Introduction

The proton-neutron (p-n) pairing is a long-standing open problem in nuclear physics, and its possible relations to various nuclear phenomena have been widely discussed [1]. However, in spite of the recent impressive experimental progress and theoretical studies, the understanding of the p-n pairing is still unsatisfactory. To address this problem, we use the nuclear density functional approach. Our ultimate goal is to develop a superfluid symmetry-unrestricted energy-density-functional (EDF) approach including the p-n mixing both in the pairing and particle-hole (p-h) channels. Indeed, in accordance with fundamental self-consistency requirements of the Hartree-Fock(-Bogoliubov) (HF(B)) equations, any generalization of quasiparticle states as mixtures of proton and neutron components must be necessarily accompanied by, somewhat less intuitive, mixing of proton and neutron single-particle (s.p.) wave functions.

Recently, as the first step in developing the superfluid EDF theory including the p-n mixing, by extending the codes HFODD [2] and HFBTHO [3], in Refs. [4,5] we have developed a s.p. EDF formalism including the p-n mixing in the p-h channel. In this p-n mixing calculation, we applied the so-called isocranking method by adding the isocranking term to the Hamiltonian: $\hat{h'}=\hat {h}-\vec{\lambda} \cdot \hat{\vec{t}}$, where $\hat {\vec{t}}$ is the isospin operator.

Our model is based on a local Skyrme EDF extended to include the p-n mixing by following the general rule given by Perlinska et al. [1]. Starting from the local density matrix $\rho({\bf r},tt')$ ($t$ and $t^\prime$ are the isospin indices), we built the isoscalar $\rho_0({\bf r})= \sum_{tt'} \rho({\bf r},tt')\hat{\tau}^0_{t't}$ and isovector $\vec{\rho}({\bf r})= \sum_{tt'} \rho({\bf r},tt')\hat{\vec{\tau}}_{t't}$ densities by contracting $\rho({\bf r},tt')$ with the isospin identity matrix $\hat{\tau}_0$ and isospin Pauli matrices $\hat{\vec{\tau}}$, respectively. The isoscalar density $\rho_0({\bf r})$ and isovector $z$ component are the sum and difference of neutron and proton densities, respectively. These densities are included in the conventional EDF calculations. The $x$ and $y$ components of the isovector densities are new elements, which we take into account to extend the EDFs, and which are nonzero only for the p-n mixed s.p. states.

In the following, we present selected numerical results obtained in $A=40$ and $A=54$ nuclei for the SkM* EDF parameters set[6]. The applications are divided into two classes, without and with the Coulomb interaction. The reason is that we have extended our Skyrme EDFs such that they are invariant under the rotation in the isospin space. If the Coulomb interaction is switched off, the total and s.p. energies should be independent of the isospin direction of the system, which allows us to validate numerical implementation of the code. The Coulomb interaction, when included, is calculated exactly both in the direct and exchange channels.