Formulation of functional theory for pairing with particle number restoration

The restoration of particle number within energy density functional theory is analyzed. It is shown that the standard method based on configuration mixing leads to a functional of both the projected and nonprojected densities. As an alternative that might be advantageous for mass models, nuclear dynamics, and thermodynamics, we propose to formulate the functional in terms directly of the one-body and two-body density matrices of the state with good particle number. Our approach does not contain the pathologies recently observed when restoring the particle number in an energy density functional framework based on transition density matrices and can eventually be applied with functionals having arbitrary density dependencies.

Figure 1

Particle number restored deformation energy curve of ${}^{18}\mathrm{O}$ as a function of quadrupole deformation ${\beta }_2$ and calculated within standard MR-EDF technique using projection after variation (PAV) with SIII and a delta pairing interaction before (blue dashed curve) and after (black solid curve) regularization. The red filled circles correspond to the results obtained using directly Eq. (19) (see text). To compare with previous work [8], the Coulomb exchange contribution has been subtracted from the energy.

Figure 2

Particle number restored deformation energy surface of ${}^{18}\mathrm{O}$ calculated using Eq. (19). The dotted curve is obtained by making the replacement (20) in Eq. (19).

Figure 3

Same as Fig. 1, but when the SLy4 effective interaction is used in the particle-hole channel. The dashed line corresponds to the nonregularized MR-EDF result directly obtained by gauge angle integration using 199 points in the discretization. The filled circles correspond to the result obtained using the symmetry-conserved EDF using Eq. (19) and the prescription (20).