388
8 No-Core Gamow Shell Model
and unbound for f < 1.6, to acknowledge the opening of the 4 n → 3 n + n channel
[128]. For each scaling factor, one uses results of the no-core Gamow shell model
with two neutrons in the continuum to generate natural orbitals and keeps only
those natural orbitals whose associated eigenvalues in the scalar density matrix are
sufficiently large (η > 10
−7 in the density matrix), therefore reducing the size of
the basis by a factor ≈ 3. The first s 1/2 and p 3/2 natural orbitals are treated as
pole states while remaining orbitals are considered as continuum shells. Then in
the results denoted NCGSM-2p2h (n.o.) and NCGSM-3p3h (n.o.), one considers at
most two and three neutrons in the natural orbitals continuum shells, respectively.
This technique allows to reduce significantly the computational cost and to include
additional many-body correlations in a more efficient way.
The results denoted NCGSM-2p2h (n.o.), shown in Fig. 8.7, illustrate this point
clearly, as they are all lower in energy than the results of initial no-core Gamow
shell model with at most two neutrons in the continuum shells (NCGSM-2p2h)
obtained in the Berggren basis. However, the quality of natural orbitals for the
description of targeted state depends on how close the generating eigenstate is to the
final eigenstate. For that reason, the NCGSM-2p2h (n.o.) calculations could only be
performed for f > 1.45.
Another advantage of the natural orbitals is the possibility to remove some of
the truncations as compared to NCGSM-2p2h calculations. However, the NCGSM2p2h calculations need to include enough of correlations in the continuum for the
removal of truncations in the calculations with natural orbitals is meaningful. The
NCGSM-3p3h (n.o.) calculations which allow for three neutrons in the natural
orbitals outside the pole space, contain more correlations than the NCGSM-2p2h
(n.o.). However, also these calculations are limited to f > 1.45. It was not possible
to completely remove the truncations in the no-core Gamow shell model and hence
one had to rely on the density matrix renormalization group method to calculate
binding energies.
The density matrix renormalization group applied to no-core Gamow shell model
allows the calculation for a tetraneutron without truncations on the number of
particles in the continuum. The same shells are used as in the no-core Gamow
shell model using natural orbitals and the convergence criterion of the density
matrix renormalization group method is fixed by the parameter ε = 10
−8 [130,131]
(see Sect. 5.9). At the scaling factor f = 2.0, these results are about 1 MeV lower
than results of the no-core Gamow shell model with three particles at most in
the continuum (NCGSM-3p3h (n.o.)), indicating important missing correlations
in the NCGSM-3p3h (n.o.). This shows that configurations with four neutrons in
the continuum shells have a large contribution to the wave function even when
the system is artificially bound. In fact, the opening of new decay channels and
the presence of continuum states in the configuration mixing above the threshold
is expected to make the width explode when f → 1, especially in the density
matrix renormalization group results where all decay channels are open. This
is in qualitative agreement with the results in Ref. [121] which show a rapid
increase of the width when the strength of the phenomenological T = 3/2 three-
8 No-Core Gamow Shell Model
and unbound for f < 1.6, to acknowledge the opening of the 4 n → 3 n + n channel
[128]. For each scaling factor, one uses results of the no-core Gamow shell model
with two neutrons in the continuum to generate natural orbitals and keeps only
those natural orbitals whose associated eigenvalues in the scalar density matrix are
sufficiently large (η > 10
−7 in the density matrix), therefore reducing the size of
the basis by a factor ≈ 3. The first s 1/2 and p 3/2 natural orbitals are treated as
pole states while remaining orbitals are considered as continuum shells. Then in
the results denoted NCGSM-2p2h (n.o.) and NCGSM-3p3h (n.o.), one considers at
most two and three neutrons in the natural orbitals continuum shells, respectively.
This technique allows to reduce significantly the computational cost and to include
additional many-body correlations in a more efficient way.
The results denoted NCGSM-2p2h (n.o.), shown in Fig. 8.7, illustrate this point
clearly, as they are all lower in energy than the results of initial no-core Gamow
shell model with at most two neutrons in the continuum shells (NCGSM-2p2h)
obtained in the Berggren basis. However, the quality of natural orbitals for the
description of targeted state depends on how close the generating eigenstate is to the
final eigenstate. For that reason, the NCGSM-2p2h (n.o.) calculations could only be
performed for f > 1.45.
Another advantage of the natural orbitals is the possibility to remove some of
the truncations as compared to NCGSM-2p2h calculations. However, the NCGSM2p2h calculations need to include enough of correlations in the continuum for the
removal of truncations in the calculations with natural orbitals is meaningful. The
NCGSM-3p3h (n.o.) calculations which allow for three neutrons in the natural
orbitals outside the pole space, contain more correlations than the NCGSM-2p2h
(n.o.). However, also these calculations are limited to f > 1.45. It was not possible
to completely remove the truncations in the no-core Gamow shell model and hence
one had to rely on the density matrix renormalization group method to calculate
binding energies.
The density matrix renormalization group applied to no-core Gamow shell model
allows the calculation for a tetraneutron without truncations on the number of
particles in the continuum. The same shells are used as in the no-core Gamow
shell model using natural orbitals and the convergence criterion of the density
matrix renormalization group method is fixed by the parameter ε = 10
−8 [130,131]
(see Sect. 5.9). At the scaling factor f = 2.0, these results are about 1 MeV lower
than results of the no-core Gamow shell model with three particles at most in
the continuum (NCGSM-3p3h (n.o.)), indicating important missing correlations
in the NCGSM-3p3h (n.o.). This shows that configurations with four neutrons in
the continuum shells have a large contribution to the wave function even when
the system is artificially bound. In fact, the opening of new decay channels and
the presence of continuum states in the configuration mixing above the threshold
is expected to make the width explode when f → 1, especially in the density
matrix renormalization group results where all decay channels are open. This
is in qualitative agreement with the results in Ref. [121] which show a rapid
increase of the width when the strength of the phenomenological T = 3/2 three-