⚛️ nuclear theory

Pfaffian and minor formulas in number-conserved pairing theory

This paper presents closed-form Pfaffian and minor formulas for calculating norms and density matrices in number-conserved pairing theory for both even and odd systems, which are then applied to angular-momentum-projected BCS calculations of $^{70}$ Ge and $^{69}$ Ge to interpret their low-lying collective bands.

Original authors: G. J. Fu

Published 2026-10-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: G. J. Fu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Inside the heart of an atom, protons and neutrons do not simply sit in a static pile; they move in a complex, coordinated dance of quantum mechanics. To understand how these tiny particles behave, scientists often rely on theories that treat them as if they are paired up, much like dancers holding hands. This pairing is a fundamental force that helps hold the nucleus together and dictates its shape. However, a major challenge in this field has been that the most common mathematical tools used to describe these pairs often break a crucial rule of nature: they allow the total number of particles to fluctuate. While this approximation works well for very large systems, it fails when studying smaller, finite objects like atomic nuclei, where counting every single particle is essential for accuracy. Researchers have long needed a way to describe these paired systems while strictly keeping the particle count fixed, a task that requires navigating a labyrinth of complex calculations to predict how these nuclei vibrate, rotate, and change shape.

A new study by physicist G. J. Fu at Tongji University in Shanghai provides a powerful new set of mathematical tools to solve this specific problem. The researcher has developed precise, closed-form formulas that allow scientists to calculate the properties of atomic nuclei while strictly conserving the number of particles. These formulas act as a bridge, connecting different theoretical states of a nucleus and allowing researchers to mix them together to find the most accurate description of reality. Unlike previous methods that struggled with the complexity of broken pairs or odd numbers of particles, this new approach works seamlessly for both even and odd numbers of particles. It handles situations where pairs are intact and where they are broken, offering a unified way to calculate the "norm," or the probability of finding a nucleus in a certain state, and the matrix elements, which describe how the nucleus responds to various forces. By using these formulas, scientists can now apply advanced techniques, such as projecting the total spin of the nucleus, to reveal the true structure of these tiny worlds without losing the critical detail of particle conservation.

To test the power of these new formulas, the study applied them to two specific isotopes of the element germanium: germanium-70 and germanium-69. Germanium-70 is an even-numbered nucleus, meaning it has an equal number of protons and neutrons that can all be paired up. Germanium-69, on the other hand, has an odd number of neutrons, leaving one particle unpaired. The researcher used a method called the angular-momentum-projected number-conserved BCS theory, which combines the new formulas with a technique that forces the nucleus to have a specific, well-defined spin. The goal was to see if this approach could accurately reproduce the known energy levels and shapes of these nuclei, particularly their collective bands, which are sequences of excited states that move together like a family.

For germanium-70, the calculations revealed that the nucleus does not have just one simple shape. Instead, the lowest energy states, known as the ground band and a gamma band, arise primarily from a triaxial shape, where the nucleus is stretched in three different directions, resembling a slightly squashed sphere that is not perfectly round. In contrast, a higher-energy beta band was found to come from a flatter, axially symmetric shape. The study showed that the ground state gains significant stability by mixing these different shapes together, a process that previous methods might have missed. The results matched experimental data remarkably well, correctly predicting the energy levels and the strength of the transitions between them. For instance, the model successfully described how the nucleus emits energy as it spins, distinguishing between different types of rotational bands and explaining why certain transitions are strong while others are weak.

The study also turned its attention to germanium-69, the odd-mass nucleus with the unpaired neutron. This case is more difficult because the unpaired particle interacts with the paired core in complex ways. The researcher constructed a model where the unpaired neutron was coupled to two different underlying shapes of the germanium-68 core. The calculations showed that the low-energy states of germanium-69 are not described by a single shape but by a mixture of these two cores. As the spin of the nucleus increases, the character of the state changes, shifting from being dominated by one core shape to the other. This mixing is crucial for understanding the positive-parity states, which require contributions from both underlying structures to be accurately described. The model successfully reproduced the observed energy levels and transition strengths, confirming that the new formulas can handle the added complexity of unpaired particles without losing precision.

The implications of this work extend beyond just these two isotopes. By providing a rigorous mathematical framework that works for both even and odd particle numbers, and for both intact and broken pairs, the study opens the door to more accurate simulations of a wide range of nuclear phenomena. It allows researchers to explore quantum phase transitions and the evolution of nuclear shapes with a level of detail that was previously difficult to achieve. The formulas serve as a versatile tool for the generator coordinate method, a technique used to mix different nuclear configurations to find the true ground state. In essence, this research refines the lens through which physicists view the atomic nucleus, ensuring that the fundamental rule of particle conservation is never violated, even as they explore the most intricate and collective behaviors of matter at the smallest scales. The findings suggest that the interplay between different shapes and the presence of unpaired particles are key to unlocking the full behavior of these quantum systems, offering a clearer path to understanding the building blocks of the universe.

Technical Summary: Pfaffian and Minor Formulas in Number-Conserved Pairing Theory

Problem Statement
Pairing theories, specifically the Bardeen-Cooper-Schrieffer (BCS) and Hartree-Fock-Bogoliubov (HFB) approaches, are standard tools for modeling quantum many-body systems. However, these methods inherently break U(1) gauge symmetry, violating particle-number conservation. While this violation is often negligible in large systems treated within the grand canonical ensemble, it is critical for finite-size systems such as atomic nuclei, where accurate descriptions of intrinsic properties and quantum phase transitions require strict particle-number conservation.

