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Progress of Theoretical Physics

Geometry of the Self-Consistent Collective-Coordinate Method for the Large-Amplitude Collective Motion: Stability Condition of ";Maximally-Decoupled" Collective Submanifold

Fumihiko Sakata, Toshio Marumori, Yukio Hashimoto, Tsutomu Une

Published Aug 1, 1983• 20 citations
Publisher

Abstract

The geometry of the self-consistent collective-coordinate (SCC) method formulated within the frame-work of the time-dependent Hartree-Fock (TDHF) theory is investigated by associating the variational parameters with a symplectic manifold (a TDHF manifold). With the use of a canonical-variable parametrization, it is shown that the TDHF equation is equivalent to the canonical equations of motion in classical mechanics in the TDHF manifold. This enables us to investigate geometrical structure of the SCC method in the language of the classical mechanics. The SCC method turns out to give a prescription how to dynamically extract a “maximally-decoupled” collective submanifold (hypersurface) out of the TDHF manifold, in such a way that a certain kind of trajectories corresponding to the large-amplitude collective motion under consideration can be reproduced on the hypersurface as precisely as possible. The stability of the hypersurface at each point on it is investigated, in order to see whether the hypersurface obtained by the SCC method is really an approximate integral surface in the TDHF manifold or not.

DOI
10.1143/ptp.70.424

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arXiv:0901.3473•Symmetry Integrability and Geometry Methods and Applications•Jan 22, 2009•0 citations
The geometry of the self-consistent collective-coordinate (SCC) method formulated within the frame-work of the time-dependent Hartree-Fock (TDHF) theory is investigated by associating the variational parameters with a symplectic manifold (a TDHF manifold). With the use of a canonical-variable parametrization, it is shown that the TDHF equation is equivalent to the canonical equations of motion in classical mechanics in the TDHF manifold. This enables us to investigate geometrical structure of the SCC method in the language of the classical mechanics. The SCC method turns out to give a prescription how to dynamically extract a “maximally-decoupled” collective submanifold (hypersurface) out of the TDHF manifold, in such a way that a certain kind of trajectories corresponding to the large-amplitude collective motion under consideration can be reproduced on the hypersurface as precisely as possible. The stability of the hypersurface at each point on it is investigated, in order to see whether the hypersurface obtained by the SCC method is really an approximate integral surface in the TDHF manifold or not.
The applicability of the concept of “optimal” collective submanifold determined by the self-consistent collective-coordinate (SCC) method is investigated by using a simple model Hamiltonian. Long-time behavior of the SCC-method trajectories on the optimal collective submanifold is analyzed, by comparing the geometrical structure of the SCC-method trajectories with that of the time-dependent Hartree-Fock (TDHF) trajectories in terms of the Poincaré-mapping method. The concept of the optimal collective submanifold determined by the SCC method turns out to be applicable for a fairly large domain of the TDHF manifold where the separatrix is not included, in the sense that it can represent an approximate time-averaged trajectory of the complicated TDHF trajectories.
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.
With the purpose of clarifying characteristic difference of the optimum collective submanifolds between in nonresonant and resonant cases, we propose an improved method of solving the basic equations of the self-consistent collective-coordinate (SCC) method, which describes optimum (“maximally-decoupled”) large-amplitude collective motion within the time-dependent Hartree-Fock theory. It is shown that, in the resonant cases, there inevitably arise essential coupling terms which break the maximal-decoupling property of the collective motion, so that we have to extend the optimum collective submanifold so as to properly treat the degrees of freedom bringing about the resonances. An illustrative example is given with a simple model Hamiltonian.
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.
The integrability-condition method is regarded as a mathematical tool to describe the symmetry of collective sub-manifold. We here adopt the particle-hole representation. In the conventional time-dependent (TD) self-consistent field (SCF) theory, we take the one-form linearly composed of the TD SCF Hamiltonian and the infinitesimal generator induced by the collective-variable differential of canonical transformation on a group. Standing on the differential geometrical viewpoint, we introduce a Lagrange-like manner familiar to fluid dynamics to describe collective coordinate systems. We construct a geometric equation, noticing the structure of coset space SU(m+n)/S(U(m) x U(n)). To develop a perturbative method with the use of the collective variables, we aim at constructing a new fermion SCF theory, i.e., renewal of TD Hartree-Fock (TDHF) theory by using the canonicity condition under the existence of invariant subspace in the whole HF space. This is due to a natural consequence of the maximally decoupled theory because there exists an invariant subspace, if the invariance principle of Schredinger equation is realized. The integrability condition of the TDHF equation determining a collective sub-manifold is studied, standing again on the differential geometric viewpoint. A geometric equation works well over a wide range of physics beyond the random phase approximation.
The maximally-decoupled method has been considered as a theory to apply an basic idea of an integrability condition to certain multiple parametrized symmetries.The method is regarded as a mathematical tool to describe a symmetry of a collective submanifold in which a canonicity condition makes the collective variables to be an orthogonal coordinate-system.For this aim we adopt a concept of curvature unfamiliar in the conventional time-dependent (TD) self-consistent field (SCF) theory.Our basic idea lies in the introduction of a sort of Lagrange manner familiar to fluid dynamics to describe a collective coordinate-system.This manner enables us to take a one-form which is linearly composed of a TD SCF Hamiltonian and infinitesimal generators induced by collective variable differentials of a canonical transformation on a group.The integrability condition of the system read the curvature C = 0. Our method is constructed manifesting itself the structure of the group under consideration.To go beyond the maximaly-decoupled method, we have aimed to construct an SCF theory, i.e., υ (external parameter)-dependent Hartree-Fock (HF) theory.Toward such an ultimate goal, the υ-HF theory has been reconstructed on an affine Kac-Moody algebra along the soliton theory, using infinite-dimensional fermion.An infinite-dimensional fermion operator is introduced through a Laurent expansion of finite-dimensional fermion operators with respect to degrees of freedom of the fermions related to a υ-dependent potential with a Υ-periodicity.A bilinear equation for the υ-HF theory has been transcribed onto the corresponding τ -function using the regular representation for the group and the Schur-polynomials.The υ-HF SCF theory on an infinite-dimensional Fock space F ∞ leads to a dynamics on an infinite-dimensional Grassmannian Gr ∞ and may describe more precisely such a dynamics on the group manifold.A finite-dimensional Grassmannian is identified with a Gr ∞ which is affiliated with the group manifold obtained by reducting gl(∞) to sl(N ) and su(N ).As an illustration we will study an infinite-dimensional matrix model extended from the finite-dimensional su(2) Lipkin-Meshkov-Glick model which is a famous exactly-solvable model.