8 No-Core Gamow Shell Model
369
The fundamental limitation of the coupled-cluster and Gorkov–Green-function
methods is that they cannot deal with all nuclei. For example, the coupled-cluster
method can handle in practice only closed-shell nuclei or nuclei with one or two
particles or holes outside closed shells [59]. Moreover, the Gorkov–Green-function
method can only be used for the calculation of even nuclei whose proton or neutron
part must consist of closed shells, and immediate neighbors of even number of
nucleons [55–57]. With above restrictions, the calcium isotopes could be calculated
in the coupled-cluster approach up to the neutron dripline. Proton-rich tin isotopes,
in the vicinity of the doubly-closed shell nucleus 100 Sn, could also be calculated
in the coupled-cluster framework [53]. However, the fact that the coupled-cluster
method is restricted to closed-shell nuclei ±2 nucleons can be partly mitigated by
calculating an equivalent interaction using a core and a restricted valence space [60].
Such an interaction can then be used in the framework of standard shell model. The
advantage of such a two step approach is that dimensions of the Hamiltonian matrix
remain tractable and, moreover all nuclei in a chosen subspace can be described.
It appears then that a profitable strategy is not to solve the initial Hamiltonian
using an exact resolution method, but to transform it so that it can be used further
within shell model using a number of configurations sufficiently small to allow for
numerical diagonalization. Such a numerical strategy can be implemented via the inmedium similarity renormalization group [61–65] or the many-body perturbation
theory [66, 67]. In the in-medium similarity renormalization group method, a
unitary transformation depending on a continuous parameter is applied to the initial
Hamiltonian, so that its low and high energy excitations are increasingly decoupled
[61–65]. Consequently, the eigenstates of a truncated shell model calculation
converge to their exact value, with a few percent precision. At present, when using a
basis of harmonic oscillator states, the in-medium similarity renormalization group
method can only be applied in a shell model space generated by a single harmonic
oscillator major shell [61–65]. Consequently, certain aspects of nuclear deformation
cannot be described in this method, as multishell valence spaces would be needed
therein [68]. Nevertheless, axial nuclear deformation could be recently included
in the fp shell by using the generator coordinate method, where Hartree–Fock–
Bogoliubov solutions of several different deformations are used as basis states (see
Ref. [69] where neutrinoless beta decay is studied in A = 48 deformed nuclei.)
The in-medium similarity renormalization group method has been applied in
light and medium-heavy well-bound nuclei, such as the isotopes of oxygen, calcium,
and nickel [61, 62, 65]. The method allows also to evaluate the electromagnetic
transitions [68]. Recently, the in-medium similarity renormalization group method
has been considered in the context of the Gamow shell model to calculate the spectra
of 22 C and 24 O [70].
States of deformed nuclei can be calculated in an ab initio framework if the
basis states are generated by a Hamiltonian depending on deformation degrees of
freedom. For example, the Monte Carlo shell model [71–74] makes use of Slater
determinants generated by a deformed potential, where the rotational invariance
of the many-body Hamiltonian is restored with projection techniques [75]. Alternatively, a new ab initio shell model framework has been recently introduced to
369
The fundamental limitation of the coupled-cluster and Gorkov–Green-function
methods is that they cannot deal with all nuclei. For example, the coupled-cluster
method can handle in practice only closed-shell nuclei or nuclei with one or two
particles or holes outside closed shells [59]. Moreover, the Gorkov–Green-function
method can only be used for the calculation of even nuclei whose proton or neutron
part must consist of closed shells, and immediate neighbors of even number of
nucleons [55–57]. With above restrictions, the calcium isotopes could be calculated
in the coupled-cluster approach up to the neutron dripline. Proton-rich tin isotopes,
in the vicinity of the doubly-closed shell nucleus 100 Sn, could also be calculated
in the coupled-cluster framework [53]. However, the fact that the coupled-cluster
method is restricted to closed-shell nuclei ±2 nucleons can be partly mitigated by
calculating an equivalent interaction using a core and a restricted valence space [60].
Such an interaction can then be used in the framework of standard shell model. The
advantage of such a two step approach is that dimensions of the Hamiltonian matrix
remain tractable and, moreover all nuclei in a chosen subspace can be described.
It appears then that a profitable strategy is not to solve the initial Hamiltonian
using an exact resolution method, but to transform it so that it can be used further
within shell model using a number of configurations sufficiently small to allow for
numerical diagonalization. Such a numerical strategy can be implemented via the inmedium similarity renormalization group [61–65] or the many-body perturbation
theory [66, 67]. In the in-medium similarity renormalization group method, a
unitary transformation depending on a continuous parameter is applied to the initial
Hamiltonian, so that its low and high energy excitations are increasingly decoupled
[61–65]. Consequently, the eigenstates of a truncated shell model calculation
converge to their exact value, with a few percent precision. At present, when using a
basis of harmonic oscillator states, the in-medium similarity renormalization group
method can only be applied in a shell model space generated by a single harmonic
oscillator major shell [61–65]. Consequently, certain aspects of nuclear deformation
cannot be described in this method, as multishell valence spaces would be needed
therein [68]. Nevertheless, axial nuclear deformation could be recently included
in the fp shell by using the generator coordinate method, where Hartree–Fock–
Bogoliubov solutions of several different deformations are used as basis states (see
Ref. [69] where neutrinoless beta decay is studied in A = 48 deformed nuclei.)
The in-medium similarity renormalization group method has been applied in
light and medium-heavy well-bound nuclei, such as the isotopes of oxygen, calcium,
and nickel [61, 62, 65]. The method allows also to evaluate the electromagnetic
transitions [68]. Recently, the in-medium similarity renormalization group method
has been considered in the context of the Gamow shell model to calculate the spectra
of 22 C and 24 O [70].
States of deformed nuclei can be calculated in an ab initio framework if the
basis states are generated by a Hamiltonian depending on deformation degrees of
freedom. For example, the Monte Carlo shell model [71–74] makes use of Slater
determinants generated by a deformed potential, where the rotational invariance
of the many-body Hamiltonian is restored with projection techniques [75]. Alternatively, a new ab initio shell model framework has been recently introduced to