Statistical uncertainty quantification for multireference covariant density functional theory

This paper presents a Bayesian framework for quantifying statistical uncertainties in covariant density functional theory by sampling parameter variations and propagating posterior distributions via a subspace-projected approach, successfully reproducing observables for deformed nuclei while highlighting current limitations in describing near-spherical systems.

The Gist

Imagine you are trying to predict the weather. You have a super-complex computer model that uses physics to simulate the atmosphere. But, like any model, it's not perfect. The "ingredients" in your recipe (the parameters) might be slightly off, or the model might miss some tiny details. If you run the model once, you get one forecast. But how much should you trust it? What if the ingredients were slightly different? Would the prediction change from "sunny" to "stormy"?

This paper is about building a super-smart, ultra-fast weather forecaster for the atomic nucleus.

Here is the breakdown of what the scientists did, using some everyday analogies:

1. The Problem: The "Perfect" Recipe Doesn't Exist

In nuclear physics, scientists use something called Density Functional Theory (DFT). Think of this as a giant recipe book for building atomic nuclei.

• The Issue: There isn't just one perfect recipe. There are many versions (called EDFs), and they all have slightly different ingredients (parameters).
• The Challenge: When scientists want to study the "low-lying states" of a nucleus (its energy levels and how it vibrates), the math gets incredibly heavy. It's like trying to bake a million different cakes, one by one, to see which one tastes best. It takes too long and costs too much computer power.
• The Uncertainty: Because we don't know the exact perfect recipe, there is a "statistical uncertainty." We need to know: If we tweak the ingredients a little bit, how much does the final cake change?

2. The Solution: The "Smart Shortcut" (SP-CDFT)

The authors developed a new method called SP-CDFT.

• The Old Way: To study a nucleus, you would have to run the heavy, slow simulation for every single possible variation of the recipe. This is like baking a million cakes from scratch to find the average taste.
• The New Way (The Shortcut): Imagine you bake 14 "training cakes" using different variations of the recipe. You taste them and memorize how they feel. Now, if you want to know what a new cake (a new recipe variation) would taste like, you don't bake it. Instead, you use a mathematical shortcut (called Eigenvector Continuation) to guess the taste based on the 14 cakes you already baked.
• The Result: This shortcut is 10,000 to 100,000 times faster. It allows them to simulate one million different nuclear recipes in the time it used to take to simulate just a few.

3. The Bayesian "Detective Work"

Once they had this super-fast shortcut, they used a statistical detective method called Bayesian Analysis.

• The Clues: They gathered "clues" from the real world:
  1. How dense is nuclear matter? (Like the density of a neutron star).
  2. What do we know about the forces between protons and neutrons?
  3. What are the actual measurements of how nuclei vibrate (B(E2) values)?
• The Filter: They ran their million simulations and filtered out the ones that didn't match the real-world clues. This gave them a "probability map" of which recipes are most likely to be correct.

4. The Findings: Deformed vs. Spherical Nuclei

They tested this on four specific nuclei:

• The "Squashed" Nuclei (Deformed): Neodymium-150 and Samarium-150. These are shaped like rugby balls.
  • Result: The model worked beautifully! Once they accounted for the statistical uncertainty (the "fuzziness" of the recipe), the predictions matched the real-world data perfectly.
• The "Round" Nuclei (Near-Spherical): Xenon-136 and Barium-136. These are shaped like billiard balls.
  • Result: The model struggled here. Even with the uncertainty accounted for, the predictions were off.
  • Why? The authors suggest that for these round nuclei, the model is missing a specific type of "dance move" (quasiparticle excitations) that the current recipe doesn't include. It's like trying to predict a waltz using only a recipe for a march; the steps just don't fit.

5. The Big Picture Takeaway

This paper is a major step forward because it finally gives scientists a way to say, "We are 92% confident that our prediction is within this range."

• For Deformed Nuclei: The model is reliable.
• For Spherical Nuclei: The model needs a new ingredient (a better recipe) to work.

In summary: The scientists built a "time machine" that lets them instantly test millions of nuclear theories. They found that while their current theory is excellent for squashed, rugby-ball-shaped nuclei, it needs a little upgrade to perfectly describe the round, billiard-ball-shaped ones. This helps physicists know exactly where their theories are strong and where they need to do more homework.

Technical Summary

1. Problem Statement

Nuclear Density Functional Theory (DFT) is a powerful framework for describing nuclear properties, but it faces significant challenges regarding uncertainty quantification (UQ).

• Systematic vs. Statistical Errors: While systematic errors (arising from the functional form itself) are difficult to quantify due to the phenomenological nature of current Energy Density Functionals (EDFs), statistical errors (arising from parameter variations around optimal values) can be quantified.
• Computational Bottleneck: Extending DFT to multireference (MR) calculations (necessary for low-lying excited states, symmetry restoration, and collective correlations) is computationally expensive. Quantifying statistical uncertainties requires running thousands to millions of MR-DFT calculations with varying parameter sets, which is currently infeasible due to the high computational cost of the Generator Coordinate Method (GCM) and symmetry projection.
• Accuracy Gap: Existing models often struggle to describe near-spherical nuclei and specific transition strengths, and the lack of rigorous error bars makes it difficult to assess the reliability of predictions against experimental data.

2. Methodology

The authors propose a comprehensive framework combining Covariant Density Functional Theory (CDFT), Multireference (MR) methods, and Eigenvector Continuation (EC) to enable efficient statistical sampling.

