Subtraction method in the second random-phase approximation: First applications with a Skyrme energy functional

We make use of a subtraction procedure, introduced to overcome double-counting problems in beyond-mean-field theories, in the second random-phase-approximation (SRPA) for the first time. This procedure guarantees the stability of the SRPA (so that all excitation energies are real). We show that the method fits perfectly into nuclear density-functional theory. We illustrate applications to the monopole and quadrupole response and to low-lying $0^{+}$ and $2^{+}$ states in the nucleus ${}^{16}\mathrm{O}$. We show that the subtraction procedure leads to (i) results that are weakly cutoff dependent and (ii) a considerable reduction of the SRPA downwards shift with respect to the random-phase approximation (RPA) spectra (systematically found in all previous applications). This implementation of the SRPA model will allow a reliable analysis of the effects of two particle-two hole configurations (2p2h) on the excitation spectra of medium-mass and heavy nuclei.

Figure 1

Isoscalar monopole response for the nucleus ${}^{16}\mathrm{O}$, calculated in the standard SRPA (orange diamonds and orange area), and with the ${\mathrm{SSRPA}}_F$, with a cutoff for the correction terms at 50 (black dotted line), 60 (black dashed line), and 70 (black solid line and magenta area) MeV.

Figure 2

Same as in Fig. 1, but in the diagonal approximation ${\mathrm{SSRPA}}_D$.

Figure 3

Isoscalar monopole response for the nucleus ${}^{16}\mathrm{O}$, calculated in the SRPA without subtraction (orange diamonds and orange area), in the ${\mathrm{SSRPA}}_F$ (black solid line and magenta area), and in the ${\mathrm{SRPA}}_D$ (black dotted line and blue area), with a cutoff in the correction terms at 70 MeV.

Figure 4

Same as in Fig. 1 but for the isoscalar quadrupole response. The 2p2h cutoffs in the correction terms are now at 40, 45, and 50 MeV.

Figure 5

Same as in Fig. 2 but for the isoscalar quadrupole response.

Figure 6

Same as in Fig. 3 but for the isoscalar quadrupole response. The 2p2h cutoff for the correction terms is at 50 MeV.

Figure 7

Diagonal part of correction term for each 1p1h configuration that contributes to the monopole strength, for the ${\mathrm{SSRPA}}_F$ (red full bars) and the ${\mathrm{SSRPA}}_D$ (blue dashed bars), with a cutoff at 70 MeV in the correction terms. The horizontal axis simply labels individual 1p1h configurations in the monopole channel.

Figure 8

Same as in Fig. 7, but for the quadrupole channel, with a cutoff at 50 MeV in the correction terms.

Figure 9

First $0^{+}$ (a) and $2^{+}$ (b) states calculated with the standard SRPA, the ${\mathrm{SSRPA}}_F$, and the ${\mathrm{SSRPA}}_D$, with different cutoffs in the correction terms (in parentheses).

Figure 10

Isoscalar monopole response in the diagonal approximation with cutoff for the correction terms at 70 (black line and magenta area), 80 (green dashed line), and 90 (blue dotted line) MeV.

Figure 11

(a) Isoscalar monopole response in the standard SRPA (orange diamonds and orange area), RPA (blue dashed line), and the ${\mathrm{SSRPA}}_F$ (black solid line and magenta area). (b) Discrete spectra (binned strength) obtained with the SRPA (orange dashed bars), the RPA (blue dotted bars), and the ${\mathrm{SSRPA}}_F$ (magenta solid bars).

Figure 12

Same as in Fig. 11 but for the quadrupole strength.

Figure 13

Ratios of the moments $m_{-1}, m_0$, and $m_1$ of the quadrupole strength distribution in the SRPA (purple circles), the ${\mathrm{SSRPA}}_F$ (red squares), and the ${\mathrm{SSRPA}}_D$ (yellow diamonds) to those in the RPA for increasingly high cutoffs in the correction terms, at 40, 45, and 50 MeV.

Figure 14

Fraction of the $E0$ EWSR in the RPA (a) and in the ${\mathrm{SSRPA}}_F$ (b). The experimental fraction, from Ref. [34], is shown in panel (c).

Figure 15

Same as in Fig. 13 but for the $E2$ EWSR.

Figure 16

Comparison of values from the standard SRPA, the ${\mathrm{SSRPA}}_F$, the RPA, and experiment for the energy of the first low-lying $0^{+}$ (a) and $2^{+}$ (b) states.