Existing approaches to restore particle number, such as particle number projection via numerical integration over gauge angles, can be computationally demanding. Alternatively, methods that directly conserve particle number, such as number-conserved pair condensates, offer a more direct framework. However, calculating norms and matrix elements for these states—particularly when dealing with broken-pair configurations (odd particle numbers) or mixing different canonical bases in the Generator Coordinate Method (GCM)—remains complex. Furthermore, angular momentum projection is often required when the underlying wave functions lack good angular momentum, necessitating efficient evaluation of overlap and Hamiltonian matrices.

Methodology
This work derives closed-form analytical formulas for the norm and matrix elements of many-body operators between wave functions with exact particle numbers. The formalism addresses both even and odd particle-number systems and accommodates configurations with broken pairs (seniority $\nu > 0$ ).

The core of the methodology involves:

Generalized Broken-Pair States: The formalism extends the standard $N$ -pair condensate $|c\rangle = (\hat{P}^\dagger)^N |0\rangle$ to broken-pair states $|\alpha\rangle = (\hat{P}^\dagger)^N \hat{c}^\dagger_{\alpha_1} \cdots \hat{c}^\dagger_{\alpha_\nu} |0\rangle$ , where $\nu$ represents the number of unpaired particles.
Algebraic Derivation: Using the generalized Kronecker delta and properties of unitary transformations between canonical bases, the overlap $\langle \alpha | \hat{R} | \beta' \rangle$ (where $\hat{R}$ is a rotation operator) is expressed as a sum over combinations of pair indices.
Pfaffian and Minor Formulations: The derived expressions are presented in two equivalent forms:
- Minor-based: The norm is expressed as a sum of determinants (minors) of submatrices formed by the transformation matrices $X = U^* W'^\top$ .
- Pfaffian-based: By rewriting the Cooper pair creation operator in terms of a matrix $P$ , the overlap is expressed as a sum involving Pfaffians of submatrices of $P$ and $Q$ (the rotated pair matrix), multiplied by determinants of the single-particle transformation matrices.
Operator Matrix Elements: The formalism is generalized to $m$ -body operators, providing compact expressions for matrix elements between different broken-pair states in different canonical bases. These formulas are applicable to configuration mixing in the GCM.

Key Contributions

Closed-Form Formulas: The paper provides explicit, closed-form expressions for norms and matrix elements that are valid for both even and odd particle numbers and for states with broken pairs.
Unified Framework: The derived formulas apply to pair condensates as well as configurations with unpaired particles, facilitating the treatment of odd-mass nuclei within the same theoretical framework as even-even nuclei.
Computational Efficiency: By expressing overlaps and matrix elements in terms of Pfaffians and minors, the work offers a computationally tractable route for symmetry restoration techniques, specifically angular momentum projection, without relying on numerical integration over gauge angles.

Results and Applications
The author applies the derived formalism to the angular-momentum-projected number-conserved BCS (PNBCS) theory to study the nuclei $^{70}\text{Ge}$ and $^{69}\text{Ge}$ using the JUN45 effective interaction in the $1p1/2, 1p3/2, 0f5/2, 0g9/2$ shell model space.

$^{70}\text{Ge}$ (Even-Even):
- The calculation mixes two intrinsic Hartree-Fock (HF) configurations: a triaxial solution $(\beta, \gamma) = (0.25, 36^\circ)$ and an axially symmetric oblate solution $(\beta, \gamma) = (0.13, 60^\circ)$ .
- The results reproduce the low-lying collective bands (ground, $\gamma$ , and $\beta$ ). The ground and $\gamma$ bands are found to arise predominantly from the triaxial configuration, while the $\beta$ band is associated with the oblate configuration.
- The calculated $B(E2)$ transition strengths and excitation energies show good agreement with experimental data and full-configuration-interaction (FCI) shell-model calculations.
- The study highlights the energy gain from angular momentum projection and configuration mixing, explaining why the ground-band $0^+$ state is lower in energy than the $\beta$ -band $0^+$ state despite the oblate HF solution being lower in energy prior to projection.
$^{69}\text{Ge}$ (Odd-Mass):
- The many-body basis is constructed by coupling a single unpaired neutron to the two pair condensates of $^{68}\text{Ge}$ (representing shape coexistence).
- The PNBCS calculations successfully describe six low-lying collective bands ( $5/2^-$ , $3/2^-$ , $1/2^-$ , $11/2^+$ , and two $9/2^+$ bands).
- Analysis of core overlaps ( $W_i$ ) reveals a spin-dependent evolution of the wave function structure. For negative-parity bands, the character shifts from being dominated by one intrinsic core at low spin to the other at higher spin.
- Positive-parity bandheads show significant mixing between the two core configurations, indicating that a single intrinsic structure is insufficient to describe these states.

Significance
The paper claims that the presented formalism provides a rigorous and efficient tool for performing symmetry-restored calculations in number-conserved pairing theory. By offering analytical formulas for norms and matrix elements involving broken pairs and different canonical bases, the work facilitates the application of the Generator Coordinate Method and angular momentum projection to both even and odd systems. The successful application to $^{70}\text{Ge}$ and $^{69}\text{Ge}$ demonstrates the method's capability to describe collective structures and shape coexistence, offering a viable alternative to standard symmetry-broken approaches or computationally intensive FCI methods for medium-mass nuclei.

More like this