Isospin symmetry violation in a mean-field approach

Figure 1: The upper panel schematically shows two possible g.s. configurations of an odd-odd $N$=$Z$ nucleus, as described by the conventional deformed MF theory. These degenerate configurations are called aligned (left) and anti-aligned (right), depending on what levels are occupied by the valence particles. The lower panel shows what happens if the isospin-symmetry is restored. The aligned configuration is isoscalar; hence, it is insensitive to the isospin projection. The anti-aligned configuration represents a mixture of the $T$=0 and $T$=1 states. The isospin projection removes the degeneracy by shifting the $T$=0 component down.

Let us begin by exposing serious problems with the MF description of g.s. configurations of odd-odd (o-o) $N$=$Z$ nuclei. If, for the sake of simplicity, the Coulomb and time-odd polarization effects are disregarded and proton-neutron symmetry is conserved, a deformed MF approach naturally leads to four-fold degenerate (isospin and Kramers) single-particle (s.p.) levels. Consequently, the MF g.s. configuration of the o-o $N$=$Z$ nucleus is not uniquely defined and depends on occupation of specific levels. As shown in the upper panel of Fig. 1, the valence proton and neutron can be arranged in two distinctively different ways. Indeed, configurations of the type shown on the left-hand side (aligned configurations), are symmetric in spin-space coordinates and, therefore, anti-symmetric (isoscalar) in isospin coordinates. The anti-aligned configurations depicted on the right-hand side, have a mixed symmetry in spin-space coordinates and, therefore, also in isospin coordinates.

Because of their isoscalar character, aligned configurations are not affected by the isospin projection. On the other hand, the projection lifts the degeneracy of $T$=0 and $T$=1 components of aligned configurations as it is illustrated in the lower panel of Fig. 1. Due to the repulsive character of the nuclear symmetry energy, the isovector (isoscalar) components of the anti-aligned configurations are shifted up (down) in energy. Hence, isospin restoration changes the structure of the ground states of o-o $N$=$Z$ nuclei without affecting the ground states of e-e $N$=$Z$ nuclei. In other words, it does affect the binding-energy staggering along the $N$=$Z$ line.

Another example that nicely illuminates problems with isospin encountered in MF approaches is the case of two SD bands observed in the doubly magic nucleus $^{56}$Ni [20]. Following Ref. [20], we label them as Band 1 and Band 2. Band 1 is interpreted as a four-particle four-hole configuration formed by promoting two protons and two neutrons from the $0f_{7/2}$ shell to $0f_{5/2}$ shell. Within deformed MF model, where it is more natural to use the notion of Nilsson orbitals, Band 1 is obtained by emptying the neutron and proton [303]7/2 Nilsson extruder orbitals and occupying the prolate-driving [321]1/2 levels. This interpretation, not involving the $0g_{9/2}$ shell, is strongly supported by both state-of-the-art shell-model (SM) calculations [24,20] in the $0\hbar\omega$ $fp$ space and the self-consistent cranked-HF theory [20]. Both approaches appear to reproduce satisfactorily the excitation energy and moment of inertia (MoI) of this band. The p-h configuration of this band is depicted schematically in the upper panel of Fig. 2.

Figure 2: The upper panel schematically shows the s.p. configuration corresponding to the SD Band 1 in $^{56}$Ni, which is based on a 4p4h excitation with respect to the doubly magic core. The middle panel depicts two possible configurations of Band 2 involving a single proton (left) or neutron (right) promoted to the $0g_{9/2}$ shell. To facilitate the discussion, we also show the relevant asymptotic Nilsson quantum numbers associated with deformed orbitals involved. The lower panel illustrates the isospin splitting induced by the isospin-symmetry restoration.

Band 2, on the other hand, cannot be reproduced within the $fp$ SM space. This band has also a larger MoI as compared to the Band 1, and it is becoming yrast around spin $I\sim 12\hbar$. These two facts strongly suggest that its structure must involve at least one particle in the prolate-driving [440]1/2 Nilsson orbital that originates from the $0g_{9/2}$ shell. According to the calculations of Ref. [20], Band 2 involves one proton in $\pi$[440]1/2 state. Its configuration, which can be regarded as one proton p-h excitation with respect to the reference Band 1, is schematically shown in Fig. 2 (middle panel, left-hand side). This scenario has been supported by the full $pf-$ and $pfg_{9/2}$-SM calculations [25].

