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

## TL;DR

This paper introduces a particle-hole bosonic approach to describe pairing and quarteting correlations in nuclei, simplifying calculations and providing analytic expressions, with validation on model and realistic nuclear systems.

## Contribution

It presents a new bosonic approximation for PBCS and QCM that reformulates condensate states in terms of particle-hole excitations, simplifying computations and enabling analytic derivations.

## Key findings

- The bosonic approximation simplifies the computation of pairing correlations.
- The approach accurately reproduces results in both model and realistic nuclear systems.
- Comparison shows the renormalized bosonic method improves upon naive bosonization.

## Abstract

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/quartet condensate state 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 condenstate 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 approach 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 $^{100}$Sn.