Transition from $\gamma$-rigid to $\gamma$-soft dynamics in the interacting boson model: Quasicriticality and quasidynamical symmetry

We study the transition from the $\gamma$-rigid to $\gamma$-soft collective nuclear dynamics across the Casten triangle of the interacting boson model using mean-field techniques and confront the description with the exact diagonalization. We inspect the domain of validity of the SU(3) quasidynamical symmetry inside the Casten triangle and reveal critical behavior within the low-lying excited spectrum due to a degeneracy of $\beta$ and $\gamma$ vibrations.

Figure 1

Relative $\gamma$ rigidity $G$ from Eq. (6) of the semiclassical $N\to \infty$ potential in the deformed part of the Casten triangle.

Figure 2

Casten triangle with the regions of different ground-state shapes separated by the line of the spherical-to-deformed phase transition ${\eta }_{\mathrm{crit}}$. We indicate the contours of the ratio $R_{\mathrm{sad}}$ from Eq. (7) throughout the deformed region together with the spinodal line ${\eta }_{\mathrm{sp}}$ for $N=100$.

Figure 3

Diagonal BMF energies (in arbitrary units) of ${\beta }^n$-(solid red lines) and ${\gamma }^n$-(dashed blue lines) phonon excitations for $\eta =0.5$ and $N=100$ bosons with $n=0,\dots ,30$. Selected series of degeneracies among the mean-field states are indicated by arrows. Additionally, the saddle point energy $E_{\mathrm{sad}}$ is indicated by the dot-dashed blue line roughly delimiting the region of applicability ($E\lesssim E_{\mathrm{sad}}$) of the BMF method. The solid black lines delimit the wedge-shaped mixing regions of BMF states listed in rows (i) and (ii) of Table .

Figure 4

Casten triangle with selected ${\beta }^{n_{\beta }}\times {\gamma }^{n_{\gamma }}$ BMF degeneracy lines. The $\beta \times \gamma$ and $\beta \times {\gamma }^2$ lines are valid in the $N\to \infty$ limit (solid lines), while ${\beta }^2\times {\gamma }^3,{\beta }^3\times {\gamma }^4$, and ${\beta }^{10}\times {\gamma }^{11}$ lines correspond to $N=100$ (dashed lines). The $\eta =0.5$ line, where the most calculations in the present paper have been performed, is also indicated.

Figure 5

Evolution of energy spectra (in arbitrary units) calculated for $\eta =0.5$ and $N=20$ bosons. (a) BMF spectra both with and without mixing of $K=0$ states [Eq. (10)]. (b) Corresponding exact spectrum of a few lowest $l=0$ and $l=2$ states; inset shows the effect of the angular momentum projection (14)—the exact spectrum is supplemented by the $0_1^{+}$ and $2_1^{+}$ levels projected from the BMF condensate. $E_{\mathrm{sad}}$ is indicated by decreasing dot-dashed blue lines in both panels.

Figure 6

Exact excitation energy ratios $R(4_i/2_i),R(6_i/4_i)$ plotted for $i=1,2,3$, and $R(4_i/3_1)$ for $i=2,3$ along the $\eta =0.5$ line with $N=20$. The “rotational” SU(3) values are indicated by arrows. Staggering of the ratios due to the crossing of $\beta$ and $\gamma$ bands is observable at $\chi \approx -1.04$. We plot $R(4/2)$ for the ground and $\beta$ bands; $R(6/4)$ for the ground, $\beta$, and $\gamma$ bands; and $R(4/3)$ for the $\gamma$ band. For $\chi >-0.6$, the spectrum evidently undergoes a transformation from the rotational to the seniority directed pattern.

Figure 7

Moments of inertia (in arbitrary units) for the ground-state band and excited $\beta$ and $\gamma$ bands for $\eta =0.5$ and $N=20$ (exact calculation). The disturbance of the smooth dependence at $\chi \approx -1.04$ is due to $\beta \times \gamma$ degeneracy. The inset shows the same for $N=40$, which demonstrates mutual convergence of all three curves for high $N$.

