Nuclear entanglement and deformation

Abstract

The single-j model is a simple toy Hamiltonian which captures certain phenomenological properties of a nuclear shell with fixed angular momentum j, namely, nucleon pairing and quadrupole deformation. We characterize the ground and excited states of this Hamiltonian, in light of its symmetries, for a shell with an even number of nucleons, building up an intuitive picture of the exact solutions. We see that, while the ground state is a rotationally-symmetric state with zero total angular momentum and vanishing deformation, the excited states are organized into multiplets with definite total angular momentum J2 and are in general deformed states. The effect of the quadrupole term is to break the degeneracy between multiplets with distinct J2. Narrowing our attention to the ground state, we next investigate entanglement between single-particle nuclear orbitals via the tools of quantum information: reduced density matrices, mutual information, entanglement entropy, and entanglement of formation. Along the way we consider how “entanglement” should be defined for systems of indistinguishable fermions, meeting tools such as the natural basis of single-particle states, the Jordan-Wigner isomorphism, and the fermionic partial trace, and considering subtleties including particle number conservation and “fluffy-bunny” entanglement which complicate the much more familiar picture of entanglement in systems of distinguishable particles. Ultimately, we find that quantum correlations in the ground state of the single-j model arise from entanglement involving four orbitals, which perhaps can be thought of as entanglement between pairs of “Cooper pairs”. For ≥ 9 ,2 ,and with N = 4 nucleons, the Cooper pairs are disrupted by the quadrupole term, leading to a more complicated entanglement structure. Finally, we compare the ground state of the single-j model with the ground state of a more realistic phenomenological nuclear Hamiltonian and find a surprisingly close fit, suggesting that the single-j model is an excellent starting point for understanding entanglement in real nuclei