The conventional MF interpretation of Band 2 is not fully supported by experiment. Indeed, the MF theory predicts existence of the second band that is built upon the single neutron occupying the $\nu$[440]1/2 orbital (middle panel of Fig. 2, right-hand side). This neutron band is predicted to be placed only slightly higher in energy than the proton band, and this is due to a tiny difference in the Coulomb energies between these two configurations with, otherwise, very similar properties. Therefore, it is extremely difficult to understand within MF theory why only one of these two bands is observed experimentally. Moreover, HF calculations predict that the $\pi$[440]1/2 band is too high in excitation energy and that it crosses Band 1 above spin $I\sim 16\hbar$, that is, well above the empirical crossing. In these discussion, energetic arguments essentially exclude configurations involving two or more particles in the $0g_{9/2}$ shell.

Here again the problems with the MF approach can be traced back to the isospin-symmetry violation by the p-h excitations. Indeed, the MF approach treats proton $\vert\pi_{p-h}\rangle$ and neutron $\vert\nu_{p-h}\rangle$ p-h excitations as independent elementary excitations, and this manifestly breaks the isospin symmetry in $N$=$Z$ nuclei. To make the elementary excitation modes in $N$=$Z$ nuclei consistent with the symmetry, one needs to symmetrize or anti-symmetrize the Slater determinants corresponding to the mirror p-h excitations $\vert T=0(1)\rangle \approx \frac{1}{\sqrt{2}} (\vert\pi_{p-h}\rangle \pm \vert\nu_{p-h}\rangle )$. Such wave functions go beyond the usual MF picture. Restoration of isospin lifts the MF degeneracy between proton and neutron excitations and shifts the isoscalar (isovector) configurations down (up) in energy, as illustrated in the lower panel of Fig. 2. Of course, it is irrelevant whether the $T$=0 and $T$=1 components are projected from the proton or neutron state; the results should be identical up to the tiny polarization effects.

At present, it is not at all clear how to include the isospin correlations directly into the functional. The most natural method of taking them it into account is the isospin projected DFT approach [26,27,28,29], which is also used in this work. We note that such an approach is within the provisos of the DFT, whereby the isospin-projected energy is still a functional of the isospin-unprojected density, cf. Refs. [30,31,32].

We finish the discussion in this Section by noticing that even the isospin-projected DFT approach cannot fully account for the structure of the g.s. of an o-o $N$=$Z$ nucleus. Indeed, within the conventional DFT approach, the aligned and anti-aligned configurations differ due to different time-odd (TO) polarization effects in these two configurations. This polarization affects the position of the $T=1,T_z=0$ states with respect to the $T=1,T_z=\pm 1$ states. While the former states are shifted due to TO polarization, the latter ones - the $I=0$ ground states of e-e $N-Z=\pm 2$ isotopes - are not influenced by TO effects due to time-reversal symmetry conservation. Hence, the TO terms introduce a very specific source of the isospin symmetry violation in the intrinsic system.

Moreover, as shown in the lower part of Fig. 1, the isospin projected DFT approach always yields ground states with $T$=0 in o-o $N$=$Z$ nuclei. This is at variance with empirical data; indeed, it is well known (see, e.g., Refs. [33,34,35]) that, with two exceptions, the isospin of the g.s. changes from $T$=0 in light ($A < 40$) o-o $N$=$Z$ nuclei to $T$=1 in heavier ($A > 40$) o-o $N$=$Z$ nuclei. To be able to address this question in the nuclear DFT, one has to consider simultaneous isospin and angular-momentum projections and also consider residual proton-neutron interactions between valence particles (either perturbatively or by breaking the proton-neutron symmetry). This subject will be considered in forthcoming studies. The focus of this paper is on high-spin superdeformed and terminating states, which - due to large spin polarization and high seniority - are expected to be less affected by the effects mentioned above.