A. Theoretical Framework

• Single-Reference CDFT (SR-CDFT): Uses a relativistic point-coupling functional (PC-PK1) with nine low-energy coupling constants (LECs). The nuclear wave function is a Slater determinant.
• Multireference CDFT (MR-CDFT): To describe low-lying states, the wave function is constructed as a superposition of quantum-number projected mean-field states (restoring angular momentum $J$, parity, and particle number $N, Z$) using the Generator Coordinate Method (GCM). This involves calculating Hamiltonian and norm kernels over a grid of deformation parameters ($q$).
• Subspace-Projected CDFT (SP-CDFT): This is the core innovation. It acts as an emulator for MR-CDFT using the Eigenvector Continuation method.
  • Training Phase: A small set of "training" EDF parameter sets ($N_t$) are used to perform full MR-CDFT calculations to generate a basis of wave functions.
  • Target Phase: For a new "target" parameter set ($C_\odot$), the wave function is approximated as a linear combination of the training wave functions.
  • Efficiency: The Hamiltonian kernels for the target set are factorized. The interaction energy is decomposed into nine terms, allowing the parameter dependence to be factored out. This reduces the computational cost from $O(N_q^2 N_{target})$ to a negligible cost after the initial training, enabling millions of calculations.

B. Statistical Analysis Strategy

1. Sampling: Approximately 1 million parameter sets were generated by varying the 9 parameters of the PC-PK1 functional around their optimal values using quasi-Monte Carlo sampling.
2. Nuclear Matter Constraints: The authors first constrained the parameter space using empirical values of nuclear matter at saturation density (density $\rho_0$, binding energy $E/A$, symmetry energy $E_{sym}$, slope $L$, incompressibility $K$) and predictions from chiral effective field theory. An "implausibility function" was used to filter out parameter sets that deviated significantly from these constraints ($\sqrt{3}\sigma$ rule), resulting in ~457,000 non-implausible samples.
3. Bayesian Inference:
   • Likelihood: Constructed using experimental data for finite nuclei (specifically $B(E2)$ transition strengths) and the filtered nuclear matter properties.
   • Posterior Distributions: Derived for the model parameters ($C$) and subsequently propagated to observables ($O$) of finite nuclei.
   • Error Decomposition: Total uncertainty is treated as a combination of emulator error (estimated by benchmarking SP-CDFT against MR-CDFT) and systematic error (estimated by comparing PC-PK1 predictions to data).

3. Key Contributions

• Development of SP-CDFT: Successfully adapted the Eigenvector Continuation method to Multireference CDFT, creating a fast emulator that accelerates calculations by orders of magnitude (speed-up factors up to $10^4$ for $10^6$ samples).
• First Statistical UQ for MR-CDFT: Provided a rigorous quantification of statistical uncertainties for low-lying nuclear states (excitation energies and $B(E2)$ values) within a relativistic framework.
• Bayesian Parameter Refinement: Demonstrated how to update parameter distributions using a combination of infinite nuclear matter constraints and finite-nuclei spectroscopic data.
• Correlation Analysis: Revealed complex correlations between observables (e.g., the anti-correlation between excitation energy and $B(E2)$ strength in deformed nuclei, and the varying correlation between proton radius and $B(E2)$ depending on nuclear structure).

4. Results

The framework was applied to four nuclei: deformed $^{150}$Nd and $^{150}$Sm, and near-spherical $^{136}$Xe and $^{136}$Ba.

• Benchmarking: The SP-CDFT emulator showed high accuracy compared to full MR-CDFT:
  • Ground-state energy and proton radius errors: < 0.2%.
  • Excitation energy ($2^+_1$) errors: ~13% (relative to the small energy scale).
  • $B(E2)$ transition strengths: Reproduced with high accuracy.
• Deformed Nuclei ($^{150}$Nd, $^{150}$Sm):
  • The model successfully reproduced experimental excitation energies and $B(E2)$ values within the calculated statistical uncertainties (uncertainties up to 21% for excitation energies and 12% for $B(E2)$).
  • The posterior distributions of parameters derived from these nuclei aligned well with the original PC-PK1 values.
• Near-Spherical Nuclei ($^{136}$Xe, $^{136}$Ba):
  • The model overestimated excitation energies and failed to reproduce the experimental $B(E2)$ values accurately.
  • The posterior distributions for parameters derived from $^{136}$Xe data were significantly offset from PC-PK1 values, indicating a model deficiency.
  • The ratio $R_{42} = E(4^+_1)/E(2^+_1) < 2$ suggests these states are dominated by seniority coupling (non-collective), which the current collective MR-CDFT framework (based on shape fluctuations) does not fully capture.

5. Significance

• Validation of Framework: The study validates SP-CDFT as a viable tool for high-throughput statistical analysis in nuclear physics, making previously impossible "million-run" studies feasible on standard computing resources.
• Identification of Model Limitations: By quantifying uncertainties, the authors clearly distinguish between statistical noise and systematic model failure. The inability to describe near-spherical nuclei highlights the need to extend the model space to include quasiparticle excitations and non-collective configurations.
• Future Directions: The work establishes a pathway for refining nuclear EDFs using Bayesian methods and low-lying state data, moving the field toward high-precision nuclear structure predictions essential for understanding neutron stars and rare nuclear processes (like $0\nu\beta\beta$ decay).