Power-law decay of the fraction of the mixed eigenstates in kicked top model with mixed-type classical phase space

Chaos and quantization of the three-particle generic Fermi-Pasta-Ulam-Tsingou model. II. Phenomenology of quantum eigenstates

We undertake a thorough investigation into the phenomenology of quantum eigenstates, in the three-particle Fermi-Pasta-Ulam-Tsingou model. Employing different Husimi functions, our study focuses on both the α-type, which is canonically equivalent to the celebrated Hénon-Heiles Hamiltonian, a nonintegrable and mixed-type system, and the general case at the saddle energy where the system is fully chaotic. Based on Husimi quantum surface of sections, we find that in the mixed-type system, the fraction of mixed eigenstates in an energy shell [E-δE/2,E+δE/2] with δE≪E shows a power-law decay with respect to the decreasing Planck constant ℏ. Defining the localization measures in terms of the Rényi-Wehrl entropy, in both the mixed-type and fully chaotic systems, we find a better fit with the β distribution and a lesser degree of localization, in the distribution of localization measures of chaotic eigenstates, as the controlling ratio α_{L}=t_{H}/t_{T} between the Heisenberg time t_{H} and the classical transport time t_{T} increases. This transition with respect to α_{L} and the power-law decay of the mixed states, together provide supporting evidence for the principle of uniform semiclassical condensation in the semiclassical limit. Moreover, we find that in the general case which is fully chaotic, the maximally localized state, is influenced by the stable and unstable manifold of the saddles (hyperbolic fixed points), while the maximally extended state notably avoids these points, extending across the remaining space, complementing each other.

Mixed eigenstates in the Dicke model: Statistics and power-law decay of the relative proportion in the semiclassical limit

How the mixed eigenstates vary when approaching the semiclassical limit in mixed-type many-body quantum systems is an interesting but still less known question. Here, we address this question in the Dicke model, a celebrated many-body model that has a well defined semiclassical limit and undergoes a transition to chaos in both quantum and classical cases. Using the Husimi function, we show that the eigenstates of the Dicke model with mixed-type classical phase space can be classified into different types. To quantitatively characterize the types of eigenstates, we study the phase space overlap index, which is defined in terms of the Husimi function. We look at the probability distribution of the phase space overlap index and investigate how it changes with increasing system size, that is, when approaching the semiclassical limit. We show that increasing the system size gives rise to a power-law decay in the behavior of the relative proportion of mixed eigenstates. Our findings shed more light on the properties of eigenstates in mixed-type many-body systems and suggest that the principle of uniform semiclassical condensation of Husimi functions should also be valid for many-body quantum systems.