Quantum chaos in the nuclear collective model. II. Peres lattices

This is a continuation of our paper [Phys. Rev. E 79, 046202 (2009)] devoted to signatures of quantum chaos in the geometric collective model of atomic nuclei. We apply the method by Peres to study ordered and disordered patterns in quantum spectra drawn as lattices in the plane of energy vs average of a chosen observable. Good qualitative agreement with standard measures of chaos is manifested. The method provides an efficient tool for studying structural changes in eigenstates across quantum spectra of general systems.

Figure 1

Peres lattices for $J=0$ eigenstates of the geometric collective model in the “2D even” quantization (see Sec. 3). The points represent individual eigenstates with coordinates $E_i$ (energy) and ${⟨P⟩}_i$ (expectation value of the respective Peres operator). Two Peres operators, $L^2$ and $H^{\prime }$, are employed, the results shown in the left and right columns, respectively. Row (a) corresponds to the fully integrable case, $B=0$. Rows (b)–(d) depict the disturbance of the lattice with gradually increasing nonintegrable perturbation until reaching the most chaotic case, $B=0.24$ (e). All quantities are given in relative units, $\hslash =5\times {10}^{-3}$.

Figure 2

Peres lattices of the GCM at $B=0.62$ with averages of $L^2$ (left) and $H^{\prime }$ (right) for 2D even (row a) and 5D (row b) quantizations $\left(\hslash =5\times {10}^{-3}\right)$. In row (a), three states denoted by full symbols are identified in both lattices, demonstrating that the assignment of a given state to a regular or chaotic part of the lattice does not depend on the choice of the Peres operator.

Figure 3

Squared wave functions for the three marked states from Fig. 2 (2D even quantization). The states (a) and (c), which correspond to the square and the diamond, respectively, are taken from the regular part of the lattice (the 1995th and 1885th level, respectively) and exhibit well pronounced quantum scar effects. The state (b), associated with the bullet, belongs to the chaotic part (the 1890th level) and shows an ergodic behavior.

Figure 4

(Color online) The GCM Peres lattices for $B=1.09$ and different values of the Planck constant: $\hslash =0.02$ (a), 0.05 (b), and 0.1 (c). The 2D even quantization was employed with the same Peres operators as in the previous figures. Lower values of $\kappa$ yield denser spectra, which can therefore be evaluated only in narrower energy intervals (numerical limitations). The box (red online) encloses the same area in the three panels of each column. Insets show the dependence of the classical regular fraction $f_{\text{reg}}$ on energy (see Part I [1]) for the energy domain displayed in each row. One can observe that the $f_{\text{reg}}$ dependences are correlated with ordered and disordered areas in the corresponding lattices in both columns.

Figure 5

The $y=0$ section of the classical phase space (coordinate $x$ versus momentum $p_x$) for $B=0.62$ and $E=0.2$. Left: Poincaré section formed by crossings of ${10}^2$ randomly chosen trajectories with the plane of the section (${10}^3$ crossings for each trajectory). Right: Density plot of the Peres invariant $⟨L^2⟩$ calculated classically across the section (on a mesh of $500\times 500$ points). Dark and light regions (low and high values of the Peres invariant, respectively) contain regular trajectories (see the left panel) and simultaneously correspond to regular domains in the Peres lattice [see Fig. 2a at $E\approx 0.2$, where the regular domains are located at the lower and upper sides of the lattice].

Figure 6

Squared wave functions of four successive eigenstates (states no. $i=52,\dots ,55$ at $E$ from 0.059 to 0.076) of the integrable $B=0$ Hamiltonian (for $\hslash =0.02$) in the 2D even quantization (the first row) and the same states for $B=0.005$, 0.01, and 0.05 (the second, third, and fourth rows, respectively). The respective values of both P-averages are given in each panel. The first three states (arranged in columns) are more sensitive to the perturbation than the fourth one, in accord with the lattices in Fig. 1 (see the text).

Figure 7

The lower and higher bounds (solid and dashed lines, respectively) of the classical Peres average ${⟨L^2⟩}_{\text{c}}$ for $E=0$. The bounds almost touch each other at $B=0.24$, where the most chaotic region is located. Fluctuations of the lower bound near $E=0.5$ are caused by the appearance and disappearance of small unstable tori.

Figure 8

Peres lattice for $⟨L^2⟩$ in 2D even quantization $\left(\hslash =5\times {10}^{-3}\right)$ at $B=0.52$ (a) and 0.78 (b). We see that the regular pattern is present before and after the main peak of regularity at $B=0.62$, cf. Figure 2a. Both panels contain the same number of states. The insets depict selected wave functions (diamonds in the respective lattices) demonstrating the presence of $\beta$ and $\gamma$ vibrations (only the sector around the minimum $\gamma =4\pi /3$ is shown in both cases). The state in the upper panel $\left(i=17\right)$ is at $E=-0.449$, the lower one $\left(i=1292\right)$ at $E=0.021$.