Local density approximation for proton-neutron pairing correlations: Formalism

TLDR

The study extends coordinate‑space self‑consistent Hartree‑Fock‑Bogoliubov theory to include arbitrary proton‑neutron mixing in both particle‑hole and particle‑particle channels. The authors define HFB density matrices, construct a general quadratic energy‑density functional with full spin‑isospin structure, analyze local gauge invariance and Skyrme limits, and derive the self‑consistent HFB Hamiltonian while summarizing symmetry constraints. They obtain the self‑consistent one‑body HFB Hamiltonian, elucidate the resulting mean‑field structure, and provide a complete list of expressions needed to compute the total energy.

Abstract

In the present study we generalize the self-consistent Hartree-Fock-Bogoliubov (HFB) theory formulated in the coordinate space to the case which incorporates an arbitrary mixing between protons and neutrons in the particle-hole $(p\text{\ensuremath{-}}h)$ and particle-particle ($p\text{\ensuremath{-}}p$ or pairing) channels. We define the HFB density matrices, discuss their spin-isospin structure, and construct the most general energy-density functional that is quadratic in local densities. The consequences of the local gauge invariance are discussed and the particular case of the Skyrme energy-density functional is studied. By varying the total energy with respect to the density matrices the self-consistent one-body HFB Hamiltonian is obtained and the structure of the resulting mean fields is shown. The consequences of the time-reversal symmetry, charge invariance, and proton-neutron symmetry are summarized. The complete list of expressions required to calculate total energy is presented.