Canonical states in continuum Skyrme Hartree-Fock-Bogoliubov theory with Green's function method

The canonical single-particle energies and wave functions are obtained by diagonalizing the density matrix on a spatial mesh in continuum Skyrme Hartree-Fock-Bogoliubov theory with the Green's function method. Taking the exotic nucleus ${}^{66}\mathrm{Ca}$ as an illustrative example, the canonical energies and wave functions are compared with those obtained by the box-discretized method. Most of the results are consistent except for $s_{1/2}$ orbitals with positive energies, which have large spatial extension. By examining the canonical energies and wave functions with different box sizes, it is shown that the Green's function method can obtain the convergent canonical energy within a box size smaller than that of the box-discretized method, due to the correct asymptotic behaviors of the canonical wave functions.

Figure 1

The comparison between neutron canonical energies calculated by the continuum HFB theory with the Green's function (solid lines) and box-discretized (dash lines) methods for ${}^{66}\mathrm{Ca}$ with the box size $R=20$ fm. The occupation probabilities for the canonical orbitals are proportional to the lengths of the lines (in logarithmic scale).

Figure 2

Positive-energy canonical states in ${}^{66}\mathrm{Ca}$ obtained by the Green's function method with the box size $R=20$ fm (solid bars), in comparison with those obtained by the box-discretized method with different $R$ (dash lines).

Figure 3

Neutron canonical wave functions ${\mathrm{\Psi }}_i\left(r\right)$ (dash lines) of the $1s_{1/2}, 2s_{1/2}, 1p_{1/2}$, and $2p_{1/2}$ states for ${}^{66}\mathrm{Ca}$ obtained by diagonalizing the density matrix in the Green's function method with the box size $R=20$ fm. The corresponding canonical single-particle energies ${\varepsilon }_i$ calculated by Eq. (11) are also given. The wave function ${\mathrm{\Psi }}_{i,D}\left(r\right)$ (solid lines) corresponding to the canonical single-particle energy ${\varepsilon }_{i,D}$ is obtained by diagonalizing the HF Hamiltonian $h$ in the canonical subspace constructed by the states with $v_i^2\approx 1$ as shown in Eqs. (12, 13, 14).

Figure 4

Neutron canonical wave functions ${\mathrm{\Psi }}_i\left(r\right)$ of $4s_{1/2}, 3p_{1/2}, 2d_{3/2}$ and $1g_{7/2}$ states for ${}^{66}\mathrm{Ca}$ obtained by the Green's function [panels (a), (c), (e), and (g)] and box-discretized [panels (b), (d), (f), and (h)] methods calculated with the box size $R=15$, 18, and 20 fm. The inserts in logarithmic scale show the canonical wave functions within $R\in [10,20]$ fm.