Symmetry-restored Skyrme-random-phase-approximation calculations of the monopole strength in deformed nuclei

Background: Within the energy density functional (EDF) approach, the use of mean-field wave functions deliberately breaking (some) symmetries of the underlying Hamiltonian is an efficient and widely utilized way to incorporate static correlations. However, the restoration of broken symmetries is eventually mandatory to recover the corresponding quantum numbers and to achieve a more precise description of nuclear properties.

Purpose: While symmetry-restored calculations are routinely performed to study ground-state properties and low-lying excitations, similar applications to the nuclear response are essentially limited to either formal studies or to schematic models. In the present paper, the effect of angular momentum restoration on the monopole and quadrupole responses of doubly open-shell nuclei is investigated.

Methods: Based on deformed Skyrme-randomphaseapproximation (RPA) calculations, the exact angular momentum projection (AMP) is implemented in the calculation of the multipole strength functions, thus defining a projection after variation (PAV-RPA) scheme. The method is employed for the first time in a realistic study to investigate the effect of AMP on the coupling of monopole and quadrupole modes in ${}^{24}\mathrm{Mg}$ resulting from its intrinsic deformation.

Results: The monopole PAV-RPA response function shows, in addition to the giant resonance peaks, a tremendous amount of strength in the low-energy part whose properties and nature are investigated and discussed. In the quadrupole channel, the AMP leads to a suppression of all the strength but the one corresponding to the isoscalar giant quadrupole resonance.

Conclusions: The nature of the anomalous low-lying monopole strength is interpreted as a contamination of the excited states via the coupling to the (noninfinitesimal) rotational motion in deformed RPA phonons. Such a spurious strength was also observed in projected generator coordinate method (PGCM) calculations based on a similar PAV approach, but was shown to disappear in its full variation after projection (VAP) counterpart. While the spurious strength could be properly subtracted in the present work, this work motivates the implementation of the full VAP-RPA in the future.

Figure 1

(a) Monopole (upper panel) and quadrupole (lower panel) RPA responses in ${}^{24}\mathrm{Mg}$ for different $N_{\mathrm{sh}}$ values ($E_{\mathrm{cut}}=100$ MeV). (b) The same but for different $E_{\mathrm{cut}}$ values ($N_{\mathrm{sh}}=11$).

Figure 2

Angular momentum decomposition of the HF ground state in ${}^{24}\mathrm{Mg}$ ($N_{\mathrm{sh}}=11$).

Figure 3

(a) Monopole (upper panel) and quadrupole (lower panel) RPA responses in ${}^{24}\mathrm{Mg}$ ($N_{\mathrm{sh}}=11$ and $E_{\mathrm{cut}}=100$ MeV). (b) The same for PAV-RPA responses. The upper-right panel additionally contains the overlap between RPA excited states and the rotated ground state [Eq. (17)] up to a normalization factor.

Figure 4

${}^{24}\mathrm{Mg}$ (a) one-body intrinsic matter density of the HF ground state, (b) intrinsic RPA transition matter density to the ISGMR phonon at 23.9 MeV, (c) intrinsic RPA transition matter density to the ISGQR phonon at 17.5 MeV, and (d) intrinsic RPA transition matter density to the phonon at 10.8 MeV.

Figure 5

(a) Polar coordinate representation of the intrinsic RPA transition density to the rotational phonon at 10.8 MeV in ${}^{24}\mathrm{Mg}$. (b) The same for the combination of two densities obtained by rotating the HF state by $\beta \approx \pm {24}^{\circ }$.

Figure 6

${}^{24}\mathrm{Mg}$ monopole (upper panel) and quadrupole (lower panel) RPA and PAV-RPA responses ($N_{\mathrm{sh}}=11$ and $E_{\mathrm{cut}}=100$ MeV). In the monopole case the subtracted PAV-RPA response is also shown.