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Progress of Theoretical Physics

A Schematic Model of Large Amplitude Collective Motions with an Exact Classical Solution. I: Classical Theory of Two-Level Pairing Hamiltonian

Shinji Iida, M. Yamamura

Published Sep 1, 1983• 2 citations
Publisher

Abstract

An exact solution of two-level pairing Hamiltonian is analytically given in the framework of the Hamiltonian form of classical mechanics. Following the shapes of the trajectories, whole phase space for pairing vibration can be divided into three sub-spaces. The solution, which is expressed in terms of the elliptic function, shows discontinuities at the boundaries of the three sub-spaces.

DOI
10.1143/ptp.70.790

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An exact solution of two-level pairing Hamiltonian is analytically given in the framework of the Hamiltonian form of classical mechanics. Following the shapes of the trajectories, whole phase space for pairing vibration can be divided into three sub-spaces. The solution, which is expressed in terms of the elliptic function, shows discontinuities at the boundaries of the three sub-spaces.
It is investigated how classical pictures of the whole phase space are connected with the original quantal system by using the two-level pairing model.Discontinuous shape changes of classical trajectories are shown to be related with the sudden changes of patterns in the transition strength between energy eigenstates.By introducing the collective and intrinsic coordinates representing pairing rotations and vibrations, respectively, these discontinuities can be interpreted as the breakdown of the excited pairing •rotational band scheme.
A model Hamiltonian describing a two-level system with a crossing plus a pairing force is investigated using technique of large-amplitude collective motion. The collective path, which is determined by the decoupling conditions, is found to be almost identical to the one in the Born-Oppenheimer approximation for the case of a strong pairing force. For the weak pairing case, the obtained path describes a diabatic dynamics of the system. PACS number(s): 21.60.-n, 36.40.Sx Nuclei and metallic clusters are finite fermionic many-body systems which support many kinds of collective motion. While properties of high-frequency vibrations (giant resonances) may be well reproduced by the random-phase approximation (RPA), some low-frequency vibrations exhibit an anharmonic nature that the RPA cannot describe. Shape coexistence and fission phenomena in nuclei and clusters also have such a large amplitude nature. In order to investigate these kinds of large amplitude collective motion, one would like to reduce the number of degrees of freedom to a few judiciously chosen collective coordinates. In order to determine the collective coordinates self-consistently, we use the theory of adiabatic
Large amplitude collective motion is investigated for a model pairing Hamiltonian containing an avoided level crossing. A classical theory of collective motion for the adiabatic limit is applied utilizing either a time-dependent mean-field theory or a direct parametrization of the time-dependent Schr\"odinger equation. A modified local harmonic equation is formulated to take account of the Nambu-Goldstone mode. It turns out that in some cases the system selects a diabatic path. Requantizing the collective Hamiltonian, a reasonable agreement with an exact calculation for the low-lying levels is obtained for both weak and strong pairing force. This improves on results of the conventional Born-Oppenheimer approximation.
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.