We present a pedagogical introduction to the recent analytical approach for the treatment pairing and quarteting correlations [1].

We then present a new bosonic approximation for the projected-BCS (PBCS) and Quartet Condensation Model (QCM) [2]. In each case, the starting point is the reformulation of the pair/quartet condensate state in terms of particle-hole excitations with respect to the completely occupied Fermi sea. 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. In the case of a picket fence model of doubly degenerate states, we find a very good agreement between the fermionic and the renormalized bosonic case, regarding both the ground state energy and the pair mixing amplitudes.

[1] V.V. Baran, D.S. Delion, Phys. Rev. C 99, 031303(R)
[2] V.V. Baran, D.S. Delion, arXiv:1902.00065