Abstract:
We discuss a mapping procedure from a space of colorless three-quark clusters into a space of elementary baryons and illustrate it in the context of a three-color extension of the Lipkin model recently developed. Special attention is addressed to the problem of the formation of unphysical states in the mapped space. A correspondence is established between quark and baryon spaces and the baryon image of a generic quark operator is defined both in its Hermitian and non-Hermitian forms. Its spectrum (identical in the two cases) is found to consist of a physical part containing the same eigenvalues of the quark operator in the cluster space and an unphysical part consisting only of zero eigenvalues. A physical subspace of the baryon space is also defined where the latter eigenvalues are suppressed. The procedure discussed is quite general and applications of it can be thought also in the correspondence between systems of 2n fermions and n bosons.

Abstract:
We discuss a multistep variational approach for the study of many-body correlations. The approach is developed in a boson formalism (bosons representing particle-hole excitations) and based on an iterative sequence of diagonalizations in subspaces of the full boson space. Purpose of these diagonalizations is that of searching for the best approximation of the ground state of the system. The procedure also leads us to define a set of excited states and, at the same time, of operators which generate these states as a result of their action on the ground state. We examine the cases in which these operators carry one-particle one-hole and up to two-particle two-hole excitations. We also explore the possibility of associating bosons to Tamm-Dancoff excitations and of describing the spectrum in terms of only a selected group of these. Tests within an exactly solvable three-level model are provided.

Abstract:
The ground state of a general pairing Hamiltonian for a finite nuclear system is constructed as a product of collective, real, distinct pairs. These are determined sequentially via an iterative variational procedure that resorts to diagonalizations of the Hamiltonian in restricted model spaces. Different applications of the method are provided that include comparisons with exact and projected BCS results. The quantities that are examined are correlation energies, occupation numbers and pair transfer matrix elements. In a first application within the picket-fence model, the method is seen to generate the exact ground state for pairing strengths confined in a given range. Further applications of the method concern pairing in spherically symmetric mean fields and include simple exactly solvable models as well as some realistic calculations for middle-shell Sn isotopes. In the latter applications, two different ways of defining the pairs are examined: either with J=0 or with no well-defined angular momentum. The second choice reveals to be more effective leading, under some circumstances, to solutions that are basically exact.

Abstract:
Low-energy spectra of 4$n$ nuclei are described with high accuracy in terms of four-body correlated structures ("quartets"). The states of all $N\geq Z$ nuclei belonging to the $A=24$ isobaric chain are represented as a superposition of two-quartet states, with quartets being characterized by isospin $T$ and angular momentum $J$. These quartets are assumed to be those describing the lowest states in $^{20}$Ne ($T_z$=0), $^{20}$F ($T_z$=1) and $^{20}$O ($T_z$=2). We find that the spectrum of the self-conjugate nucleus $^{24}$Mg can be well reproduced in terms of $T$=0 quartets only and that, among these, the $J$=0 quartet plays by far the leading role in the structure of the ground state. The same conclusion is drawn in the case of the three-quartet $N=Z$ nucleus $^{28}$Si. As an application of the quartet formalism to nuclei not confined to the $sd$ shell, we provide a description of the low-lying spectrum of the proton-rich $^{92}$Pd. The results achieved indicate that, in 4$n$ nuclei, four-body degrees of freedom are more important and more general than usually expected.

Abstract:
The ground state correlations induced by a general pairing Hamiltonian in a finite system of like fermions are described in terms of four-body correlated structures (quartets). These are real superpositions of products of two pairs of particles in time-reversed states. Quartets are determined variationally through an iterative sequence of diagonalizations of the Hamiltonian in restricted model spaces and are, in principle, all distinct from one another. The ground state is represented as a product of quartets to which, depending on the number of particles (supposed to be even, in any case), an extra collective pair is added. The extra pair is also determined variationally. In case of pairing in a spherically symmetric mean field, both the quartets and the extra pair (if any) are characterized by a total angular momentum J=0. Realistic applications of the quartet formalism are carried out for the Sn isotopes with the valence neutrons in the 50-82 neutron shell. Exact ground state correlation energies, occupation numbers and pair transfer matrix elements are reproduced to a very high degree of precision. The formalism also lends itself to a straightforward and accurate description of the lowest seniority 0 and 2 excited states of the pairing Hamiltonian. A simplified representation of the ground state as a product of identical quartets is eventually discussed and found to improve considerably upon the more traditional particle-number projected-BCS approach.

