Chiral solutions

The planar HF solutions were easily obtained by applying small cranking frequency increments to the non-rotating state. For chiral bands, analogous task was more difficult, because these bands start at finite frequencies, which in the present case are not lower than $\approx0.5\,\mathrm{MeV}/\hbar$. Several level crossings may occur between $\omega =0$ and such a high frequency, and it is difficult to spot the required s.p. configuration at high frequencies. A hint on how to follow the $\pi h_{11/2}^1~\nu h_{11/2}^{-1}$ configuration diabatically comes from the classical prediction that the chiral band branches off from the planar solution (at the point corresponding to the critical frequency). One can thus restart iterations from the planar band by applying cranking frequency with non-zero component on the medium axis.

As the first step we performed a kind of perturbative search along the planar bands, which turned out to be a very reliable test of where the chiral solution may exist. Such a search gives us also some understanding of why the chiral solutions do not appear in several cases. The procedure we applied was the following. To each converged point of the planar band, a small additional component, $\omega _m$, of the angular frequency along the medium axis was added, with $\omega_s$ and $\omega_l$ unchanged. The resulting s.p. Routhian (3) was diagonalized only once. Then, it was checked whether in the resulting non-selfconsistent state the angular momentum and rotational frequency vectors were parallel, as required by the Kerman-Onishi necessary condition of self-consistency; see Section 2. We can guess that in nuclei stiff against deformation changes, the direction of $\vec {I}$ is the only degree of freedom, and thus the Kerman-Onishi condition is also sufficient. If $\vec {I}$ is parallel to $\vec {\omega}$ in the non-selfconsistent state after one diagonalization, then it is very probable that further iterations may lead to a converged chiral solution. Indeed, this was always the case, and never a chiral solution was obtained, in spite of several attempts, if that simple test gave negative result.

Figure 9: (color online). Ratios $I_m/\omega _m$ (open circles) and $I_s/\omega _s\approx I_l/\omega _l$ (plus symbols) obtained in the perturbative search for the HF chiral solutions along the planar bands in the $N=75$ isotones; see text. Results for the SLy4 and SkM* forces with no time-odd fields are shown. Ratios $I_m/\omega _m$ (solid line) and $I_s/\omega _s=I_l/\omega _l$ (dashed line) corresponding to the classical-model chiral and planar bands, respectively, are also plotted for comparison.

The condition for $\vec {I}$ and $\vec {\omega}$ being parallel can be written in the form

$$\frac{I_s}{\omega_s}=\frac{I_m}{\omega_m}=\frac{I_l}{\omega_l}~.$$ (24)

Note that the $I_s/\omega_s$ and $I_l/\omega_l$ ratios must be very close to each other in the non-selfconsistent state, because the Kerman-Onishi condition is fulfilled for the self-consistent planar state. Therefore, the test consists in checking for each point of the planar band if $I_m/\omega _m$ is equal to $I_s/\omega _s\approx I_l/\omega _l$. In fact, this is reliable only if the time-odd fields are switched off, because in their presence, the $I_m/\omega _m$ ratio calculated perturbatively is significantly smaller than the self-consistent result would be. The reason is that the relevant components of the time-odd fields become active only after self-consistency is achieved for a non-zero $\omega _m$. The test was made with $\omega_m=0.05\,\mathrm{MeV}/\hbar$. The discussed ratios, calculated for all the HF planar bands found in the $N=75$ isotones, are plotted in Fig. 9. Plus symbols and open circles denote the HF values of $I_m/\omega _m$ and $I_s/\omega _s\approx I_l/\omega _l$, respectively. To guide the eye, in the same Figure we also plotted the ratios $I_m/\omega _m$ and $I_s/\omega _s=I_l/\omega _l$ corresponding to the classical-model chiral and planar bands, respectively.

It can be a priori expected that chiral solutions do not appear in the oblate minima in $^{134}$Pr and $^{136}$Pm, because of insufficient triaxiality. Indeed, the calculated values of the $I_m/\omega _m$ and $I_s/\omega _s\approx I_l/\omega _l$ ratios exhibit a complicated behavior, and do not become equal to one another at any point. In $^{130}$Cs, as well as in the triaxial minimum in $^{136}$Pm, the two ratios clearly approach each other. It seems that the only reason why they do not attain equality is that the planar bands were not found up to sufficiently high frequencies, because of level crossings. Note, however, that the moment of inertia associated with the medium axis, $I_m/\omega _m$, significantly drops with angular frequency, which takes $I_m/\omega _m$ away from $I_s/\omega _s\approx I_l/\omega _l$, and defers their equalization to higher frequencies. This effect is much weaker in $^{132}$La, where the ratios do become equal, slightly above the point expected from the classical model. Indeed, self-consistent chiral solutions were found in this case, as described below.