Figure 8

Relative intraband $B(E2)$ transition rates (exact calculation) for the (a) ground-state, (b) $\beta$, and (c) $\gamma$ bands for $\eta =0.5$ and $N=20$. The transitions indicated in the legend are divided by $B(E2;2^{+}\to 0^{+})$ for the ground-state and $\beta$ bands and by $B(E2;4^{+}\to 2^{+})$ for the $\gamma$ band. For each transition, the values predicted by the Alaga rule are indicated by thin lines of corresponding type and color. The spikes at $\chi \approx -1.04$ originate from the $\beta \times \gamma$ crossing (see text).

Figure 9

$B(E2)$ values for transitions between two-phonon $K=0$ and one-phonon bands (states indicated in each panel) obtained by both the intrinsic BMF formalism and exact numerical diagonalization for $\eta =0.5$ and $N=20$ bosons. The BMF calculation takes into account the mixing of two-phonon $K=0$ bands. The mixing of bands with different phonon number $n$ is not included, which leads to significant discrepancies for $\chi >-0.6$.

Figure 10

$B(E2)$ values of transitions from the $l=2$ states belonging to the one-phonon $\beta$ and $\gamma$ bands to the $l=0$ ground state for $N=20$ bosons calculated by both BMF approximation and exact diagonalization. The range of axes is set to allow direct comparison between panels (a) and (b) in the region ${\chi }_{\mathrm{deg}}(\beta ,\gamma )\approx -1.04$. The inset inside panel (a) displays the complete curves using a different scaling of the $y$ axis.

Figure 11

Behavior of the mixing amplitudes $A,B$ of the two-phonon ${\beta }^2$ and ${\gamma }^2$ $K=0$ bands for $\eta =0.5$ as a function of χ for $N=20$ bosons is shown in the inset, while the width of the mixing region ${\Delta }_{\chi }$, decreasing roughly as $O(1/N)$, is in the main panel. The line represents a linear fit yielding ${\Delta }_{\chi }\propto N^{-1.004}$.

Figure 12

Wave function components in the SU(3) basis for $N=30,\eta =0.5$, and $\chi =-1.3,-1.04,-0.7$ in the left, middle, and right columns, respectively. The amplitudes of the BMF states and individual exact states with different $l$ are arranged as indicated in the left-most bottom panel. The SU(3) quasidynamical symmetry is disclosed by approximate equality of the amplitudes among the rotational band members. Evidently, it is present for $\chi =-1.3$, gets disrupted by the degeneracy of $\beta$ and $\gamma$ excitations around ${\chi }_{\mathrm{deg}}(\beta ,\gamma )\approx -1.04$, and is restored (although not perfectly) for $\chi =-0.7$ (especially in $\gamma ,\beta \gamma$, and $A{\beta }^2+B{\gamma }^2$ bands).

Figure 13

Low-energy part of the $\langle {\hat{n}}_d\rangle$ and $\mathrm{var}({\hat{n}}_d)$ spectral lattices for $\eta =0.5$ and $N=30$ bosons (quantities are dimensionless). The numerical eigenstates of Eq. (1) with angular momenta $l=0,2,4,6,8$ (crosses and squares) are supplemented by the BMF approximation (dots).

Figure 14

$\langle {\hat{n}}_d\rangle$ and $\mathrm{var}({\hat{n}}_d)$ lattices of the whole spectrum of numerical eigenstates for $\eta =0.5$ and $N=30$ bosons (quantities are dimensionless). Panels (a)–(d) correspond to $\chi =-1.3$, (e)–(h) to $\chi =-1.04$, and (i)–(l) to $\chi =-0.4$. The panels in the first and third row display the $l=0$ spectrum separately, while those in the second and fourth row show a synoptical view upon the $l=0,2,4$ lattices. The solid vertical lines correspond to $E_{\mathrm{sad}}$ for $N=30$. For comparison, black dots corresponding to a few lowest BMF states (10) are inserted, see Fig. 13.