Pairing and the HFB continuum

The structure of HFB continuum indicated schematically in Fig. 1 has been a subject of many works [18,19,20,21,51,52,53,54,55,56,57]. Within the real-energy HFB framework, the proper theoretical treatment of the HFB continuum is fairly sophisticated since the scattering boundary conditions must be met. One way of tackling this problem is the coordinate-space Green's function technique[20,55,56]. If the outgoing boundary conditions are imposed, the unbound HFB eigenstates have complex energies; their imaginary parts are related to the particle width[20]. The complex-energy spherical HFB problem has been formulated and implemented within the Gamow HFB (GHFB) approach of Ref.[58].

In addition to the methods that directly employ proper asymptotic boundary conditions for unbound HFB eigenstates, the quasiparticle continuum of HFB can be approximately treated by means of a discretization method. The commonly used approach is to impose the box boundary conditions in the coordinate-space calculations [19,21,59,60,61,62,57], in which wave functions are spanned by a basis of orthonormal functions defined on a lattice in coordinate space and enforced to be zero at box boundaries. In this way, referred to as the $\mathcal{L}^2$ discretization, quasiparticle continuum of HFB is represented by a finite number of box states. It has been demonstrated by explicit calculations for weakly bound nuclei[58,52,57] that such a box discretization is accurate when compared to the exact results. Alternatively, diagonalizing the HFB matrix in the Pöschl-Teller-Ginocchio basis[63] or Woods-Saxon basis [64,65,66,67,68] turned out to be an efficient way to account for the continuum effects. Finally, quasiparticle continuum can be effectively discretized by solving the HFB problem by means of expansion in a harmonic oscillator (HO) or transformed HO (THO) basis[69,70,71]. As far as the description of nonlocalized HFB states is concerned, the coordinate-space method is superior over the HO expansion method, as the HO basis states are always localized. Consequently, the discretized representation of the quasiparticle continuum is different in coordinate-space and HO basis-expansion approaches [54].

Among the quasiparticle resonances, the deep-hole states play a distinct role. In the absence of pairing, a deep-hole excitation with energy $E_i>0$ corresponds to an occupied HF state with energy $\varepsilon_i=-E_i+\lambda$. If pairing is present, it generates a coupling of this state with unbound particle states with $\varepsilon_i\approx E_i+\lambda$ that gives rise to a quasiparticle resonance with a finite width[20,21,72]. Quasiparticle resonance widths can be directly calculated with a high precision using coordinate-space Green's function technique [20,55,56] and GHFB[58]. For approaches based on the $\mathcal{L}^2$ discretization, several approximate methods have been developed to deal with HFB resonances. The modified stabilization method based on box solutions [73,74,57] has been used to obtain precisely the resonance energy and widths. Based on the box solutions, the HFB resonances are expected to be localized solutions with energies weakly affected by changes of the box size. The stabilization method allows to obtain the resonance parameters from the box-size dependence of quasiparticle eigenvalues.

Besides the stabilization method, a straightforward smoothing and fitting method that utilizes the density of box states has been successfully used. In this technique, resonance parameters are obtained by fitting the smoothed occupation numbers obtained from the dense spectrum discretized HFB solutions.

Figure 3: Occupation numbers of the discretized neutron quasiparticle continuum states calculated for $^{70}$Zn in Ref.[57]. The corresponding Breit-Wigner envelopes are indicated by dashed lines. The $-\lambda_n$ threshold is marked by a dotted line.

Figure 3 displays occupation probabilities $v_i^2$ for the discretized neutron quasiparticle states in $^{70}$Zn as a function of quasiparticle energy $E_i$. To extract resonance parameters from the discrete distribution of $v_i^2$, one can first smooth it using a Lorentzian shape function and then perform a fit using a Breit-Wigner function[57].

Various ways of computing widths of high-energy deep-hole states have been compared in Ref.[57]. By comparing with the exact GHFB results, it has been concluded that the stabilization method works fairly well for all HFB resonances, except for the very narrow ones. The smoothing-fitting method is also very effective and can easily be extended to the deformed case. The perturbative Fermi golden rule[20] has been found to be unreliable for calculating widths of neutron resonances. (For more discussion on limitations of the perturbative treatment, see Ref.[20]).

Pairing correlations can profoundly modify properties of the system in drip line nuclei due to the presence of the vast continuum space available for pair scattering. One example is the appearance of the pairing-antihalo effect [75,76,61,77,37,78], in which pairing correlations in the weakly-bound even-particle system change the asymptotic behavior of particle density thus reducing its radial extension. While neutron radii of even-even nuclei are expected to locally increase when approaching the two-neutron drip line [79,72,65,77,37] the size of the resulting halo is fairly modest, especially when compared with spatial extensions of neighboring odd-neutron systems.

Pairing correlations impact the limits of the nuclear existence: the odd-even staggering of the nuclear binding energy does result in the shift between one-neutron and two-neutron drip lines. The pairing coupling to the positive-energy states is an additional factor influencing the nuclear binding[20,21]. In particular, because of strong coupling to the neutron continuum, the neutron chemical potential may be significantly lowered thus extending the range of bound nuclei, and this effect is expected to depend on the character of pairing interaction. For more discussion on the impact of continuum on quasiparticle occupations, emergence of bound canonical HFB states from the continuum, and contributions of nonresonant continuum to the localized ground state in dripline nuclei, see Refs.[21,72,77,37,56,68].