Unified description of pairing and quarteting correlations within the particle-hole-boson approach

We study the description of single-species and isovector pairing correlations in the framework of the projected-BCS (PBCS) and the quartet condensation model (QCM) from a particle-hole perspective, and we introduce the representation of the QCM quartet condensate state in terms of particle-hole excitations with respect to the Hartree-Fock state. We also present a new bosonic approximation for both PBCS and QCM. In each case, the starting point is the reformulation of the pair and quartet condensate states in terms of particle-hole excitations with respect to the Hartree-Fock state. The main simplification of our approach is the assumption that the pair operators corresponding to both particle and hole states obey bosonic commutation relations. This simplifies tremendously the computations and allows for an analytic derivation of the averaged Hamiltonian on the condensate state as a function of the mixing amplitudes. We study both the pure bosonic approach and the renormalized version and compare the particle-hole bosonic version to the naive prescription of applying the boson approximation directly to the original condensate state. We compare the fermionic and the renormalized particle-hole bosonic approaches in the case of a picket-fence model of doubly degenerate states and in a realistic shell-model space with an effective interaction for the $N=Z$ nuclei above ${}^{100}\mathrm{Sn}$.

Figure 1

The linear behavior of the (a) normalized PBCS and (b) QCM pair mixing amplitudes for hole states $x_a$ (circles) and inverses of pair mixing amplitudes for particle states $1/x_k$ (squares) for a picket-fence model of doubly degenerate levels ${\epsilon }_k=k\mathrm{MeV},k=0,1,...,19$ for a coupling constant $G=0.5\mathrm{MeV}$. The linear fit for each set of amplitudes is shown with dashed lines.

Figure 2

The effect of the particle and hole degrees of freedom on the average level occupation fraction $\langle n\rangle$, shown for the QCM in the case of the picket-fence model for $q=4$ quartets.

Figure 3

The ground-state energies in MeV versus the ratio $G/G_{\mathrm{cr}}$ for (a) $n_p=6$ and (b) $n_p=10$ and the average level occupation fractions versus state index for (c) $n_p=6$ and (d) $n_p=10$ for the PBCS, bPBCS, and renormalized bosonic PBCS (rbPBCS) approaches of the picket-fence model in the strong pairing regime at $G/G_{\mathrm{cr}}=4$.

Figure 4

The ground-state energies (in MeV) versus the ratio $G/G_{\mathrm{cr}}$ for (a) $q=1$, (b) $q=2$, (c) $q=3$, and (d) $q=4$ the QCM, bQCM, rbQCM, and rb-naive approaches of the picket-fence model.

Figure 5

The error in the correlation energy versus the ratio $G/G_{\text{cr}}$ for the (a) rbPBCS and, respectively, (b) rbQCM approaches, relative to the fermionic cases of (a) PBCS and (b) QCM.

Figure 6

Average level occupation fractions versus state index for the QCM, bQCM, rbQCM, and rb-naive for the first ten levels of the picket-fence model for (a) $q=1$, (b) $q=2$, (c) $q=3$, and (d) $q=4$ in the strong pairing regime at $G/G_{\mathrm{cr}}=4$.

Figure 7

Mixing amplitudes $x_i$ versus state index for the QCM, bQCM, and rbQCM for the first ten levels of the picket-fence model for (a) $q=1$, (b) $q=2$, (c) $q=3$, and (d) $q=4$ in the strong pairing regime at $G/G_{\mathrm{cr}}=4$. The best fit for the amplitudes in the bosonic models was chosen as allowed by the scaling symmetry.

Figure 8

Average level occupation fractions $\langle n_i/{\mathrm{\Omega }}_i\rangle \equiv \langle N_{0,i}\rangle /\left[2(2j_i+1)\right]$ versus the state index $i$ corresponding to the exact QCM and to the renormalized bosonic approximation rbQCM for the nuclei above the ${}^{100}\mathrm{Sn}$ core, computed with the Bonn A effective interaction of Ref. [60].