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Physical Review C

Comment on ‘‘Algebraic analysis of physical and spurious states in Dyson boson mapping’’

F.J.W. Hahne, H. B. Geyer, C. A. Engelbrecht

Published Feb 1, 1988• 9 citations
Publisher

Abstract

By stressing the application of boson mapping techniques to the truncated collective branch of nuclear excitations, the usefulness of two recently developed methods for the treatment of spurious states are compared. It is shown why the identification proposed by Geyer et al. is superior to the use of the ``Majorana'' operator S as proposed by Park.

DOI
10.1103/physrevc.37.885

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By stressing the application of boson mapping techniques to the truncated collective branch of nuclear excitations, the usefulness of two recently developed methods for the treatment of spurious states are compared. It is shown why the identification proposed by Geyer et al. is superior to the use of the ``Majorana'' operator S as proposed by Park.
Implementation of Dyson boson mapping is discussed in connection with effective Hamiltonians. A feature of the mapping technique, when implemented in an ideal boson basis, is the possible appearance of spurious states. These spurious states typically signal the overcompleteness of the basis. Without truncation, no contamination of the physical states and spectrum takes place. However, in practice one may be required to select from the ideal boson basis the dominant components for a given interaction. It is shown that the correspondence between a perturbative expansion, aˊla\'a\mathit{la}aˊla Bloch-Horowitz, and Dyson boson mapping allows for the identification of spurious states. The proposed method is applied to the mapping of a bi-fermionic Hamiltonian.
Dyson boson mapping of a system of 2n fermions distributed among k single-particle states uses a space generated by two-index ideal bosons. We define a unitary group U(k) to classify this ideal-boson space. The physical subspace is shown to correspond to the antisymmetric irreducible representation [12n{1}^{2n}12n]. This classification enables one to introduce a ``Majorana'' operator S which is a linear combination of one- and two-body Casimir operators of U(k). A zero eigenvalue of S characterizes the physical subspace while positive eigenvalues identify all other irreducible representations that occur. The Hermitian boson image of the Hamiltonian does not connect physical and spurious states because it is a function of the generators of U(k). All the nonphysical eigenvalues and eigenstates of a physical boson Hamiltonian (Dyson boson image of a physical fermion Hamiltonian) which could be Hermitian or non-Hermitian, can be removed without affecting the physical eigenvalues and eigenvectors, by adding a suitable multiple of S to the Hamiltonian.
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.
A comparison between the finite Dyson-type and the infinite Holstein-Primakoff-type boson expansion is presented, considering both particle-hole and pairing type excitations. The significance of truncating to the truly collective branches is emphasized, and the convergence of the Holstein-Primakoff expansion is discussed. It is shown that in both expansions there is no need to construct physical basis states. Complete equivalence between the two types of expansion is exhibited by treating these excitations within models which are schematic but not totally unrealistic.NUCLEAR STRUCTURE Boson expansion theory, convergence, comparison of different methods, finite Dyson expansion, infinite Holstein-Primakoff expansion, pairing and particle-hole collective motions.
The skeleton boson realizations of fermion collective subalgebras proposed by Kim and Vincent are discussed. The Dyson boson mapping is found to play an essential role in the possibility of such realizations. A simple but instructive example is given to show that a straightforward application of the ``skeletonization procedure'' to the Holstein-Primakoff boson mapping may lead to serious errors in reproducing the fermion matrix elements. A possible way of constructing the modified skeleton Holstein-Primakoff realization of the collective subalgebra is suggested.