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Physical Review C

Physical boson basis states in the boson expansion theories

Y. K. Gambhir, Javid A. Sheikh, Peter A. Ring, Peter Schuck

Published Apr 1, 1985• 16 citations
Publisher

Abstract

The use of physical boson basis states is stressed for the calculations in the boson space. The explicit form of physical boson basis states in terms of bosons is derived for the nonunitary boson mapping of Dyson. The ambiguity in the normalization introduced due to the use of bi-orthonormal basis states is satisfactorily resolved, resulting in a Hermitian matrix. This Hermitian matrix is found to coincide with the Hamiltonian matrix in the fermion space. The model cases where the use of a boson basis is justified are shown to be consistent with our view of using physical boson basis states.

DOI
10.1103/physrevc.31.1519

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Physical Review C•Mar 13, 2006•1 citation
The use of physical boson basis states is stressed for the calculations in the boson space. The explicit form of physical boson basis states in terms of bosons is derived for the nonunitary boson mapping of Dyson. The ambiguity in the normalization introduced due to the use of bi-orthonormal basis states is satisfactorily resolved, resulting in a Hermitian matrix. This Hermitian matrix is found to coincide with the Hamiltonian matrix in the fermion space. The model cases where the use of a boson basis is justified are shown to be consistent with our view of using physical boson basis states.
An algorithm for obtaining a recursive algebraic expression for the various expansion coefficients appearing in the physical boson basis states is presented, using the Dyson boson mapping. It is observed that only the first few terms of the physical boson basis states, expressed in the present form, contribute to the matrix element of the Hamiltonian. Explicit numerical calculations have been performed for Ni and Sn isotopes in the obtained physical boson basis states. The pairing Hamiltonian has been used as the model interaction. The results are more than 90% in agreement with exact shell model calculations.
This Brief Report deals with the solution of the shell-model Hamiltonian in the boson representation. In the present approach, the fermion Hamiltonian is exactly mapped to the boson space using the Dyson boson mapping. The problem of mapping the basis states is resolved by employing the intrinsic boson basis. The method has been applied to a model Hamiltonian consisting of a monopole pairing and a deformation force. It is observed that only s and d bosons are dominant in the intrinsic wave function for the model case studied. This is synonymous to the interacting boson model approach.
The nonunitary character of the Dyson boson mapping is explored in a model case with the monopole pairing interaction. It is shown that because of the very special form of this interaction, the calculations in the intrinsic state formalism with the particle-particle representation of the Hamiltonian reproduce the exact results of the original fermion space. In contrast, we show that with the particle-hole representation of the Hamiltonian the intrinsic state approach produces erroneous results. Of course, upon employing the physical boson states we recover the exact results in the particle-hole represenation as well.
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.