Mean-field proton-neutron pairing correlations with the Gogny D1S energy density functional

This paper investigates proton-neutron pairing correlations using the Gogny D1S functional within a generalized Hartree-Fock-Bogoliubov framework, finding that the functional exhibits numerical instabilities in large single-particle spaces and that self-consistent energy minima in $sd$-shell nuclei correspond to vanishing proton-neutron pairing.

The Gist

The Tale of the Unruly Dance Partners: A Summary

Imagine you are organizing a massive, high-stakes ballroom dance competition. In this ballroom, there are two main groups of dancers: the Protons and the Neutrons.

For decades, nuclear physicists have been studying how these dancers move. Usually, they assume the dancers follow strict rules: Protons only dance with other Protons, and Neutrons only dance with other Neutrons. This is a very organized, predictable way to run a ballroom.

However, this paper explores a much more chaotic and interesting scenario: What happens if we allow the Protons and Neutrons to mix and dance with each other? This "mixed dancing" is what scientists call Proton-Neutron Pairing.

Here is the breakdown of what the researchers discovered, using the ballroom as our guide.

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1. The "Gogny" Choreography (The Problem)

To predict how these dancers move, scientists use a set of "choreography manuals" called Energy Density Functionals. The most famous manual used in this study is called Gogny D1S.

The Gogny manual is excellent for standard dances (where Protons stay with Protons). But the researchers found that the manual has a "glitch" in its instructions. When you tell the dancers to start mixing (Proton-Neutron dancing), the manual’s instructions for certain movements become so intense and nonsensical that the dancers lose control entirely. Instead of a beautiful dance, the ballroom descends into a riot—the math literally "breaks," and the computer can't find a stable way to describe the movement.

2. The "B1" Backup Plan (The Comparison)

To see if the problem was the dance or the manual, the researchers tried a different, older manual called B1.

The B1 manual is much simpler—it doesn't have the complex, "density-dependent" instructions that the Gogny manual has. When the researchers applied the B1 manual to the mixed dancing, everything worked perfectly! The dancers moved smoothly, the math stayed stable, and the ballroom remained orderly.

The Lesson: The chaos wasn't caused by the Protons and Neutrons mixing; it was caused by a specific "instruction" in the Gogny manual (the zero-range density term) that reacts badly when the groups mix.

3. The "Stiff" vs. "Soft" Dances (The Results)

Even though the Gogny manual was "unstable" in large rooms, the researchers used a smaller, controlled practice room to see how the dancers would behave. They looked at different types of "pairings":

• The Isoscalar Dance (The Stiff Dance): This is the most intense mixed dance. The researchers found it is very "stiff." If you try to force the dancers into this specific pattern, the energy required is massive. It’s like trying to force a group of people to move in perfect, rigid synchronization—it takes an incredible amount of effort.
• The Isovector Dance (The Soft Dance): This is a slightly more relaxed version of mixed dancing. It’s "softer," meaning the system can handle it with a bit more ease.
• The Like-Particle Dance (The Standard): Dancing with your own kind (Proton-Proton) is the most natural and "easy" state for these nuclei.

4. The Grand Conclusion

The researchers concluded that our current "choreography manuals" (like Gogny D1S) are a bit outdated. They were written for a world where Protons and Neutrons stay in their own lanes.

If we want to truly understand the heart of the atom—where these particles are constantly mixing and interacting—we need to write new manuals. We need instructions that can handle the "mixed dancing" without causing a mathematical riot.

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In short: The paper tells us that while we have great maps for how nuclei behave in simple situations, our current maps "break" when we try to look at the complex, mixed-particle interactions. We need better maps to navigate the true complexity of the subatomic world.

