Pairing-plus-quadrupole model

In the previous sections, we have considered models having either
degenerate (Sec. 3.2) or arbitrarily fixed (Secs. 3.3
and 3.4) single-particle spectra. However, in real systems,
the single-particle energies do depend on the numbers of particles,
and the energy-spacing filters (7) extract from masses
the single-particle energies which include the particle-number
dependence, i.e, at every *N*, the distance to the next available
single-particle level is obtained.

In order to account for such effects, in this section we analyze the
results of the exact diagonalization of the pairing-plus-quadrupole
(PPQ) Hamiltonian,

in a single-

Exact ground-state energies for all particle numbers
*N*=0,1,,20 and *G*=0 are plotted in Fig. 9. All
the even systems have the ground-state spins of
=0,
while in the odd systems, the ground-state spins are
=|*N*-/2+/2; this corresponds to
rotational bands based on oblate sequences of deformed
single-particle levels (see the discussion in Ref. [30]).

Even in a large energy scale of Fig. 9, the effect of the twofold Kramers degeneracy and the OES effect are clearly seen. Note that in this model the pure QQ interaction generates a (weak) OES effect. Indeed, in a relatively small phase space, the QQ interaction has a tangible pairing component.

In order to quantify
the OES and mean-field effects, Fig. 10 shows values of
the three-mass filter (1) obtained for different pairing
strengths *G*. Ground-state energies of even and odd systems are used
to calculate
for each value of *G*. For even systems, the
ground-state spins equal 0 for all values of *N* and *G*, while for odd
systems, the values of
=19/2 (for all *N*) replace at
large *G* the values of
=|*N*-/2+/2,
which characterize the *G*=0 solutions. This corresponds to a gradual
transition from deformed to spherical shapes and from rotational to
seniority-like excitation spectra.

In Fig. 10, one can clearly see that the OES increases almost
linearly with *G*, while the pattern of alternating larger and
smaller values of
is almost independent of *G*.
Both these features of
are explicated by using filters
(6) and (8), which give values of
and
plotted in Fig. 11. One can very well see the
almost linear dependence of the OES on the pairing strength *G*,
Fig. 11(a), and a very weak *G*-dependence (apart from *N*=10)
of the
single-particle energy spacings, Fig. 11(b).

The PPQ model exhibits several features pertaining to two kinds of
phase transitions. First, the static pairing correlations set in at
critical values of the pairing strength *G*. Depending on the number
of particles, this phase transition occurs
at about =0.03-0.05 for even particle number.
Second, the transition from deformed
to spherical shapes occurs at slightly higher values of *G*, i.e., at
about =0.08, 0.10, 0.12, and 0.12 for *N*=4, 6, 8, and 10,
respectively. In addition, deformed equilibrium shapes are oblate for
*N*=2, 4, 6, and 8, and prolate for *N*=10. (Note that apart from a
linear dependence of energies on *N*, the PPQ Hamiltonian is exactly
symmetric with respect to the particle-hole transformation
*N*
-*N* [31].)

Since the system is finite (and fairly small for that matter), the phase transitions are hardly visible in the exact results of Fig. 11. However, when filters (6) and (8) are applied to the mean-field (HFB) ground-state energies (Fig. 12), the phase transitions become visible as sudden increases in (pairing transition), and degeneracies of (shape transition).

As one can see, a comparison of the exact and HFB ground-state energies (Figs. 11 and 12) is very instructive when it is based on comparing the corresponding filters (6) and (8). It turns out that in the PPQ model the HFB method reproduces quite well the OES and the single-particle properties simultaneously. Some deviations occur only near the phase transitions, where it is well known that the mean-field approximation is not accurate.

When analyzing exact solutions for systems interacting with
two-body interactions, or when similarly analyzing the experimental data,
one does not have *a priori* access to the single-particle
energies or to the single-particle energy-spacings. In fact,
the single-particle energies are concepts that appear naturally
only in the mean-field approximation. Therefore, in order
to assess the meaning of numbers obtained from the energy-spacing
filter (8), one should compare them with the energy-spacings
calculated directly from the mean-field spectra, i.e., with the
differences

[cf. Eq. (9)] where is the

In Fig. 13(b) we show the differences (65)
calculated from the single-particle spectra of canonical HFB energies
obtained in the PPQ model. (In fact, since all of the PPQ+HFB
equilibrium solutions conserve the axial symmetry, the PPQ+HFB method
reduces to the simple BCS approximation, and the canonical energies
are equal to the eigenenergies of the mean-field Hamiltonian.) It is
clear that the energy-spacing filters (applied either to the exact or
to the HFB total energies) give results similar to the differences of
canonical energies only for deformed shapes. Whenever the mean-field
solutions become spherical, differences (65) collapse to
zero, as expected, while the energy-spacing filters give non-zero
values. This result is easy to understand: in the spherical limit,
the results of the PPQ model should resemble those of the seniority model.
While the single-particle energies are degenerate (the Hilbert space consists
of one *j*-shell only), the energy-spacing filter should be proportional
to the pairing strength *G* (see Sec. 3.2).
On the other hand, before the transition to sphericity, the results
presented in Figs. 11(b), 12(b), and
13(b) are encouragingly similar. Note that deviations
obtained at *N*=10 should again be attributed to the differences
in predicted shapes (prolate vs. oblate).

Finally, in Fig. 13(a) we show the values of the HFB ``equivalent'' gap parameters (56) calculated in the PPQ+HFB model. It is seen that significantly underestimates the magnitude of the OES effect, and moreover, it exhibits some particle-number dependence which is absent in the exact results.

In light of the above discussion, the very weak average dependence
of
on *N* (except for the OES, of course),
shown in Fig. 10, can be given by a very simple explanation.
In the weak pairing limit,
is small, and its overall particle-number
dependence is much weaker than the even-odd effect (see
Fig. 6).
On the other hand, in the limit of strong pairing,
is expected to
approach the seniority limit in which
depends only on the
number parity but not on *N*. Note that the results of
the non-degenerate model shown in Fig. 6
are very far from the spherical limit since *G*/*d*<1.