on the choice of basis. There is a need for an exact formalism to derive these effective models and a systematic way to judge when perturbative truncations are valid.

2. Methodology

The authors develop a rigorous, exact downfolding formalism based on the path integral representation of the partition function.

• Formalism: Starting from a generic many-body Hamiltonian, the system is split into a Target Space ($T$) and a Rest Space ($R$). The fields are separated into $f$ (target) and $c$ (rest).
• Exact Integration: The rest space degrees of freedom are integrated out exactly using the path integral formalism. This yields an exact effective action ($S_{eff}$) for the target space fields.
• Diagrammatic Expansion: The resulting effective action is expanded in a Taylor series around zero fields. This generates an infinite series of $n$-particle interaction vertices ($G^{(n)}$) involving the target space fields.
  • The coefficients of these vertices are determined by the coupling between $T$ and $R$ (vertices $A$ and $B$) and the connected Green's functions of the rest space ($g^{(k)}$).
  • The formalism accounts for all orders of interactions, including retarded (frequency-dependent) effects and higher-order many-body terms (3-body, 4-body, etc.).
• Perturbative Truncation (PCD): The authors propose Perturbatively Controlled Downfolding (PCD). They establish two criteria to justify truncating the expansion to single- and two-particle terms (standard effective models):
  1. Rest Space Convergence: The connected Green's functions of the rest space involving $>2$ particles must be negligible (rapid convergence of the rest space expansion).
  2. Coupling Hierarchy: The coupling vertices must satisfy a hierarchy where terms generating 3-fermion interactions (specifically $A_{3:1}$ and $A_{1:3}$) are negligible compared to density-density ($A_{2:2}$) and hopping ($A_{1:1}$) terms.
• Recovering cRPA: The authors demonstrate that the widely used cRPA is a specific approximation within this exact formalism, obtained by resumming specific diagram classes (RPA bubbles) and neglecting mixed propagation terms and higher-order vertices.

3. Key Contributions

• Exact Formalism: Derivation of a mathematically rigorous, exact downfolding theory that produces an effective action equivalent to the full system for any target space observables.
• Diagrammatic Rules: Development of a complete set of diagrammatic rules (Fig. 1) to visualize and calculate the contributions to the effective action, distinguishing between single-particle hybridization, retarded interactions, and multi-particle vertices.
• Reconstruction of cRPA: Showing explicitly how cRPA emerges from the exact theory, identifying its underlying assumptions (orthogonality of bases, neglect of mixed propagation, and specific vertex truncations) and highlighting missing corrections (e.g., hybridization corrections).
• Basis Dependence Analysis: Demonstrating that the validity of the perturbative truncation is strongly basis-dependent. In an atomic orbital basis, the hierarchy of couplings ($A_{2:2} \gg A_{3:1}$) often holds, whereas in a band basis, anomalous vertices can become large, invalidating simple truncations.
• Double Counting Clarification: The formalism clarifies the double-counting problem, noting that the downfolded model and the subsequent many-body solution do not double-count contributions if the downfolding is derived from the partition function, unlike some Hamiltonian-based approaches.

4. Results

The authors applied their formalism to three material systems:

1. fcc Nickel (Ni):

   • Target: $3d$orbitals; **Rest:**$4s$and$4p$ orbitals.
   • Findings: In the atomic orbital basis, $A_{2:2}$ (density-density coupling) is dominant ($\sim 10$ eV), while $A_{3:1}$ and $A_{1:3}$ are negligible (symmetry forbidden or orthogonal).
   • Conclusion: The conditions for PCD are met. The effective model is well-described by single- and two-particle terms, justifying the use of cRPA-like approaches, though hybridization ($A_{1:1}$) induces significant retarded hopping.

2. Infinite-Layer Cuprate SrCuO$_2$:

   • Target: $d_{x^2-y^2}$ orbital; Rest: Cu-$3d$and O-$2p$ orbitals.
   • Findings: Large kinetic coupling ($A_{1:1}$) exists between the target and the "Emery" oxygen orbitals. While $A_{2:2}$ is still dominant, the large hybridization leads to a spectral function that cannot be mapped to a simple single-band Hamiltonian without frequency dependence.
   • Conclusion: Standard static downfolding fails; the retarded nature of the interaction is crucial.

3. SrVO$_3$ (Multi-orbital Metal):

   • Target: $t_{2g}$ orbitals; Rest: $e_g$ and O-$p$ orbitals.
   • Comparison: The authors compared cRPA, cFRG, and their PCD scheme.
   • Key Finding: cRPA overestimates the Hubbard $U$ compared to PCD. More importantly, including hybridization terms (retarded hopping) in the effective model drastically changes the physical outcome.
     • Models without hybridization predicted a Mott insulating state.
     • Models with hybridization (captured by PCD) predicted a Fermi-liquid metallic state.
   • This highlights that neglecting the kinetic coupling between target and rest spaces can lead to qualitatively wrong predictions of the ground state.

5. Significance

• Controlled Approximations: The paper provides a "gold standard" for downfolding. It offers a systematic way to check if a chosen target space and basis set are suitable for a simplified effective model.
• Beyond cRPA: It reveals that cRPA is an approximation that misses specific diagrammatic classes (mixed propagation, hybridization corrections) which can be physically significant, particularly in determining whether a material is metallic or insulating.
• Basis Selection: It emphasizes that the choice of basis (atomic vs. band) is not merely a technical detail but fundamentally alters the convergence of the effective theory. Atomic orbitals often provide a better hierarchy for perturbative downfolding.
• Future Tooling: The authors propose the development of automated post-ab initio tools that incorporate these convergence checks, moving the field toward fully predictive, error-controlled effective models for complex materials.

In summary, this work bridges the gap between exact many-body theory and practical effective model construction, providing the theoretical tools to validate when and how simplified models can accurately represent real materials.