Abstract:
We describe the ground state of the isovector pairing Hamiltonian in self-conjugate nuclei by a product of collective quartets of different structure built from two neutrons and two protons coupled to total isospin T=0. The structure of the collective quartets is determined by an iterative variational procedure based on a sequence of diagonalizations of the pairing Hamiltonian in spaces of reduced size. The accuracy of the quartet model is tested for N=Z nuclei carrying valence nucleons outside the $^{16}$O, $^{40}$Ca, and $^{100}$Sn cores. The comparison with the exact solutions of the pairing Hamiltonian, obtained by shell model diagonalization, shows that the quartet model is able to describe the isovector pairing energy with very high precision. The predictions of the quartet model are also compared to those of the simpler quartet condensation model in which all the collective quartets are assumed to be identical.

Abstract:
We analyze the role of maximally aligned isoscalar pairs in heavy $N=Z$ nuclei by employing a formalism of quartets. Quartets are superpositions of two neutrons and two protons coupled to total isospin $T=0$ and given $J$. The study is focused on the contribution of spin-aligned pairs carrying the angular momentum $J=9$ to the structure of $^{96}$Cd and $^{92}$Pd. We show that the role played by the $J=9$ pairs is quite sensitive to the model space and, in particular, it decreases considerably by passing from the simple $0g_{9/2}$ space to the more complete $1p_{1/2}$,$1p_{3/2}$,$0f_{5/2}$,$0g_{9/2}$ space. In the latter case the description of these nuclei in terms of only spin-aligned $J=9$ pairs turns out to be unsatisfactory while an important contribution, particularly in the ground state, is seen to arise from isovector $J=0$ and isoscalar $J=1$ pairs. Thus, contrary to previous studies, we find no compelling evidence of a spin-aligned pairing phase in $^{92}$Pd.

Abstract:
Working within an exactly solvable 3 level model, we discuss am extension of the Random Phase Approximation (RPA) based on a boson formalism. A boson Hamiltonian is defined via a mapping procedure and its expansion truncated at four-boson terms. RPA-type equations are then constructed and solved iteratively. The new solutions gain in stability with respect to the RPA ones. We perform diagonalizations of the boson Hamiltonian in spaces containing up to four-phonon components. Approximate spectra exhibit an improved quality with increasing the size of these multiphonon spaces. Special attention is addressed to the problem of the anharmonicity of the spectrum.

Abstract:
Isoscalar (T=0,J=1) and isovector (T=1,J=0) pairing correlations in the ground state of self-conjugate nuclei are treated in terms of alpha-like quartets built by two protons and two neutrons coupled to total isospin T=0 and total angular momentum J=0. Quartets are constructed dynamically via an iterative variational procedure and the ground state is represented as a product of such quartets. It is shown that the quartet formalism describes accurately the ground state energies of realistic isovector plus isoscalar pairing Hamiltonians in nuclei with valence particles outside the 16O, 40Ca and 100Sn cores. Within the quartet formalism we analyse the competition between isovector and isoscalar pairing correlations and find that for nuclei with the valence nucleons above the cores 40Ca and 100Sn the isovector correlations account for the largest fraction of the total pairing correlations. This is not the case for sd-shell nuclei for which isoscalar correlations prevail. Contrary to many mean-field studies, isovector and isoscalar pairing correlations mix significantly in the quartet approach.

Abstract:
We carry out a microscopic analysis of the ground and excited states of the Na_8 metal cluster within the jellium model. We perform a series of configuration interaction calculations on a Hartree-Fock basis and construct eigenstates of the Hamiltonian which carry up to 4-particle 4-hole components. Based on the analysis of the dipole transition strengths, we single out those states which can be interpreted as the collective dipole plasmon and its double excitations. These modes are found to possess a high degree of harmonicity, deviations from the harmonic limit remaining, however, of the order of 10%.