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Physical Review C

Boson mapping of the shell model algebra obtained from a seniority-dictated similarity transformation

H. B. Geyer

Published Dec 1, 1986• 19 citations
Publisher

Abstract

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-$j$ shell fermion operators in terms of seniority bosons. It is shown that the results of Otsuka, Arima, and Iachello, or generalizations thereof which include $g$ bosons or even bosons with $J>4$, can be obtained in an economic and transparent way without resorting to any comparison of matrix elements.

DOI
10.1103/physrevc.34.2373

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arXiv:nucl-th/9410012•Oct 5, 1994•0 citations
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.
We discuss the construction of ideal space seniority images of single fermion operators which proceeds via the generalized Dyson mapping and a subsequent similarity transformation. Particular attention is paid to the implications of a consistent treatment of single-particle and collective degrees of freedom. The results are compared with expressions for the mapped single fermion operators previously obtained, respectively, within the Otsuka-Arima-Iachello framework and by using the prescriptions of nuclear field theory. We point out certain inconsistencies in existing constructions of interacting-boson-fermion model operators in general and the interacting-boson-fermion model quadrupole operator in particular.
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.
Methods of boson mapping are tested in calculations for a simple model system of four protons and four neutrons in single-j distinguishable orbits. Two-body terms in the boson images of the fermion operators are considered. Effects of the seniority v=4 states are thus included. The treatment of unphysical states and the influence of boson space truncation are particularly studied. Both the Dyson boson mapping and the seniority boson mapping as dictated by the similarity transformed Dyson mapping do not seem to be simply amenable to truncation. This situation improves when the one-body form of the seniority image of the quadrupole operator is employed. Truncation of the boson space is addressed by using the effective operator theory with a notable improvement of results.
After a brief review of various mappings of fermion pairs to bosons, we rigorously derive a general approach. Following the methods of Marumori and Otsuka, Arima, and Iachello, our approach begins with mapping states and constructs boson representations that preserve fermion matrix elements. In several cases these representations factor into finite, Hermitian boson images times a projection or norm operator that embodies the Pauli principle. We pay particular attention to truncated boson spaces, and describe general methods for constructing Hermitian and approximately finite boson image Hamiltonians, including effective operator theory to account for excluded states. This method is akin to that of Otsuka, Arima, and Iachello introduced in connection with the Interacting Boson Model, but is more rigorous, general, and systematic.