386
8 No-Core Gamow Shell Model
In the no-core Gamow shell model, the Hamiltonian matrix is either diagonalized
with the Jacobi–Davidson method or with the density matrix renormalization group
method. In order to reduce the computational challenge that are configuration
interaction calculations in the continuum, the density matrix renormalization group
method in no-core Gamow shell model has been supplemented with natural orbitals
(see Sect. 5.3).
The density matrix renormalization group calculations are performed as follows.
First the standard density matrix renormalization group run is performed, with
some given truncations, to approach the final eigenstate of the Hamiltonian.
Then, the corresponding natural orbitals are calculated and a new density matrix
renormalization group run is performed where the natural orbitals replace the
Berggren basis states and the truncations are removed. This technique leads to an
impressive gain in computational time, while efficiently incorporating many-body
correlations with the successive sweeps of the density matrix renormalization group
procedure. In the calculation of tetraneutron, natural orbitals were generated from a
truncated space which is large enough to have a decent approximation of the targeted
Hamiltonian eigenstate. A small basis of natural orbitals can be built, by including
only those whose associated eigenvalue η in the density matrix is larger than some
chosen value 0 < η min < 1. It is then possible to do a many-body calculation of the
tetraneutron without model space truncation when Berggren basis states are replaced
by a few natural orbitals.
8.3.1 Tetraneutron Energy from the Extrapolation of Overbinding
Interactions
In the exploratory no-core Gamow shell model study of a tetraneutron [128], the
model space consisted of the 0s 1/2 and 0p 3/2 resonant shells (pole space), and
associated nonresonant partial waves whose energies are selected along the contours
in the complex-momentum plane. It has been checked that the positions of the poles
are unimportant as long as the poles are embedded in their associated complex
continua. This model space is then augmented by 30 nonresonant p 1/2 partial waves
along a real-momentum contour, and seven harmonic oscillator shells for each of
the partial waves d, f and g. Details of the calculation can be found in Ref. [128].
When allowing only two neutrons into the continuum, the energies and widths
of the J π = 0 + tetraneutron, calculated for various two-body chiral interactions
(N 3 LO [4], N 2 LO opt [129] and N 2 LO sat [17]) with different renormalization cutoffs
are all mutually consistent as shown in Table 8.3. For a qualitative comparison, in
the N 2 LO sat interaction one only used its two-body part. Similar simplification has
been done in the N 3 LO interaction in previous sections. The small effect of the Λcutoff variations on the calculated results indicates a weak influence of the missing
induced three- and four-body forces on the tetraneutron properties.
In the following, the N 3 LO two-body interaction with a renormalization cutoff
Λ = 1.9 fm
−1 is used. The role of the continuum in shaping the tetraneutron is
illustrated by scaling the interaction by a factor f = 2.0 so that the system is
8 No-Core Gamow Shell Model
In the no-core Gamow shell model, the Hamiltonian matrix is either diagonalized
with the Jacobi–Davidson method or with the density matrix renormalization group
method. In order to reduce the computational challenge that are configuration
interaction calculations in the continuum, the density matrix renormalization group
method in no-core Gamow shell model has been supplemented with natural orbitals
(see Sect. 5.3).
The density matrix renormalization group calculations are performed as follows.
First the standard density matrix renormalization group run is performed, with
some given truncations, to approach the final eigenstate of the Hamiltonian.
Then, the corresponding natural orbitals are calculated and a new density matrix
renormalization group run is performed where the natural orbitals replace the
Berggren basis states and the truncations are removed. This technique leads to an
impressive gain in computational time, while efficiently incorporating many-body
correlations with the successive sweeps of the density matrix renormalization group
procedure. In the calculation of tetraneutron, natural orbitals were generated from a
truncated space which is large enough to have a decent approximation of the targeted
Hamiltonian eigenstate. A small basis of natural orbitals can be built, by including
only those whose associated eigenvalue η in the density matrix is larger than some
chosen value 0 < η min < 1. It is then possible to do a many-body calculation of the
tetraneutron without model space truncation when Berggren basis states are replaced
by a few natural orbitals.
8.3.1 Tetraneutron Energy from the Extrapolation of Overbinding
Interactions
In the exploratory no-core Gamow shell model study of a tetraneutron [128], the
model space consisted of the 0s 1/2 and 0p 3/2 resonant shells (pole space), and
associated nonresonant partial waves whose energies are selected along the contours
in the complex-momentum plane. It has been checked that the positions of the poles
are unimportant as long as the poles are embedded in their associated complex
continua. This model space is then augmented by 30 nonresonant p 1/2 partial waves
along a real-momentum contour, and seven harmonic oscillator shells for each of
the partial waves d, f and g. Details of the calculation can be found in Ref. [128].
When allowing only two neutrons into the continuum, the energies and widths
of the J π = 0 + tetraneutron, calculated for various two-body chiral interactions
(N 3 LO [4], N 2 LO opt [129] and N 2 LO sat [17]) with different renormalization cutoffs
are all mutually consistent as shown in Table 8.3. For a qualitative comparison, in
the N 2 LO sat interaction one only used its two-body part. Similar simplification has
been done in the N 3 LO interaction in previous sections. The small effect of the Λcutoff variations on the calculated results indicates a weak influence of the missing
induced three- and four-body forces on the tetraneutron properties.
In the following, the N 3 LO two-body interaction with a renormalization cutoff
Λ = 1.9 fm
−1 is used. The role of the continuum in shaping the tetraneutron is
illustrated by scaling the interaction by a factor f = 2.0 so that the system is