Identifying strong correlation using only the Kohn-Sham… — Plain-Language Explanation

The Big Problem: The "Overcrowded Room" of Electrons

Imagine a crowded dance floor (the material) where the electrons are the dancers. In some materials, the music is calm, and the dancers move smoothly. In strongly correlated systems (such as certain metals and oxides), however, the dancers are packed so tightly at the center of the floor (the "Fermi level") that they constantly bump into each other.

In physics, this "bumping" is called electron-electron interaction. When the crowd is so dense, the dancers' movements become chaotic and unpredictable. Standard computer models (called DFT) usually fail here because they assume the dancers move independently of one another. They cannot handle the chaos of the "strongly correlated" crowd, leading to inaccurate predictions about the material's energy and behavior.

The Old Solution: Adding a "Bouncer"

Traditionally, when standard models failed, scientists had to add complex and expensive "bouncers" (mathematical corrections like DFT+U or DFT+DMFT) to force the dancers to respect their personal space. These methods explicitly calculate the repulsion between electrons but are computationally intensive and complicated.

The New Idea: Breaking Symmetry

This paper proposes a clever shortcut. Instead of adding a bouncer, the authors suggest letting the dancers break the rules of symmetry.

Imagine the dance floor is perfectly round and symmetrical. If everyone tries to dance in a circle, they get stuck in a jam. But if the dancers spontaneously split into two groups—one group moving clockwise, the other counter-clockwise (this is spin symmetry breaking)—the crowd in the middle becomes thinner.

The Analogy: By breaking perfect symmetry, the "near-degenerate states" (the overcrowded, identical spots where dancers get stuck) are eliminated. The energy gap between occupied and empty spots widens.
The Result: The crowd in the middle of the floor becomes much less dense. Since the crowd is less dense, the dancers have less to worry about bumping into each other. The system transforms from a "chaotic, strongly correlated mess" into a "calm, normally correlated system" that standard models can easily handle.

The "Correlation Meter" ( $\Gamma$ )

How do you know if a material needs this symmetry-breaking trick? The authors invented a simple correlation meter called $\Gamma$ (Gamma).

How it works: You look at the density of dancers in the middle of the floor (the density of states at the Fermi level) and compare it to a "normal, calm crowd" (a homogeneous electron gas).
The Readout:
- $\Gamma \le 1$ : The crowd is normal. Standard models work well. No special tricks needed. (Examples: Copper, Silver).
- $\Gamma \gg 1$ : The crowd is dangerously dense. The material is strongly correlated. Standard models will fail unless you allow symmetry breaking. (Examples: Iron, Nickel, Gadolinium).

What They Found

The team tested this idea on a list of materials, including metals like Iron (Fe) and Nickel (Ni), as well as oxides like Nickel Oxide (NiO).

For "normal" materials: When they tried to break symmetry, the system simply snapped back to symmetry. The density of dancers did not change much, and the energy did not drop. These materials are naturally calm.
For "strongly correlated" materials: When they allowed symmetry breaking, the density of dancers in the middle dropped significantly.
- The Energy Gain: The total energy of the system dropped significantly (became more negative), meaning the material became much more stable.
- The Gap: In some cases (such as with NiO), this symmetry breaking even opened a "band gap," turning a metal into an insulator, which matches real-world experiments.

The Conclusion

The paper argues that symmetry breaking is not just a mathematical trick; it is a physical reality.

By allowing electrons to break symmetry (such as when forming magnetic patterns), the system naturally reduces the "overcrowding" that leads to strong correlation. This enables simple, standard computer models to accurately describe materials that were previously thought to require complex and expensive methods.

They also found a strong connection: the higher the $\Gamma$ value (the denser the crowd), the more energy is saved by breaking symmetry. This gives scientists a quick and easy way to check if a material is "strongly correlated" by simply looking at the electron density, without having to run the most expensive simulations first.

In short: If the electron crowd is too dense, let them break symmetry to thin out the crowd. Once the crowd is thin, the standard rules of physics work again.

1. Problem Statement

Strongly correlated electron systems (e.g., transition metal oxides, rare-earth compounds) pose a significant challenge for standard Density Functional Theory (DFT).

The Limitation: Standard DFT approximations (LDA, GGA, meta-GGA) are based on "normally correlated" systems (such as the homogeneous electron gas). They often fail to capture the energetic stabilization due to strong electron-electron interactions in symmetric configurations, leading to inaccurate total energies and preventing the prediction of phenomena such as Mott insulator phases or band gap openings.
Current Solutions: Traditional approaches to address this problem include adding explicit interaction terms (e.g., DFT+U, DFT+DMFT) or using multi-reference wavefunction methods. However, these are computationally expensive or require empirical parameters.
The Core Question: Can the energetic effects of strong correlation be captured within standard DFT without explicit interaction terms simply by allowing a symmetry breaking (specifically spin symmetry breaking) in the non-interacting Kohn-Sham (KS) system?

