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Progress of Theoretical Physics

Breking of Separability Condition for Dynamical Collective Subspace: Onset of Quantum Chaos in Large-Amplitude Collective Motion

Fumihiko Sakata, Y. Yamamoto, Toshio Marumori, Shinji Iida, Hidehiko Tsukuma

Published Nov 1, 1989• 6 citations
Publisher

Abstract

It is the purpose of the present paper to study “global structure” of the state space of an N-body interacting fermion system, which exhibits regular, trasientand stochastic phasesdepending on strength of the interaction. An optimum representation called a dynamical representationplays an essential role in this investigation. The concept of the dynamical representation has been introduced in the quantum theory of dynamical subspacein our previous paper, in order to determine self-consitently an optimum cokkective subspace as well as an optimum collective Hamiltoninan. In the theory, furthermore, dynamical conditions called separabilityand stability conditionshas been provided in order to identify the optimum collective subspace as an approximate invariant subspaceof the Hamiltonian. Physical meaning of these conditions are clarified from a viewpoint to relate breakingof them with bifurcationof the collectivity and an onset of quantum chaosfrom the regularcollective motion, by illustrating the general idea with numerical results obtained for a simple soluble model. It turns out that the onset of the stochastic phase is associated with dissolution of the quantum numbers to specify the collective subspace and this dissolution is induced by the breaking of the separability condition in the dynamical representation.

DOI
10.1143/ptp.82.965

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It is the purpose of the present paper to study “global structure” of the state space of an N-body interacting fermion system, which exhibits regular, trasientand stochastic phasesdepending on strength of the interaction. An optimum representation called a dynamical representationplays an essential role in this investigation. The concept of the dynamical representation has been introduced in the quantum theory of dynamical subspacein our previous paper, in order to determine self-consitently an optimum cokkective subspace as well as an optimum collective Hamiltoninan. In the theory, furthermore, dynamical conditions called separabilityand stability conditionshas been provided in order to identify the optimum collective subspace as an approximate invariant subspaceof the Hamiltonian. Physical meaning of these conditions are clarified from a viewpoint to relate breakingof them with bifurcationof the collectivity and an onset of quantum chaosfrom the regularcollective motion, by illustrating the general idea with numerical results obtained for a simple soluble model. It turns out that the onset of the stochastic phase is associated with dissolution of the quantum numbers to specify the collective subspace and this dissolution is induced by the breaking of the separability condition in the dynamical representation.
It is the purpose of the present paper to illustrate the quantum theory of dynamical collective subspace, which has been proposed in a previous paper to describe the large-amplitude collective motion, by using a simple four-level model with the pairing interactions. The quantum theory provides us a prescription to dynamically specify a set of "global" collective operators as well as the "optimum" dynamical collective subspace spanned by them. This prescription is applied to the simple model, and the results thus obtained are compared with exact numerical solutions. It is shown that the results by our theory are in excellent agreement with the exact solutions, not only in the low-lying collective states in the vicinity of the "vacuum" with the stable mean-field but also in the higher excited collective states far from it. Through the investigation, it is demonstrated that our theory is quite powerful in disclosing microscopic structure of transfiguration of collectivity near the critical point of the "phase transition". It is the essential statement derived by our theory that the conventional (local) collective operators based on the Tamm-Dancoff approximation or the random phase approximation should be replaced by relevant (global) collective operators with changeable microscopic structure depending on the amplitude of collective motion, in order to properly specify a set of collective excited states.
By placing emphasis on conceptual correspondence to the “classical” theory which has been developed within the framework of the time-dependent Hartree-Fock theory, a full quantum theory appropriate for describing large-amplitude collective motion is proposed. A central problem of the quantum theory is how to determine an optimal representation called a dynamical representation; the representation is specific for the collective subspace where the large-amplitue collective motion is replicated as satisfactoryly as possible. As an extension of the classical theory where the concept of an approximate integral surface plays an important role, the dynamical representation is properly characterized by introducing a concept of an approximate invariant subspace of the Hamiltonian.
A systematic microscopic theory of large-amplitude collective motion, which has been recently developed within the framework of the time-dependent Hartree-Fock (TDHF) theory under INS-TSUKUBA joint research project, is reformulated in terms of classical mechanics in the TDHF symplectic manifold. This reformulation clarifies an essential concept in the theory, i.e., an approximate integral surface in the TDHF manifold, which is identified as a dynamical collective submanifold for the large-amplitude collective motion. It is shown that the approximate integral surface must satisfy three dynamical conditions; (a) a stationary condition (i.e., a maximal-decoupling condition), (b) a stability condition and (c) a separability condition between collective and “local” non-collective modes of motion. These three dynamical conditions make clear the concept of “local” intrinsic modes of motion and guarantee that the TDHF-trajectory representing the large-amplitude collective motion under consideration is successfully reproduced on the approximate integral surface.
By applying the microscopic theory of dynamical collective submanifold, which has been formulated by the present authors, to a system with a simple model Hamiltonian, characteristic features of large-amplitude motion described by the time-dependent Hartree-Fock (TDHF) theory are visualized. Our theory dynamically defines an “optimum” submanifold out of the TDHF symplectic manifold (phase space) in such a way that the Hamiltonian on the submanifold is stationary with respect to variations perpendicular to it. The theory also classifies the submanifold into three different regions in terms of stablility and separability conditions. It is displayed that these three regions characterize collective, dissipative and stochastic motions in the TDHF theory.
This is the third in a series of papers which intends to develop a new microscopic theory capable by itself to determine the “maximaldecoupled” collective subspace in the many-particle Hilbert space as well as the collective Hamiltonian specifying the subspace. The fundamental principle to formulate the theory is called the invariance principle of the Schrödinger equation. The main purpose of this paper is to show that, when we positively utilize the invariance principle, we can directly obtain a quantum theory of collective motion going beyond the random phase approximation.
We study the quantum-classical correspondence for systems with interacting spin particles that are strongly chaotic in the classical limit. This is done in the presence of constants of motion associated with the fixed angular momenta of individual spins. Our analysis of the Lyapunov spectra reveals that the largest Lyapunov exponent agrees with the Lyapunov exponent that determines the local instability of each individual spin moving under the influence of all other spins. Within this picture, we introduce a rigorous and simple test of ergodicity for the spin motion, and use it to identify when classical chaos is both strong and global in phase space. In the quantum domain, our analysis of the Hamiltonian matrix in a proper representation allows us to obtain the conditions for the onset of quantum chaos as a function of the model parameters. From the comparison between the quantum and classical domains, we demonstrate that quantum quantities, such as the local density of states (LDoS) and the shape of the chaotic eigenfunctions written in the noninteracting many-body basis, have well-defined classical counterparts. We also find a relationship between the Kolmogorov-Sinai entropy and the width of the LDoS, which is useful for studies of many-body dynamics.