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arXiv:nucl-th/9712008

An efficient projection algorithm and its application to the spurious center-of-mass motion problem

Nakada, H., Takashi Sebe, Tomoki Otsuka

Published Dec 2, 1997• 0 citations
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Abstract

Based on the correspondence of projection to an eigenvalue problem via the underlying group structure, we generalize the projection algorithm proposed by Morrison et al. This algorithm is eligible for large-scale computations, because we can avoid the accumulation of errors. As an example, we apply it to the spurious c.m. motion in a multi-$\hbarω$ shell-model calculation. In comparison with Lawson's method, the present method is exact and nevertheless does not need longer CPU time. It is pointed out that the exact treatment of the spurious c.m. motion is crucial to some physical quantities.

DOI
10.48550/arxiv.nucl-th/9712008

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Based on the correspondence of projection to an eigenvalue problem via the underlying group structure, we generalize the projection algorithm proposed by Morrison et al. This algorithm is eligible for large-scale computations, because we can avoid the accumulation of errors. As an example, we apply it to the spurious c.m. motion in a multi-ℏω\hbarωℏω shell-model calculation. In comparison with Lawson's method, the present method is exact and nevertheless does not need longer CPU time. It is pointed out that the exact treatment of the spurious c.m. motion is crucial to some physical quantities.
A new approximate projection method is proposed to treat the spurious state problem in the Dyson boson description of nuclear collective motion. In this method, successive orders of approximate projection are used to construct the various physical states in the Dyson boson representation of the shell model. Even in the lowest orders of approximation, the approximately projected boson states obtained turn out to be quite accurate in producing physical quantities. Furthermore, this approximate projection method is applicable not only to vibrational or transitional nuclei but also to rotational nuclei.
An explicit construction is found for a unitary operator which insures the free motion of the center of mass of any many-particle wave function on which it is allowed to act. The transformation is used to calculate recoil correction terms for the internal energy and external interactions of nuclei, and some numerical evaluations are given for cases of interest. The many-body harmonic-oscillator problem is exactly soluble when one uses the transformation, and one is thus enabled to give a more general discussion of the spurious states.
A theoretical analysis is given of the equation of motion method, due to Alben et al. [Phys. Rev. B 12, 4090 (1975)], to compute the eigenvalue distribution (density of states) of very large matrices. The salient feature of this method is that for matrices of the kind encountered in quantum physics the memory and CPU requirements of this method scale linearly with the dimension of the matrix. We derive a rigorous estimate of the statistical error, supporting earlier observations that the computational efficiency of this approach increases with the matrix size. We use this method and an imaginary-time version of it to compute the energy and specific heat of three different, exactly solvable, spin-1/2 models, and compare with the exact results to study the dependence of the statistical errors on sample and matrix size.
It is shown that one can find an orthogonal transformation that will enable one to split the motion of a many-body system to a center-of-mass motion and an internal motion. A particular orthogonal transformation has been chosen which retains the independent-particle aspect of a harmonic-oscillator shell-model Hamiltonian. It is suggested that one could easily study and eliminate the states with spurious motions of the center of mass by a direct transformation of the shell-model wave function into the new coordinate system.
Two methods to calculate the density matrix of one• body or more•than•one•body which does not include the spurious center•of•mass excitation are presented.One is the method to calculate the density matrix for the many-body wave function of general type and the other is applicable to the case in which the center~of-mass motion separates from the internal motion and is in the (Os) orbit of harmonic oscillator.The applications of these methods are illustrated for some many-body wave function.§ 1. IntroductionTo represent the characteristic features of the many-body wave function of nuclei, the density matrices are often utilized.Experimentally, for example, in the electron scattering from nuclei, the nuclear structure is analyzed by the terms of the nuclear density and the current density, which are the special forms of the one-body density matrix.Theoretically, the one-body density matrix plays a crucial role in the current nuclear theories based on the mean field approximation, such as Hartree-Fock theory.And, in recent nuclear studies, interest is taken in not only the nuclear density which is the diagonal part of the one-body density matrix, but the non-local part of it.The non-local part of the one-body density matrix is often visualized with the use of the Wigner transform of it. 1),2)Especially, in the time-dependent Hartree-Fock theory, the Wigner transform of the one-body density matrix is interpreted as the classical distribution function and obeys the Vlasov equation approximately.3)These works clarify the importance of the non-local part of the one-body density matrix in the dynamics.Usually the spatial coordinate of the density matrix is measured from some spatially fixed origin.Due to the finiteness of the nuclear system, however, we need to measure the coordinate from the center of mass of the system; otherwise the spurious center-of-mass motion affects the density matrix.This effect of center-of-mass motion becomes important for light systems.For the diagonal part of the one-body transition density matrix, the effect of center-of-mass motion was discussed by Tassie and Barker for the case of harmonic oscillator shell mode1. 4) And for the diagonal part of the one-body density matrix in momentum representation, that is the nuclear momentum distribution, the effect of spurious center-of-mass motion has been discussed. 5) For the general non-local density matrix, however, there seems not to be presented any method to treat the spurious center-of-mass effect.In this paper, we present two methods to get the center-of-mass free density matrix.One is the method, applicable to the general wave function, which is especially suitable for the case where the wave function is a (linear superposition of) Slater determinant.The other is applicable to the cases in which the center-of-mass motion separates from other internal motions and is in the (Os) orbit of harmonic oscillator.For such wave functions, the latter method is more easy to calculate though the former method is also applicable.
The problem of construction of projection operators on eigensubspaces of symmetry operators is considered. This problem arises in many approximate methods for solving time-independent and time-dependent quantum problems, and its solution ensures proper physical symmetries in development of approximate methods. The projector form is sought as a function of symmetry operators and their eigenvalues characterizing the eigensubspace of interest. This form is obtained in two steps: (1) identification of algebraic structures within a set of symmetry operators (e.g., groups and Lie algebras) and (2) construction of the projection operators for individual symmetry operators. The first step is crucial for efficient projection operator construction because it allows for use of information on irreducible representations of the present algebraic structure. The discussed approaches have potential to stimulate further developments of variational approaches for electronic structure of strongly correlated systems and in quantum computing.