Narrow Results

Our contribution aims to celebrate the immeasurable contribution that Tom Kuo has provided to the understanding of the structure of atomic nuclei, and also of the infinite nuclear matter, in terms of the fundamental principles governing the realistic nuclear potential. The authors want to testify Tom Kuo’s heritage and impact on their approach to the study of nuclear systems by reviewing some recent findings on the role of the two-body component of shell-model effective $\beta$-decay operators. The focus is spotted on the so-called Pauli-blocking effect that plays a nonnegligible role in nuclei characterized by a large number of valence nucleons.

The understanding of the convergence properties of the shell-model effective Hamiltonian, within the framework of the many-body perturbation theory, is a long-standing problem. The infinite summation of a certain class of diagrams, the so-called “bubble diagrams,” may be provided calculating the Kirson–Babu–Brown induced interaction, and provides a valid instrument to study whether or not the finite summation of the perturbative series is well-grounded. Here, we perform an application of the calculation of the Kirson–Babu–Brown induced interaction to derive the shell-model effective Hamiltonian for $p$-shell nuclei starting from a modern nucleon–nucleon potential, obtained by way of the chiral perturbation theory. The outcome of our calculation is compared with a standard calculation of the shell-model Hamiltonian, where the core-polarization effects are calculated only up to third-order in perturbation theory. The results of the two calculations are very close to each other, evidencing that the perturbative approach to the derivation of the shell-model Hamiltonian is still a valid tool for nuclear structure studies.

Complexities of using an SU(4) and SU(3) basis in the $1p0f$ shell are discussed. SU(3) representations are detailed for 2–5 particles. Earlier calculations by McGrory are discussed with the view of using the SU(3) model to describe multi-particle multi-hole states.

Embedded random matrix ensembles generated by random interactions (of low body rank and usually two-body) in the presence of a one-body mean field, introduced in nuclear structure physics, are now established to be indispensable in describing statistical properties of a large number of isolated finite quantum many-particle systems. Lie algebra symmetries of the interactions, as identified from nuclear shell model and the interacting boson model, led to the introduction of a variety of embedded ensembles (EEs). These ensembles with a mean field and chaos generating two-body interaction generate in three different stages, delocalization of wave functions in the Fock space of the mean-field basis states. The last stage corresponds to what one may call thermalization and complex nuclei, as seen from many shell model calculations, lie in this region. Besides briefly describing them, their recent applications to nuclear structure are presented and they are (i) nuclear level densities with interactions; (ii) orbit occupancies; (iii) neutrinoless double beta decay nuclear transition matrix elements as transition strengths. In addition, their applications are also presented briefly that go beyond nuclear structure and they are (i) fidelity, decoherence, entanglement and thermalization in isolated finite quantum systems with interactions; (ii) quantum transport in disordered networks connected by many-body interactions with centrosymmetry; (iii) semicircle to Gaussian transition in eigenvalue densities with $k$-body random interactions and its relation to the Sachdev–Ye–Kitaev (SYK) model for majorana fermions.

The quark shell model has been successful in describing properties of hadrons. Because of color the quark shell model with 3n valence quarks has many more states which are singlet in color than the nuclear shell model with n valence nucleons. However, the quark interaction has been shown to favor two quarks coupled to spin zero and isospin zero and color , called diquarks. We show that the color singlet states in the quark shell model which have the maximal number of diquarks consistent with the Pauli symmetry are in one to one correspondence with the states of the nuclear shell model. We also investigate the implications of the quark interactions on the nuclear shell model interaction.

The understanding of the convergence properties of the shell-model effective Hamiltonian, within the framework of the many-body perturbation theory, is a long-standing problem. The infinite summation of a certain class of diagrams, the so-called “bubble diagrams,” may be provided calculating the Kirson–Babu–Brown induced interaction, and provides a valid instrument to study whether or not the finite summation of the perturbative series is well-grounded. Here, we perform an application of the calculation of the Kirson–Babu–Brown induced interaction to derive the shell-model effective Hamiltonian for p-shell nuclei starting from a modern nucleon–nucleon potential, obtained by way of the chiral perturbation theory. The outcome of our calculation is compared with a standard calculation of the shell-model Hamiltonian, where the core-polarization effects are calculated only up to third-order in perturbation theory. The results of the two calculations are very close to each other, evidencing that the perturbative approach to the derivation of the shell-model Hamiltonian is still a valid tool for nuclear structure studies.

Complexities of using an SU(4) and SU(3) basis in the 1p0f shell are discussed. SU(3) representations are detailed for 2–5 particles. Earlier calculations by McGrory are discussed with the view of using the SU(3) model to describe multi-particle multi-hole states.