Particle-number restoration within the energy density functional formalism

We give a detailed analysis of the origin of spurious divergences and finite steps that have been recently identified in particle-number-restoration calculations within the nuclear energy density functional framework. We isolate two distinct levels of spurious contributions to the energy. The first one is encoded in the definition of the basic energy density functional itself, whereas the second one relates to the canonical procedure followed to extend the use of the energy density functional to multi-reference calculations. The first level of spuriosity relates to the long-known self-interaction problem and to the newly discussed self-pairing interaction process that might appear when describing paired systems with energy functional methods using auxiliary reference states of Bogoliubov or BCS type. A minimal correction to the second level of spuriosity to the multi-reference nuclear energy density functional proposed in [D. Lacroix, T. Duguet, and M. Bender, Phys. Rev. C 79, 044318 (2009)] is shown to remove completely the anomalies encountered in particle-number-restored calculations. In particular, it restores sum rules over (positive) particle numbers that are to be fulfilled by the particle-number-restored formalism. The correction is found to be on the order of several hundreds of keVs up to about 1 MeV in realistic calculations, which is small compared to the total binding energy but often accounts for a substantial percentage of the energy gain from particle-number restoration and is on the same energy scale as the excitations one addresses with multi-reference energy density functional methods.

Figure 1

Particle-number-restored deformation energy surface of ${}^{18}\mathrm{O}$ calculated with SLy4 and a density-dependent pairing interaction and the corresponding single-particle spectra of protons and neutrons as a function of the axial quadrupole deformation for $L=5$ and 199 discretization points of the integral over the gauge angle (lowest panel). There are clear anomalies that appear when either a proton or neutron single-particle level crosses the Fermi energy. The dimensionless quadrupole deformation ${\beta }_2$ is defined in Eq. (66).

Figure 2

Schematic view of the analytical structure of the transition densities defined in Eqs. (38, 39, 40) and of the PNR functional energy kernel $\mathcal{E}[\phi ]$ in the complex plane. Poles marked with filled circles are within the standard circular integration contour of radius $R=1$, whereas those outside are marked with open circles. The cross marks the location of the SR energy functional at $z=1$. The operator $e^{i\phi \hat{N}}$ produces a rotation in gauge space, whereas $e^{\eta \hat{N}}$ is a shift transformation as defined in Eq. (45).

Figure 3

Dispersion of the proton and neutron number of the unprojected SR state and the particle-number-projected SR using 3, 5, or 7 discretization points of the gauge-space integrals as a function of their deformation. For 5 points the projected state is sufficiently converged; for 7 and more points (not shown) the dispersion cannot be distinguished from numerical noise.

Figure 4

Spectrum of poles $z_{\mu }=|u_{\mu }/v_{\mu }|$ for protons (top panel) and neutrons (middle panel), which for levels in the vicinity of the Fermi energy resembles a stretched and slightly distorted Nilsson diagram. The dashed red line at $z=1$ denotes the radius of the standard integration contour $R=1$. The bottom panel shows the particle-number-projected quadrupole deformation energy for $L=5$ and 199 discretization points for the integral in gauge space. The insert shows a close-up of the steps at small deformation.

Figure 5

Correction for neutrons (top panel) and protons (middle panel) and energy gain from projection without and with correction for ${}^{18}\mathrm{O}$ as a function of quadrupole deformation for 5 and 199 discretization points for the integrals in gauge space. The corrected energy gain in independent on the discretization of the integrals when 5 or more angles are used. All panels share the same energy scale.

Figure 6

Corrected (solid line) and uncorrected (dotted and dashed lines) particle-number-projected quadrupole deformation energy for ${}^{18}\mathrm{O}$, calculated with $L=5$ and 199 discretization points of the integral in gauge space. The corrected curves are identical.

Figure 7

Spurious energy from the single-particle orbits that correspond to the spherical neutron $d_{5/2^{+}}$ level in ${}^{18}\mathrm{O}$ as a function of quadrupole deformation (see text).

Figure 8

Same as Fig. 7, but for the proton $1/2^{+}$ and $1/2^{-}$ levels that give the dominant contributions to the proton correction at prolate deformation and cross at the Fermi energy at ${\beta }_2=0.67$.

Figure 9

Projected energy for ${}^{18}\mathrm{O}$ at the deformation ${\beta }_2=0.371$ as a function of the radius $R_p$ of the integration contour calculated without and with correction using 5 and 199 angles. The energy scale on the left gives the absolute energy, the scale on the right the energy gain from projection. The insert magnifies the curves around $R_p=1$. The regularized PNR energy in independent on the discretization of the integrals when 5 or more angles are used. The integration contour for projection on neutron number is $R_n=1$ in all cases.

Figure 10

Weight $c_Z^2(R_p=1)=|\langle {\Psi }^Z|{\Phi }_1\rangle |{}^2$ of the normalized proton-number-projected states in the SR HFB state (upper panel), the weighted spurious energy $c_Z^2(R_p=1){\mathcal{E}}_{\mathrm{CG}}^Z(R_p=1)$ (middle panel), the nonregularized weighted PNR energies $c_Z^2(R_p=1){\mathcal{E}}^Z(R_p=1)$ and regularized $c_Z^2(R_p=1){\mathcal{E}}_{\mathrm{REG}}^Z(R_p=1)$ (lower panel). All results are obtained using the same SR state calculated for ${}^{18}\mathrm{O}$ at a deformation of ${\beta }_2=0.371$ as auxiliary state. The neutron number is not restored.

Figure 11

Weight $c_Z^2(R_p=2.5)=|\langle {\Psi }^Z|{\Phi }_{R_p}\rangle |{}^2$ of the normalized proton-number-projected states into the radially shifted SR HFB state at $R_p=2.5$ (upper panel), the weighted spurious energy $c_Z^2(R_p=2.5){\mathcal{E}}_{\mathrm{CG}}^Z(R_p=2.5)$ (upper middle panel), the nonregularized ${\mathcal{E}}^Z(R_p=2.5)$ and regularized ${\mathcal{E}}_{\mathrm{REG}}^Z(R_p=2.5)$ PNR energies weighted by $c_Z^2(R_p=2.5)$ (lower middle panel) and by $c_Z^2(R_p=1)$ (lower panel). All results are obtained using the same SR state calculated for ${}^{18}\mathrm{O}$ at a deformation of ${\beta }_2=0.371$ as auxiliary state. The neutron number is not restored.

Figure 12

Weight of the normalized state projected on various values of $Z$ in the SR vacuum (top panel) and decomposition of the energy into $Z$ components for three different radii of the integration contour for protons (bottom panel) for ${}^{18}\mathrm{O}$ at a deformation of ${\beta }_2=0.371$. All states are projected on the same neutron number $N=10$ with an integration contour of radius $R_n=1$, using $L=199$ integration points for both protons and neutrons. Corrected PNR energies are the same for all values of $R_p$ within numerical accuracy.

Figure 13

Spectrum of poles $z_{\mu }=|u_{\mu }/v_{\mu }|$ for protons and neutrons, the uncorrected and corrected energy gain from projection and the particle-number-projected quadrupole deformation energy for $L=9$ and 99 discretization points of the integral in gauge space for ${}^{76}\mathrm{Kr}$.

Figure 14

Spectrum of poles $z_{\mu }=|u_{\mu }/v_{\mu }|$ for protons and neutrons, the correction to the particle-number-restored EDF separately for protons and neutrons, the uncorrected and corrected energy gain from projection, and the particle-number-projected quadrupole deformation energy without and with correction for $L=13$ and 99 discretization points of the integral in gauge space for ${}^{186}\mathrm{Pb}$.