Technical Summary

Technical Summary: Mean-field proton-neutron pairing correlations with the Gogny D1S energy density functional

1. Problem Statement

The study addresses a critical limitation in the application of standard Energy Density Functionals (EDFs), specifically the widely used Gogny D1S, when moving beyond traditional mean-field approximations. Most Gogny-type functionals were optimized under the assumption of isospin symmetry, where proton-neutron ($pn$) mixing in the Hartree-Fock-Bogoliubov (HFB) transformation is neglected. This restriction effectively excludes $pn$-pairing correlations (both isovector $T=1$ and isoscalar $T=0$ channels) from the mean-field description.

The authors investigate what happens when these symmetry restrictions are lifted. Specifically, they aim to determine if the Gogny D1S functional remains numerically stable and physically consistent when the HFB framework is generalized to allow for $pn$-mixing, which activates new degrees of freedom in the pairing field.

2. Methodology

• Theoretical Framework: The authors utilize the Hartree-Fock-Bogoliubov (HFB) framework. They generalize the Bogoliubov transformation to allow for the mixing of proton and neutron single-particle states, thereby explicitly including $T=1$ and $T=0$ $pn$-pairing channels.
• Computational Tool: The TAURUS numerical code was extended. While originally designed for two-body Hamiltonians, it was modified to handle density-dependent terms and the necessary rearrangement terms that arise when applying the variational principle to density-dependent functionals.
• Comparative Analysis: To isolate the cause of potential instabilities, the authors compare the Gogny D1S functional against the Brink-Boecker B1 interaction. The B1 interaction is a Hamiltonian-based model that lacks the zero-range density-dependent term but includes a zero-range spin-orbit term, providing a "stable" baseline for comparison.
• Constraints and Observables: Constrained HFB calculations were performed in reduced configuration spaces (specifically the $sd$-shell) to explore Total Energy Curves (TECs). They varied collective coordinates such as axial quadrupole deformation ($\beta_2$) and various pairing parameters ($\delta^{T}_{\tau\tau'}$) to map the energy landscape.

3. Key Contributions

• Code Extension: Successful implementation of density-dependent Gogny-type functionals within a generalized HFB scheme that allows for $pn$-mixing.
• Stability Diagnosis: Identification of a fundamental instability in the Gogny D1S functional triggered by the zero-range density-dependent term when $pn$-pairing channels are active.
• Systematic Mapping: Detailed mapping of the energy response of $sd$-shell nuclei to different types of pairing correlations (pp, nn, isovector $pn$, and isoscalar $pn$).

4. Results

• Numerical Instability: When using large single-particle spaces (high number of harmonic oscillator shells), the Gogny D1S functional becomes completely unstable if $pn$-mixing is allowed. The HFB energy and gradient exhibit erratic, non-convergent behavior. In contrast, the B1 interaction remains stable, proving that the instability is specifically linked to the zero-range density-dependent term in the Gogny functional.
• Energy Landscapes: In constrained calculations (using smaller, regularized bases), the authors found that the self-consistent HFB minima always correspond to vanishing $pn$-pairing.
• Pairing Stiffness: The total energy increases rapidly as $pn$-pairing is introduced. The isoscalar ($T=0$) channel is the "stiffest" (energy increases most rapidly), followed by the isovector ($T=1$) $pn$ channel, then the like-particle (pp/nn) channels.
• Isospin Effects: In $N=Z$ nuclei, the isovector $pn$-pairing channel becomes more competitive, whereas in neutron-rich isotopes, the like-particle (nn) pairing dominates the energy landscape.

5. Significance

This paper serves as a cautionary guide for nuclear theorists. It demonstrates that functionals optimized at the standard mean-field level may be "pathological" when used in Beyond-Mean-Field (BMF) calculations (such as the Generator Coordinate Method or symmetry restoration) that involve $pn$-pairing fluctuations.

The findings highlight the necessity of developing new generations of functionals—such as those with finite-range density dependence (e.g., D2 or DG Gogny interactions)—that are explicitly designed to be stable and reliable in generalized HFB environments where proton-neutron mixing is a fundamental degree of freedom.