Perturbative Hartree-Fock-Bogoliubov model for many-body pairing correlations

We develop a perturbative model to treat the off-diagonal components in the Hartree-Fock-Bogoliubov (HFB) transformation matrix, which are neglected in the Bardeen-Cooper-Schrieffer (BCS) approximation. Applying the perturbative model to a weakly bound nucleus ${}^{84}\mathrm{Ni}$, it is shown that the perturbative approach reproduces well the solutions of the HFB method both for the quasiparticle energies and the radial dependence of quasiparticle wave functions. We find that the nonresonant part of the continuum single-particle state can acquire an appreciable occupation probability when there exists a weakly bound state close to the Fermi surface. This result originates from the strong coupling between the continuum particle state and the weakly bound state, and it is absent in the BCS approximation. The limitations of the BCS approximation are pointed out in comparison with the HFB and the present perturbative model.

Figure 1

The matrix elements of the pairing potential $\Delta (r)$. These are plotted as a function of the single-particle energy ${\epsilon }_j$ for the state $|{\phi }_j\rangle$ with a fixed bound state $\langle {\phi }_i|$, which has the same total and orbital angular momenta j and l as the state ${\phi }_j$. The open circles, filled squares, and open triangles are for the neutron $1s_{1/2}$ state at −31.6 MeV, the $3s_{1/2}$ state at −0.634 MeV, and the $1g_{9/2}$ state at −4.11 MeV, respectively. The diagonal components are denoted by arrows.

Figure 2

The deviation of the neutron $s_{1/2}$ quasiparticle energy, obtained with the BCS method (dashed line) and the perturbative method p-HFB (solid line), from the corresponding HFB energy as a function of the single-particle energy.

Figure 3

The quasiparticle wave functions for the neutron $s_{1/2}$ states obtained with the three methods: HFB, BCS, and p-HFB. The left and right panels are for the lower (V) and upper (U) components of the wave function, respectively. The solid line is the solution of the HFB method, while the dotted line denotes the wave function in the BCS approximation. The result of the perturbative method, p-HFB, is shown by the dashed line. The top, middle, and bottom panels are for the $5s_{1/2},4s_{1/2}$, and $3s_{1/2}$ states, respectively.

Figure 4

The occupation probability of HFB and BCS quasiparticle states for the $g_{9/2}$ configuration. It is plotted as a function of the HF energy ε of the dominant single-particle state in the wave function. The solid and dotted lines are results of the HFB method and the BCS approximation, respectively, while the dashed line is obtained by the perturbative method, p-HFB.

Figure 5

Same as Fig. 3, but for the $1g_{9/2}$ (upper panel) and $3g_{9/2}$ (lower panel) states.