Finite-range pairing in nuclear density functional theory

TL;DR

This work addresses ultraviolet divergences and continuum sensitivity in zero-range pairing within nuclear density functional theory by introducing a finite-range, Gaussian-folded pairing functional. The folding yields a fourfold-separable form that remains computationally efficient and is compatible with Skyrme and Fayans energy-density functionals. Through systematic $\chi^2$ calibrations, the authors find optimal folding radii around $0.7$–$0.9$ fm, which improve convergence, suppress pairing isomers, and enhance portability across numerical implementations. They demonstrate in $^{120}$Sn and $^{170}$Yb that finite-range pairing reduces box-size and cutoff dependencies and provides smoother, more reliable behavior in the presence of the continuum, enabling more robust large-scale nuclear-structure calculations.

Abstract

Pairing correlations are ubiquitous in low-energy states of atomic nuclei. To incorporate them within nuclear density functional theory, often used for global computations of nuclear properties, pairing functionals that generate nucleonic pair densities and pairing fields are introduced. Many pairing functionals currently used can be traced back to zero-range nucleon-nucleon interactions. Unfortunately, such functionals are plagued by deficiencies that become apparent in large model spaces that contain unbound single-particle (continuum) states. In particular, the underlying computational schemes diverge as the single-particle space increases, and the results depend on how marginally occupied states are incorporated. These problems become more pronounced for pairing functionals that contain gradient-density dependence, such as in the Fayans functional. To remedy this, finite-range pairing functionals are introduced. In this study, this is done by folding the pair density with Gaussians. We show that a folding radius of about 1\,fm offers the best compromise between quality and stability, and substantially reduces the pathological behavior in different numerical applications.

Figures (10)

• Figure 1: Cut of the density distributions along $x$- and $y$-plane of a Fermi gas with density-dependent zero-range pairing simulated in a numerical box in three dimensions with box length 11.2 fm and with average density 0.07/fm$^3$. The pairing parameters were taken from typical nuclear functionals and are $V_\mathrm{prot}=601$ MeV/fm$^3$, $V_\mathrm{neut}=567$ MeV/fm$^3$, and switching density $\rho_{0,\mathrm{pair}}=0.212/\mathrm{fm}^3$. The left panel shows the BCS result and the right panel the HFB result.
• Figure 2: Density of neutron s.p. states in $^{120}$Sn computed with HFB in spherical (dashed line) and axial (solid line) geometry using the functional Fy(IVP). The green dashed line shows the 1D spherical result and the red line the 2D axial result. The density of neutron s.p. states has been smoothed by a Gaussian of 0.5 MeV width to render the graphical representation better visible.
• Figure 3: Total energy of $^{120}$Sn computed with the functional SV-bas and HFB pairing as a function of constrained occupation amplitude $v_\alpha$ for the neutron $2p_{1/2}$ state which lies in the continuum.
• Figure 4: The global quality measure $\chi^2$ as a function of the folding radius for two functionals: (a) Skyrme EDF and (b) Fayans EDF Fy(IVP3). Both EDFs were optimized with a cutoff in pairing space of $E_\mathrm{cut}=15$ MeV.
• Figure 5: Trends of key observables for BCS with $\tilde{R}_{\mathcal{F}} = 0$ (panels (a) and (b)) and HFB (panels (c)-(h)) for three different folding radii, $\tilde{R}_{\mathcal{F}}$ = 0, 1, and 2 fm as functions of s.p. energy. Test case is $^{120}$Sn computed with SV-bas and cutoff energy $\varepsilon_\mathrm{cut}=100$ MeV. The orbital angular momentum $\ell$ of the s.p. states is indicated by color code. Left panels: s.p. r.m.s. radii are plotted as a function of s.p. energy. The radius scale extends up to 18 fm, corresponding to the radius of the numerical box. Right panels: neutron s.p. pairing gaps $\Delta_\alpha$ plotted as a function of s.p. energy. The faint vertical lines indicate the continuum threshold. The horizontal dotted line indicates the total charge r.m.s. radius of $^{120}$Sn.