In-medium bound states and pairing gap

The two-particle Green's function and $T$ matrix including pphh correlations in infinite nuclear matter are evaluated by a diagonalization of the effective total Hamiltonian. This diagonalization procedure corresponds to the same eigenvalue problem as for the pphh Random Phase Approximation. The effective Hamiltonian is nonHermitian and, for specific density domains and partial waves, yields pairs of complex conjugated eigenvalues and eigenfunctions representing in-medium bound states of two nucleons. The occurrence of these complex poles of the two-particle in-medium Green's function indicates the well known pairing instability. It is shown that the corresponding bound-state wave functions contain momentum dependencies of the BCS gap function, so that the latter can be found from a single diagonalization procedure for the effective Hamiltonian matrix. The approach is illustrated by calculations for ${}^{1}S_0$ and ${}^{3}PF_2$ gap functions in neutron matter which essentially coincide with the results found by a direct solving of the BCS gap equation. However the developed approach shows a new interesting feature, i.e., the gap closure and a phase transition point at very low density in the case of coupled channels ${}^{3}SD_1$ in symmetric nuclear matter. This finding goes beyond the conventional BCS treatment and is discussed in the context of transition from Bose-Einstein condensation of quasideuterons to the formation of BCS pairing.

Figure 1

The eigenvalues of the total Hamiltonian near $2e_F$ threshold (filled circles) for $k_F=0.5{\mathrm{fm}}^{-1}$ (a) and $k_F=1.1{\mathrm{fm}}^{-1}$ (b). The initial partition of the hh and pp continuum is represented by red line with bars.

Figure 2

Imaginary (a) and real (b) parts of the on-shell $T$ matrix found with the partial phase shift (c) via a diagonalization procedure (solid curves) and from the direct solution of the integral equation for the reaction matrix (filled circles) for ${}^{1}S_0$ channel at $k_F=0.5{\mathrm{fm}}^{-1}$. The dashed curve in (b) reflects the complex-bound-state contributions.

Figure 3

The bound-state wave function of the Hamiltonian $H$ (solid curve) in comparison with characteristic function $A\varphi \left(k\right)$ (filled circles) at $k_F=0.8$ (a) and 1.2 ${\mathrm{fm}}^{-1}$ (b).

Figure 4

The absolute value of the gap $\mathrm{\Delta }\left(k\right)$ found from the bound-state wave function (solid curves) in comparison with direct solutions of the gap equation (filled circles) for the ${}^{1}S_0$ channel at $k_F=0.8{\mathrm{fm}}^{-1}$ (a) and 1.2 ${\mathrm{fm}}^{-1}$ (b).

Figure 5

The momentum dependence of the bound-state wave function of the RPA Hamiltonian $H$ (solid curve) in comparison with characteristic function $A\varphi \left(k\right)$ (filled circles) for the coupled channels ${}^{3}PF_2$ in neutron matter at $k_F=2.5{\mathrm{fm}}^{-1}$.

Figure 6

The absolute value of the gap $\mathrm{\Delta }\left(k\right)$ found from the bound-state wave function (dashed curves) in comparison with solutions of the gap equation (solid curves) for the ${}^{3}PF_2$ channel at $k_F=2.5{\mathrm{fm}}^{-1}$ (a) and the partial gaps for the partial $P$ (b) and $F$ (c) waves.

Figure 7

Behavior of the real-valued bound-state energies (solid and dash-dotted curves) and the real part of the complex-valued bound-state energy $E_b$ (dash-dot-dotted curve) for small $k_F$ values in the spin-triplet $NN$ channel at $K=0$. The critical point is denoted by a filled circle.

Figure 8

The 3D trajectory (solid curve) of bound-state energies below and above the critical value $k_F^C$ which is denoted by a filled circle. A projection of this curve on the plane ${\mathrm{\Gamma }}_0=0$ (dashed curve) reflects the same picture as in Fig. 7.

Figure 9

Wave functions in the form (30) of the first (solid curve) and second (dashed curve) real bound states at different values of the Fermi momentum $k_F=0.15{\mathrm{fm}}^{-1}$ (a), 0.175 ${\mathrm{fm}}^{-1}$ (b), and 0.182 ${\mathrm{fm}}^{-1}$ (c). The dash-dotted curves at each layer corresponds to the deuteron wave function in vacuum while the vertical lines reflect the positions of the Fermi momentum.

Figure 10

The gap value $\mathrm{\Delta }\left(k_F\right)$ found from the approximate solution of the gap equation (dashed curve) and imaginary part of the eigenvalue $E_b$ for the coupled channels ${}^{3}SD_1$ in symmetric nuclear matter.

Figure 11

The total pairing gap $\mathrm{\Delta }\left(k\right)$ (a) and the partial $S$- (b) and $D$-wave (c) gaps found from the solution of the gap equation (dashed curve) and from the bound-state wave function for the coupled channels ${}^{3}SD_1$ in symmetric nuclear matter at $k_F=1.2{\mathrm{fm}}^{-1}$.