Classical and quantum properties of the semiregular arc inside the Casten triangle

We investigate classical and quantum signatures of increased regularity in the Alhassid-Whelan semiregular arc inside the symmetry triangle of the interacting boson model. A significant bunching/antibunching pattern of quantum levels, similar to that observed along the O(6)-U(5) transition, is found in the $0^{+}$ energy spectrum and related to a crossover of two specific families of classical regular orbits slightly above $E=0$. We also discuss the degeneracy of β and γ bandheads in a region close to the arc and the relation to regularity in the geometric model.

Figure 1

Sections of potential (4) in the plane $y=0$ for control parameters corresponding to dynamical symmetries (panel a) and a contourplot of the potential for $\eta =0.5$ and $\chi =-0.91$ (panel b). Bohr coordinates β and γ are the radius and polar angle, respectively, in the plane $x\times y$. Energy unit is arbitrary.

Figure 2

Phase space portraits disclosed by $E_{\mathrm{cl}}=0$ Poincaré sections at $\eta =0.5$ and $\chi \in [0,-\sqrt{7}/2]$. Each panel contains $~{10}^4$ passages of 120 trajectories with $E_{\mathrm{cl}}=l=0$ and randomly generated initial condition through the plane defined by setting $y=0$. The crossing of the AW arc can be noticed at $\chi \approx -0.9$.

Figure 3

Regularity of classical IBM for $\eta =0.5$ and $E_{\mathrm{cl}}=0$ as a function of χ determined by two methods: (i) the fraction of regular area in the Poincaré section (solid line) and (ii) the fraction of regular trajectories obtained by the SALI method (dashed line). The peak at $\chi \approx -0.9$ corresponds to AW arc. Oscillations on the dashed curve are caused by a finer step of the SALI evaluation.

Figure 4

Energy dependences of regularity in classical IBM for $\eta =0.5$ at selected values of χ determined by the SALI method. Energy unit is arbitrary.

Figure 5

The evolution of $0^{+}$ energy levels (panel b) and the corresponding dependence of the Brody parameter ω on χ at $\eta =0.5$ (panel a). The minimum of ω at $\chi \approx -1$ is connected with the bunching of $E\in [0,1]$ levels. Energy unit is arbitrary. The boson numbers in each panel differ only to make panel (b) more legible.

Figure 6

Evolutions of $0^{+}$ spectra for $N=40$ [Hamiltonian (1) with $a=4$] along paths shown in Fig. 7. Energy unit is arbitrary. A bunching pattern slightly above $E=0$ is observed in the AW arc (panel $k=3$) and gradually disappears as the path deviates from the arc (panels with $k\ne 3$).

Figure 7

A set of paths (10) “parallel” to the AW arc (6). The paths are labeled by $k=0,1,\dots ,5$, with $k=0$ corresponding to the SU(3)-U(5) edge and $k=3$ representing the arc (solid curve). Only the $\chi ⩽0$ half of the extended Casten triangle is shown.

Figure 8

Washing out of the bunching pattern at ${\chi }_{\mathrm{reg}}(\eta )$ ($k=3$) with increasing angular momentum $l$ for $N=30$. (Energy unit is arbitrary.)

Figure 9

Comparison of the bunching of $0^{+}$ states in O(6)-U(5) transition (upper panel) and along the AW arc (lower panel); both spectra calculated for $N=40$. (Energy unit is arbitrary.)

Figure 10

Poincaré sections $y=0$ for $\eta =0.5$ and ${\chi }_k,k=2,3,4$ (columns) at energies $E_{\mathrm{cl}}=E_1,\dots ,E_{10}$ (rows). The middle column corresponds to the AW arc. Notice that sections for $E_{\mathrm{cl}}<E_{\mathrm{sad}}$ are expanded compared to those for $E_{\mathrm{cl}}⩾E_{\mathrm{sad}}$.

Figure 11

Poincaré sections and the central periodic orbits corresponding to the principal regular islands in the vicinity of the AW arc. In both panels $\eta =0.5$. Panel (a) corresponds to $E_{\mathrm{cl}}=E_4<E_{\mathrm{sad}},\chi ={\chi }_2$, panel (b) to $E_{\mathrm{cl}}=E_7>0,\chi ={\chi }_3$ (cf. the respective sections of Fig. 10). The dashed lines at $y=0$ demarcate the plane of section and the arrows show the position of individual orbits in the Poincaré sections.

Figure 12

The AW arc [Eq. (6), the dashed curve] and the locus of degeneracy of β and γ bandheads [Sec. 6, the dotted curve] inside the Casten triangle. The points mark the change of stability of γ-vibrations (orbits of type 1 in Fig. 11) at $E_{\mathrm{cl}}=-\left|V_{\min }\right|/2$ (crosses) and at $E_{\mathrm{cl}}=0$ (asterisks), see Subsec. 5b. Numerical errorbars are smaller than the pointsize.

Figure 13

Oscillating contributions to the state density calculated for the central periodic orbits from Fig. 11 at $\chi ={\chi }_{\mathrm{reg}}$ and $\eta =0.35$ (a), 0.5 (b), 0.65 (c). Maslow index ${\mu }_p=4$ for all orbits makes no change of the cosine. Grey zones indicate the regions corresponding to the bunching of quantum levels. (Energy unit is arbitrary.)

Figure 14

Regularity $f_{\mathrm{reg}}^{(S)}$ at $E_{\mathrm{cl}}=0$ as a function of $\overline{B}=B/\sqrt{\left|\mathit{AC}\right|}$ and $\overline{B}(\eta ,\chi )$ in the GCM and IBM, respectively, determined by the SALI method.

Figure 15

Comparison of $E_{\mathrm{cl}}=0$ Poincaré sections $y=0$ in the IBM at $\eta =0.75$ (lefthand column) and GCM (righthand column) at the most pronounced minima and maxima of $f_{\mathrm{reg}}(\overline{B})$ from Fig. 14.