Classical chaos in the geometric collective model

Abstract

Collective vibrations and rotations of atomic nuclei are investigated from the classical viewpoint within the geometric collective model. It is shown that the model, even in its truncated form, exhibits a very sophisticated interplay of regular and chaotic modes of motions. We quantify the proportion of regular and chaotic orbits in the phase space and analyze its sensitive dependence on the model control parameters, energy, and angular momentum. A quasi-regular region is observed at low excitation energies in a bounded domain of the control parameter on the deformed side of the shape phase diagram, and another one at higher energies around the transition between spherical and deformed shapes. We also demonstrate a tendency for overall suppression or enhancement of chaos with angular momentum, depending on the values of control parameters.

Excited-state quantum phase transitions in systems with two degrees of freedom: II. Finite-size effects

“…4 This can be achieved by substitutions x  → x ′ = x/ √ R 0 and y  → y ′ = √ R 0 y performed in Eq. (17), which leads to:…”

“…(17). Before showing results of our numerical analysis (Section 4.2), we make some general remarks on the state density in separable systems (Section 4.1).…”

“…Eq. (22) can be easily used to calculate the smooth componentρ λ (E) of the eigenstate density of the separable CW Hamiltonian (17). We know that the smooth component of the harmonic-oscillator state density is constant, independent of energy:ρ y = const.…”

“…A separable Hamiltonian of this form is realized within the family of Creagh-Whelan potentials, see Eq. (17). The componentĤ y represents just a harmonic oscillator with spectrum given by E y (n y ) =hΩ y (n y + 1/2), where Ω y = √ 2D and n y = 0, 1, 2, .…”

“…An efficient heuristic (hence imperfect) criterion to estimate the ''critical'' energy E ch , where first chaotic orbits occur, was discussed in Ref. [17]. For system with Hamiltonian…”