2. Methodology

The authors propose a hypothesis and test it with a specific computational framework:

Hypothesis: Strong correlation in an interacting system arises from nearly degenerate states between occupied and unoccupied states near the Fermi level ( $\epsilon_F$ ). In the non-interacting KS system, allowing symmetry breaking (e.g., magnetic ordering) lifts these degeneracies, reduces the density of states (DOS) at $\epsilon_F$ , and transforms the system from a "strongly correlated" symmetric state into a "normally correlated" symmetry-broken state.
Computational Setup:
- Software: VASP (Projector-Augmented-Wave method) with the r2SCAN meta-GGA functional.
- Test Set: A diverse selection of materials, including strongly correlated metals/oxides (Ni, Fe, NiO, VO $_2$ , Co, Pd, EuB $_6$ , SmB $_6$ , Gd) and normally correlated metals (Cu, Ag, Zn, Cd).
- Procedure:
  1. Calculation of the ground state for the Symmetric (NM) configuration (non-magnetic, no spin polarization).
  2. Calculation of the ground state for the Symmetry-Broken (SB) configuration (magnetic, allowing spin polarization).
  3. Calculation of the energy difference per atom: $\Delta E = E_{symm} - E_{symm.brok}$ .
  4. Analysis of the density of states (DOS) at the Fermi level for both configurations.
The Correlation Parameter ( $\Gamma$ ):
To quantify correlation purely from the KS-DOS, the authors define a dimensionless parameter $\Gamma$ :
$\Gamma = \frac{D(\epsilon_F)}{D_{unif}(\epsilon_F)}$
Where:
- $D(\epsilon_F)$ is the DOS at the Fermi level of the target material (calculated via r2SCAN).
- $D_{unif}(\epsilon_F)$ is the DOS of a Uniform Electron Gas (UEG) with the same electron density ( $n$ ) and the same cell volume.
- Interpretation:
  - $\Gamma \le 1$ : Normally correlated (DFT is reliable).
  - $\Gamma \gg 1$ : Strongly correlated (DFT fails in the symmetric phase; symmetry breaking is likely required).

3. Main Contributions

Redefinition of Strong Correlation: The paper argues that strong correlation is effectively encoded in the nearly degenerate states of the KS spectrum. By symmetry breaking, these degeneracies are lifted, reducing the "correlation demand" on the functional.
Introduction of $\Gamma$ : A novel, computationally inexpensive metric derived exclusively from the KS-DOS that distinguishes systems requiring explicit correlation treatment from those that do not.
Validation of Symmetry Breaking: It is shown that for strongly correlated systems, the symmetry-broken (magnetic) state not only significantly lowers the total energy but also reduces the DOS at the Fermi level to a range where standard DFT becomes reliable (i.e., the problem is transformed into a "normally correlated" one).
Extension to $\Gamma_g$ : In the supplementary information, the authors introduce a Gaussian-smoothed version ( $\Gamma_g$ ) that integrates states over an energy window $\delta$ . This accounts for the imbalance between occupied and unoccupied states and shows a stronger linear correlation with $\Delta E$ (Pearson coefficient up to 0.96) compared to the raw $\Gamma$ .

4. Results

Energy Differences ( $\Delta E$ ):
- Strongly Correlated Systems: Showed a significant energy lowering upon symmetry breaking (e.g., Gd: $\Delta E \approx 8.98$ eV/atom; Fe: $0.91$ eV/atom; NiO: $0.49$ eV/atom).
- Normally Correlated Systems: Showed $\Delta E = 0$ (converged back to the non-magnetic state), indicating no energetic advantage from symmetry breaking.
Density of States (DOS) Analysis:
- In symmetric states of strongly correlated materials, there is a high DOS at $\epsilon_F$ (often due to $d$ - or $f$ -orbitals).
- Upon symmetry breaking, the DOS at $\epsilon_F$ decreases significantly. In cases like NiO, a band gap opens, transforming the metal into an insulator.
Performance of $\Gamma$ :
- Strongly Correlated: $\Gamma > 1$ (e.g., Gd: 23.07, Ni: 9.14, Fe: 7.22).
- Normally Correlated: $\Gamma \le 1$ (e.g., Cu: 0.67, Ag: 0.53, Zn: 0.70).
- Pd Case: Palladium is experimentally non-magnetic but lies near the Stoner instability. The calculation showed $\Gamma \approx 4.4$ and a small $\Delta E$ , correctly identifying it as a system on the edge of strong correlation/symmetry breaking.
Correlation between $\Gamma$ and $\Delta E$ :
- A qualitative trend exists: Higher $\Gamma$ correlates with larger $\Delta E$ .
- While a simple linear fit for $\Gamma$ versus $\Delta E$ exhibited a low correlation coefficient (0.23 without Gd), the generalized $\Gamma_g$ (accounting for energy width) showed a strong linear dependence (0.96), validating the physical link between high DOS near $\epsilon_F$ and the energy gain from symmetry breaking.

5. Significance and Implications

Methodological Guidance: The parameter $\Gamma$ $Γ$ offers researchers a practical "litmus test." If a calculation yields $\Gamma \gg 1$ $Γ ≫ 1$ , it signals that the symmetric DFT solution is unreliable and one should either:
1. Allow symmetry breaking (magnetic ordering) within standard DFT to restore the correct physics.
2. If symmetry must be preserved, switch to explicit many-body methods (DFT+U, DMFT).
Physical Insight: The work bridges the gap between the Stoner criterion (instability driven by $I \times D(\epsilon_F)$ ) and strong correlation. It suggests that a high DOS at the Fermi level is a hallmark of strong correlation that can be "tamed" by symmetry breaking, effectively transforming a difficult many-body problem into a manageable single-reference problem.
Computational Efficiency: By relying on standard DFT with symmetry breaking, the method avoids the high computational costs and parameter tuning associated with DFT+U or DMFT, offering a more automated pathway to describing correlated materials.

In summary, the paper concludes that symmetry breaking in the Kohn-Sham system is not merely a numerical artifact but a physical mechanism that resolves strong correlation by lifting nearly degenerate states, and it provides the parameter $\Gamma$ as a robust tool to identify when this mechanism is necessary.

Identifying strong correlation using only the Kohn-Sham density of one-electron states