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Physical Review C

Deriving an interacting boson model from the fermion SO(8) model

G.-K. Kim, C. Martin Vincent

Published Apr 1, 1987• 19 citations
Publisher

Abstract

To see how closely an interacting-boson-model-like model can be approached, we systematically develop a similarity transformation that unitarizes the simple correspondence version of the Dyson boson mapping of the Ginocchio SO(8) model. For a large region of Hamiltonian parameter space, a one-body plus two-body Hermitian boson Hamiltonian of the interacting-boson-model form is achieved. The similarity transformation consists of a dynamical part and a normalizing part. The dynamical transformation results in a one-body plus two-body Hermitian Hamiltonian and simple transition operators. The normalizing transformation commutes with the dynamically transformed Hamiltonian. This enables the transition rates to be calculated without introducing dual bases. However, the resulting boson model differs from the phenomenological interacting boson model in that transition rates are not calculable from the absolute squares of matrix elements. Dynamically transformed operators are given explicitly, and normalizing transformations for SO(6) and seniority limits are obtained in closed form.

DOI
10.1103/physrevc.35.1517

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To see how closely an interacting-boson-model-like model can be approached, we systematically develop a similarity transformation that unitarizes the simple correspondence version of the Dyson boson mapping of the Ginocchio SO(8) model. For a large region of Hamiltonian parameter space, a one-body plus two-body Hermitian boson Hamiltonian of the interacting-boson-model form is achieved. The similarity transformation consists of a dynamical part and a normalizing part. The dynamical transformation results in a one-body plus two-body Hermitian Hamiltonian and simple transition operators. The normalizing transformation commutes with the dynamically transformed Hamiltonian. This enables the transition rates to be calculated without introducing dual bases. However, the resulting boson model differs from the phenomenological interacting boson model in that transition rates are not calculable from the absolute squares of matrix elements. Dynamically transformed operators are given explicitly, and normalizing transformations for SO(6) and seniority limits are obtained in closed form.
We derive an approximate boson-quasifermion mapping of a single-j shell-model algebra, applicable to the seniority classification scheme, as an expansion in the ratio of the seniority quantum number to the level degeneracy, with selective summation of higher order terms. The procedure used is an extension of the algebraic technique first applied to the purely bosonic part of the mapping by Bonatsos, Klein, and Li. Only the maps of the generators are needed to derive a Hamiltonian for the interacting boson fermion model (IBFM), and these can be obtained in Hermitian form without having to go through the intermediary of a mapping of the fermion operators. Application of the results is illustrated by a fresh discussion of the so-called exchange interaction of the IBFM. A consistent mapping of the single fermion in terms of bosons and quasifermions is also derived that includes a consideration of the unusual properties of the quasifermions.
The qualitative ideas put forward by Geyer and Lee are given quantitative content by constructing a similarity transformation which reexpresses the Dyson boson images of the single-jjj shell fermion operators in terms of seniority bosons. It is shown that the results of Otsuka, Arima, and Iachello, or generalizations thereof which include ggg bosons or even bosons with J>4J>4J>4, can be obtained in an economic and transparent way without resorting to any comparison of matrix elements.
For an even number of nucleons, we consider collective motion based on any dynamical symmetry subalgebra \^A of the valence-shell bifermion algebra; the generators of \^A are coherent linear combinations of bifermion operators of the types ${a}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$${a}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$, ${a}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$a, and aa. A boson algebra is obtained by Dyson mapping the whole valence-shell bifermion algebra, and then ``skeletonizing'' the resulting expressions by deleting all noncollective boson operators from them. This skeleton boson algebra is shown to have the same commutation relations as the collective bifermion algebra, provided that the bifermion algebra is self-conjugate. The resulting boson Hamiltonian is not Hermitian, but does resemble the Hamiltonian of the interacting boson model in containing only one- and two-body terms. Moreover, the skeleton boson algebra (when restricted to a collective boson subspace that is free of spurious states) is shown to be physically equivalent to the collective bifermion algebra. It is shown that by working solely with collective bosons one can identify spurious boson states unambiguously, although the procedure lacks convenience. The similarity transformation that connects conjugate versions of the skeleton boson mapping is discussed, and a general condition for hermiticity of the boson image of a Hermitian operator is given.
A similarity transformation is an equivalence relation between square matrices which preserves determinant, trace and eigenvalues, playing a key role in quantum mechanics in simplifying complex hamiltonian systems and improving analytical results attainable from the use of perturbation theory. As a prototypical example, the conventional BCS theory of superconductivity is usually derived from a similarity transformation of the original electron-phonon hamiltonian, written in second quantized version. Here we discuss the general method for writing the similarity transformation operator in second quantized form, allowing one to recast a hamiltonian describing an interacting fermion-boson system into an effective theory in which only the desired degrees of freedom are kept after the transformation.