Adequacy of the boson basis and the identification of spurious states

Abstract

It is shown that it is not necessary to resort to an explicit mapping of the fermion basis into the boson space when bifermion excitations are treated in a boson picture. One may use the usual ideal boson basis in the boson space, Under certain circumstances this may lead to the occurrence of spurious states. However, these do not mix with the physical states and they can be identified by means of a simple procedure which is developed within the framework of the Dyson boson method. The results are illustrated by means of three interesting cases in the Sp(4) model.

Boson analyses in the Ge isotopes

“…Although the above constraints lead to the correct enumeration of ideal boson basis states, one problem remains, namely that, as discussed by Geyer et al [6], the boson basis is generally overcomplete in as much as it may contain (independent) boson states with fermion counterpart states which are linearly dependent.…”

“…[6], it follows from the invariance of the physical subspace of the ideal boson basis under the action of arbitrary physical operators that the mapped operator O~has the property that, for any spurious braeigenstate (/spur~o f H~, (4'spur~eB~Wphys) = 0 (3.5) while (@phys / eB / 4'spur) 3 (3.6) where~/ phys) is any nonspurious, i.e. , physical state.…”

“…Although physical states and energies emerge unharmed from the diagonalization [6], it may lead to spurious states and, since some of these states may occur in the lower part of the spectrum, they must be properly identified and eliminated. +X, ) St(i)st'('), (3.4) with A", A", and Af arbitrary.…”

“…The operators appearing in the extended pairing model Hamiltonian (3.1), are generators of SO(5)SO (5) (enumerated by the single-particle levels) [6]. Although this allows an analysis with methods of group theory, one still encounters the standard diKculty of de6ning state labels in addition to those which siinply follow from the chain of subalgebras.…”

“…Although this allows an analysis with methods of group theory, one still encounters the standard diKculty of de6ning state labels in addition to those which siinply follow from the chain of subalgebras. A simpler computational &ame-work is obtained by erst representing bifermion excitations in terms of bosons, mapping the Hamiltonian, and then diagonalizing it in a boson Fock space spanned by all those bosons appearing in the collective realization [6] Here each SO (5) A. Identi8cation of spurious states As already stated, it is well known that for a closed collective subalgebra, physical eigenstates and eigenvalues are not contaminated when a boson mapped Hamiltonian is diagonalized in the complete boson Fock space [6]. Spurious states can, however appear in a nontrivial way and their identification is thus of utmost importance in app1icatioas of the boson method.…”