Planar solutions

We began our self-consistent TAC calculations by finding planar solutions corresponding to the classical band A. The first point of each band of this kind was obtained by restarting the HF iterations from the previously converged non-rotating solution and by applying the initial cranking frequency vector with non-zero components on its short and long intrinsic axes. Once convergence was achieved, the obtained solution served in turn as the starting point for the next value of the rotational frequency. We proceeded in this way with the frequency step of 0.05MeV/$\hbar$. We followed each band diabatically, i.e., by exciting particles near the Fermi level whenever an empty and an occupied s.p. level of the same parity were about to cross, so that always the states with the same physical properties were occupied.

Figure 6: Similar as in Fig. 2 but for the proton and neutron s.p. Routhians from the HF TAC calculations with the SkM* force. The thin vertical line is drawn at $\omega _{\text{crit}}^{\text{HF}}$. The Routhians to the left and to the right of this line correspond to the planar and chiral bands, respectively.

In the solutions corresponding to a planar rotation, the cranking frequency vector had non-zero components on the short and long intrinsic axes in the self-consistent results. To give some insight into the s.p. structure of those solutions, in Fig. 6 we give the proton and neutron s.p. Routhians for the planar band in $^{132}$La obtained with the SkM* force and no time-odd fields. Parts of the plots to the left of the thin vertical lines are relevant for the planar results. Contrary to the PAC Routhians of Fig. 2, now both the lowest and the highest $h_{11/2}$ levels split with rotational frequency. This is consistent with the picture that the valence particle and hole angular momenta, $\vec {j}^p$ and $\vec {j}^h$ , aligned on the short and long axes, respectively, now both have non-zero projections on the tilted axis of rotation.

Figure 7: Angular-momentum alignments of the lowest proton (positive values) and highest neutron (negative values) $h_{11/2}$ levels on the short, medium and long intrinsic axes from the HF TAC calculations in $^{132}$La. The thin vertical line is drawn at $\omega _{\text{crit}}^{\text{HF}}$. The curves to the left and to the right of this line correspond to the planar and chiral bands, respectively. The HF results with the SLy4 and SkM* forces are shown for the $N$, $G$, and $T$ variants of calculation defined in Sec. 3.2.

In order to examine the angular momenta of the valence nucleons, $\vec {j}^p$ and $\vec {j}^h$, in Fig. 7 we plot their projections onto the short (dotted line), medium (solid line) and long (dashed line) intrinsic axes, for all the self-consistent solutions in $^{132}$La. The positive and negative alignments are those of the lowest proton and highest neutron $h_{11/2}$ levels; the latter can be considered as representing $-\vec {j}^h$. Parts of the plots to the left of the thin vertical lines concern the planar bands. It can be seen that, indeed, the proton particle and the neutron hole align their angular momenta on the short and long axes. Furthermore, those alignments change rather weakly with rotational frequency which means that the wave-functions are strongly confined by deformation (deformation alignment).

Figure 8: (color online). Intrinsic-frame trajectories of the angular frequency vector along the HF planar bands in the $N=75$ isotones, compared to the classical solutions. The HF results with the SLy4 and SkM* forces are shown for the $N$, $G$, and $T$ variants of calculation defined in Sec. 3.2.

It is worth emphasizing that the intrinsic-frame trajectories of $\vec {\omega}$ along the self-consistent bands almost exactly follow the classical ones, in all the considered cases. This is illustrated in Fig. 8, where the dashed lines represent classical A bands with parameters of Table 1, and the HF results are marked with open circles. For the oblate bands in $^{136}$Pm the classical line is not shown because the parameters $\mathcal{J}_s$ and $s_s$ could not be unambiguously extracted from the PAC calculations; see Section 3.4. However, the HF results follow a curve that very much resembles the classical hyperbola. For the triaxial minima in $^{134}$Pr and $^{136}$Pm, no self-consistent planar bands could be obtained because of multiple level crossings.

Energy in function of spin also shows a striking agreement between the classical and self-consistent results for the planar bands. This can be traced in the case of $^{132}$La in Fig. 1, by following the same symbols as those in Fig. 8. Some deviations are visible only for rather high angular momenta.