Nuclear symmetry breaking rearranges where correlations live

In nuclear physics, researchers often simplify a complicated many-particle system by describing it with a mean-field state: a manageable approximation that treats particles as moving in an average environment. Such a state can break symmetries respected by the underlying Hamiltonian. In nuclear calculations, this can be useful, but it means the approximate state may lack definite particle number or angular momentum. Projection methods can restore those quantum numbers afterward.

A familiar interpretation is that symmetry breaking allows the mean-field state to take in correlations while retaining a simple mathematical structure, and that restoring symmetry adds further correlations. But what does “taking in correlations” mean at the level of the particles and groups of particles? The authors examine that question in the single-j model, a simplified but physically motivated model of nucleons in one valence shell, interacting through pairing and quadrupole forces.

The team combines Hartree-Fock-Bogoliubov (HFB) mean-field calculations with projections that restore angular momentum, particle number, or both. They then use fermionic quantum-information measures to describe correlations among single-particle orbitals. The choice of orbitals matters: some correlation measures depend on the basis used to describe the system. The authors therefore analyze the states in a natural-orbital basis, which diagonalizes the one-body density matrix and helps remove dependence on arbitrary orientation of a deformed state.

Their measures distinguish several questions. Mutual information tracks total correlations between two subsystems; quantum discord captures quantum correlations, including correlations beyond entanglement; and two-orbital entropy indicates how a pair of orbitals is correlated with the rest of the system. These are properties of a chosen partition of the state, not direct measurements of observable quantities such as energy.

In the HFB states, the model’s behavior depends on the competition between pairing and quadrupole interactions. In the spherical region, paired orbitals carry strong correlations, while the system has zero total angular momentum. As the quadrupole interaction drives the state into a deformed region, correlations within those pairs generally decrease, and the pattern depends on the particle number and orbital pair. For one particle-number setting studied, the calculations show a further transition at which pairing correlations disappear and the HFB state becomes a Slater determinant. That additional feature is not found in all the cases examined.

This pattern challenges a simple, universal picture of symmetry breaking as a way to maximize correlations within a mean-field state. Here, rotational symmetry breaking is associated with reduced correlations within pairs, while the particle-number behavior can differ. The authors suggest that the interplay between these symmetries helps explain why the correlation pattern is not straightforward.

Projection changes where correlations are located. Restoring angular momentum in the deformed region makes a paired-orbital subsystem mixed and creates correlations between that pair and the rest of the system. Correlations within pairs decrease as correlations involving larger subsystems grow. When both angular momentum and particle number are restored, the authors likewise find correlations among orbital pairs, even though pairwise quantum discord vanishes under their analysis. In the spherical region, particle-number restoration can also change the structure: the paired-orbital state becomes mixed, with correlations extending beyond an individual pair.

The paper calls some correlations in the mean-field description “fake” in the sense that they arise within the restricted product-of-pairs structure and are redistributed when symmetries are restored. This does not mean the calculations show correlations disappearing from the full physical system. Rather, the measures assign them to different subsystems after the state and its symmetry properties change.

These findings are model-based, not a direct study of an experimentally measured nucleus. The single-j model captures selected features of nuclear interactions but is still a simplification. The authors also note that their correlation measures depend on how the system is partitioned, and that comparing the calculated correlation structure with an exact ground state is future work. The results therefore refine a conceptual picture within this model; they do not establish that the same detailed redistribution occurs in every nucleus.

The work is available as an arXiv preprint. A preprint is a public research manuscript and may not have completed peer review. Its conclusions should be read as the authors’ findings in this specific theoretical framework, rather than as settled consensus.