After the first diagonalization of the perturbative test, the HF iterations were continued in each case to achieve self-consistency. In $^{130}$Cs, $^{134}$Pr, and $^{136}$Pm, as well as for low rotational frequencies in $^{132}$La, the iterations converged back to planar solutions. The same result was obtained for different initial orientations of $\vec {\omega}$ with respect to the intrinsic frame. This provides a strong argument that, for the concerned configuration, no self-consistent chiral solutions exist at low frequencies. In $^{132}$La, for $\omega$ high enough, converged solutions were obtained with $\vec {I}$ having non-zero components on all the three intrinsic axes, which corresponds to chiral rotation. To examine the chiral solutions independently of the planar ones, the found fragments of chiral bands were used as starting points to obtain solutions for lower and higher frequencies. Calculations were performed with $\omega$ step of $0.02\,\mathrm{MeV}/\hbar$. At a certain value of decreasing $\omega$, the planar orientation of $\vec {I}$ was regained in the intrinsic frame, and the solution merged into the previously found planar one. In a natural way, that junction value of $\omega$ can be regarded as the Skyrme-HF result for the critical frequency, and is denoted in the following as $\omega _{\text{crit}}^{\text{HF}}$. Values of $\omega _{\text{crit}}^{\text{HF}}$ obtained in the present calculations are collected in Table 1, and discussed in Section 4. On the side of highest frequencies, chiral solutions were obtained up to a certain value of $\omega$, and all attempts to go higher caused the iterations to fall into a different minimum. This is probably due to multiple smooth crossings of occupied and empty levels, particularly in neutrons.

Figure 10: Trajectories of the angular frequency vectors following the HF chiral bands in $^{132}$La (plus symbols), compared to the classical solutions (solid lines). Projections onto the $s$-$m$ and $l$-$m$ planes of the intrinsic frame are shown. 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.

The HF results presented so far corroborate the main prediction of the classical model, that chiral solution exists only above a certain critical frequency, at which it branches off from the planar one. Also the intrinsic-frame trajectory of $\vec {\omega}$ along the HF chiral band is almost a straight line parallel to the medium axis, as in the classical case. This is demonstrated in Fig. 10, which shows projections of $\vec {\omega}$ on the $s$-$m$ and $l$-$m$ intrinsic planes for the HF (plus symbols) and classical (line) results. The only difference is that $\omega _{\text{crit}}^{\text{HF}}$ is a bit higher than $\omega _{\text{crit}}^{\text{clas}}$, and the HF line is shifted along the planar band to higher frequencies. Although the chiral solutions have been found in a rather narrow $\omega$ interval, of about $0.1\,\mathrm{MeV}/\hbar$, the accompanying increase in $\omega _m$ is significant, from zero to about $0.4\,\mathrm{MeV}/\hbar$. This is so because $\omega_s$ and $\omega_l$ are relatively large and almost constant along the chiral solution.

Figure 6 shows the proton and neutron s.p. Routhians for the chiral solution obtained in $^{132}$La with the SkM* force and no time-odd fields. A thin vertical line is drawn at the value of $\omega _{\text{crit}}^{\text{HF}}$. The Routhians to the left and to the right of this line correspond to the planar and chiral bands, respectively. Note first that the planar and chiral Routhians do indeed coincide at $\omega _{\text{crit}}^{\text{HF}}$. The chiral Routhians do not seem to exhibit any particular behavior. At high frequencies, the Routhian occupied by the neutron hole enters into a region of high level density and the corresponding s.p. state mixes with other negative-parity states. Thus, it is doubtful whether the valence neutron hole can be identified with a single state, and we do not examine its s.p. properties. The marking in open circles is only tentative. However, the $h_{11/2}$ proton particle is still well separated.

The alignments of the angular momentum, $\vec {j}^p$, of the $h_{11/2}$ proton particle on the short, medium, and long intrinsic axes for the SLy4 and SkM* forces and $N$, $G$, $T$ time-odd fields are shown in Fig. 7. As in Fig. 6, the vertical line separates the planar and chiral bands. The plot confirms the stiff character of those alignments in the chiral solutions. Since the $\omega_s$ and $\omega_l$ components of the cranking frequency vector hardly change along the chiral band, also the considered alignments on those axes, $j^p_s$ and $j^p_l$, are nearly constant. Only the projection on the medium axis, $j^p_m$, increases due to the increase in $\omega _m$ from zero to about $0.4\,\mathrm{MeV}/\hbar$.