Spurious finite-size instabilities in nuclear energy density functionals: Spin channel

Background: It has been recently shown that some Skyrme functionals can lead to nonconverging results in the calculation of some properties of atomic nuclei. A previous study has pointed out a possible link between these convergence problems and the appearance of finite-size instabilities in symmetric nuclear matter (SNM) around saturation density.

Purpose: We show that the finite-size instabilities not only affect the ground-state properties of atomic nuclei, but they can also influence the calculations of vibrational excited states in finite nuclei.

Method: We perform systematic fully-self consistent random phase approximation (RPA) calculations in spherical doubly magic nuclei. We employ several Skyrme functionals and vary the isoscalar and isovector coupling constants of the time-odd term $\mathbf{s}·\mathrm{\Delta }\mathbf{s}$. We determine critical values of these coupling constants beyond which the RPA calculations do not converge because the RPA stability matrix becomes nonpositive.

Results: By comparing the RPA calculations of atomic nuclei with those performed for SNM we establish a correspondence between the critical densities in the infinite system and the critical coupling constants for which the RPA calculations do not converge.

Conclusions: We find a quantitative stability criterion to detect finite-size instabilities related to the spin $\mathbf{s}·\mathrm{\Delta }\mathbf{s}$ term of a functional. This criterion could be easily implemented in the standard fitting protocols to fix the coupling constants of the Skyrme functional.

Figure 1

Evolution of Landau parameters $G_l$ and $G_l^{\prime } (l=0,1)$ as a function of the density in SNM for the six Skyrme functionals considered in this article.

Figure 2

Evolution of the lowest-energy vibrational states for different multipolarities $J^{\pi }$ in ${}^{56}\mathrm{Ni}$ as a function of the isoscalar coupling constant $C_0^{\mathrm{\Delta }s}$ (a) and the isovector one (b), for the SAMi functional.

Figure 3

Evolution phonons in ${}^{56}\mathrm{Ni}$ as a function of the multiplicative factor $\gamma$ for T44 (a), SLy5 (b), BSk27 (c), and SIII (d). The caption is the same as that for Fig. 2.

Figure 4

Evolution of the critical coupling constants for the SIII functional for the different numbers of shells included in the calculation. With lines we have shown some proposed functions to describe the dependence of the critical coupling constants with the shell numbers.

Figure 5

Ratio among the critical coupling constant and its nominal value using the best extrapolation as defined in the text and for the different nuclei considered in our analysis.

Figure 6

Evolution of the low-lying spectrum in ${}^{132}\mathrm{Sn}$ (calculated with 16 oscillator shells) with respect to $C_0^{\mathrm{\Delta }s}$ for the SLy5 functional. we present the spectra calculated using (a) $C_0^{\mathrm{\Delta }s}=9.22$ MeV ${\mathrm{fm}}^5$, (b) $C_0^{\mathrm{\Delta }s}=46.10$ MeV ${\mathrm{fm}}^5$, (c) $C_0^{\mathrm{\Delta }s}=55.3$ MeV ${\mathrm{fm}}^5$, and (d) $C_0^{\mathrm{\Delta }s}=59.91$ MeV ${\mathrm{fm}}^5$.

Figure 7

Instabilities in SNM for the functionals considered in the present article. The dashed-dotted horizontal line stands for the saturation density of the functional. See text for details.

Figure 8

Evolution of the critical densities in the $S=1$ channel in asymmetric matter for different values of the asymmetry parameter $Y$ for the BSk27 functional.

Figure 9

Position of the poles in the $(\rho ,q)$ plane. The minimum of this curve defines the point $({\rho }_c,q_c)$.

Figure 10

Critical density ${\rho }_c/{\rho }_{\mathrm{sat}}$. The uncertainty band comes from the numerical extraction and the number of shells employed in the calculations. Solid bands refer to the isoscalar channel, while dashed bands refer to the isovector one. The vertical black lines refer to the best value as given in Table 3.