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At deformations
and
, the projected energy curve exhibits
unphysical jumps. By comparing with Fig. 7, one concludes
that at these deformations
the neutron 1d
poles
cross the integration contour.
Obviously, the residue contributions of these poles
cause the sudden jumps in the deformation energy. The 1d
pole introduces a positive contribution at
, while
the 1d
pole introduces another positive contribution at
. Based on this observation, two other
sets of PAV calculations were carried out.
The first calculation (open circles) was done by excluding
contributions from the 1d
poles, as is the case
for the
ground state configuration. This can be accomplished by
reducing the integration radius from
to a
smaller value of about
, cf. Fig. 7. At
small deformations,
, the
new results are
identical to those obtained with the unit circle, and at the larger
deformations (prolate or oblate), the energy
curve smoothly continues without
any jump. Thus, in this example, an appropriate shift of the
integration contour allows us to obtain smooth and unique projected
energy. The second PAV curve (open squares) has been
obtained by always
including the lowest 1d
pole, i.e., by continuously
varying the integration radius as a
function of
, to ensure that
it always stays between the first and second 1d
pole
(cf. Fig. 7). The resulting
energy curve coincides at large
deformations with the standard PAV result, and then smoothly
continues to
, where the 1d
poles (
=3)
disappear.
Figure 10 also presents the fully self-consistent VAP
results. Similar to the PAV case, two sets of calculations were performed.
The solid (open) stars correspond
to including
(excluding) the contributions from the lowest 1d poles. In
both cases, one obtains smooth curves, which, beyond the spherical
point, differ from one another.
In this rather simple case of O, both in the PAV and VAP
calculations, one can avoid unphysical jumps of the projected energy
curve by making a specific selection of ``active" poles that are
considered during contour integration.
Such selection of residues can, in principle, become a part of the definition of
the projected energy. The variational principle can then be invoked
to pick the selection that yields the lowest projected energy. In the
discussed case of
O, the PAV and VAP energies obtained by
excluding the 1d
poles are the lowest, and they are smooth functions of deformation.
Therefore, such a selection can be adopted for the final PNP energy in
this nucleus. It is clear, however,
that one cannot a priori tell which
selection of poles leads to the lowest projected energy. For example,
in heavier oxygen isotopes, the lowest energy is
obtained by including some of the 1d
poles.
Let us now consider a more complicated case of the HFB+LN
calculations for the neutron rich nucleus Mg.
The total HFB energy (without the corrective
LN
term) is shown in Fig. 11 as a
function of
. Solid squares denote the result
of PAV PNP calculations
on the top of HFB+LN.
At
, the PAV curve exhibits a small jump, after which
its behavior changes character. This is clearly related to the proton
1d
pole crossing the unit circle,
cf. Fig. 8. Otherwise, the PAV deformation energy is quite
smooth as a function of deformation, despite the fact that
three pairs of neutron poles cross the unit circle in the deformation
range considered. This apparent lack of sensitivity to
neutron poles can be traced back to the fact that
they cross the integration contour at or
near points where they pairwise cross one another. Therefore, the
increasing degeneracy factor
makes the poles disappear at the
crossing points; hence, the total PAV curve behaves smoothly.
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In a search for the most sensible method of calculating the
projected energy curve within the PAV approach, we can employ a
prescription whereby the number of the lowest poles is
kept fixed within the integration contour. This can be realized by
keeping between the poles or at the pole
crossing point. Since at the crossing points
the poles vanish, at least in SIII calculations,
this results in a smooth energy curve.
Such an option is shown in Fig. 11 with solid squares.
The resulting curve is indeed very smooth; however,
at
there appears an unphysical bump, which makes
this option as unacceptable as the other one.
We have also attempted calculating the energy curve within the VAP approach. In principle, the VAP approach could have generated problems related to the fact that the density-dependent terms of fractional order may lead to large negative contributions (see Sec. 3.6). In practice, this is never the case because the VAP method [23] is not implemented through an explicit minimization of the energy, but is carried out by solving variational equations that have been derived with the same incorrect treatment of cuts in the complex plane. In this context, it is worth emphasizing that the appearance of poles never leads to infinite total energies, but to discontinuities in the total energy. Therefore, there is no danger that the minimization procedure may attract a solution towards a pole.
The main problem in implementing the VAP method is related to the
fact that unprojected quantities, e.g., particle
or pairing
densities, lose their
usual
physical meaning [23] in VAP. They depend on the internal
normalization
of the density
that is
not related to the particle number
onto which the state is projected.
Neither the total VAP energy nor other projected observables depend
on the normalization
. However, depending on the choice of the internal
normalization
, one obtains different canonical
occupation probabilities; hence, the associated poles
are not
distributed in the same way as in the unprojected HFB case. Depending
on the internal normalization
, different poles
enter the
integration contour, and the convergence procedure cannot be easily
controlled.
Additional problems arise when two poles are nearly degenerate.
Although at the point of degeneracy the poles disappear, when the
distance between the poles is small but nonzero and the integration
contour is between them, one faces significant instabilities of the
constrained VAP problem. During the iteration of VAP equations,
one or both poles
enter or leave the integration contour. The poles create jumps in the
projected energy and, which is even more important, they create jumps in the
deformation. The numerical algorithm enters a `ping-pong' regime,
which cannot be overcome, and one cannot converge to any solution.
Figure 11 offers a good illustration of this problem.
The converged VAP energies for Mg are shown with
solid circles. The converged solution can be found
only in limited regions of
deformation. The 1d and 1f neutron
poles close to the contour
spoil the convergence in the regions of
,
0.25, and
. The same is true for the proton 1d states around
and 0.1. As a result, the VAP procedure could be
solved only within small deformation
intervals around
, 0, and
0.4.