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arXiv:nucl-th/0412042

Bosonization and IBM

Fabrizio Palumb

Published Dec 11, 2004• 1 citation
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Abstract

We derive a boson Hamiltonian from a Nuclear Hamiltonian whose potential is expanded in pairing multipoles and determine the fermion-boson mapping of operators. We use a new method of bosonization based on the evaluation of the partition function restricted to the bosonic composites of interest. By rewriting the partition function so obtained in functional form we get the euclidean action of the composite bosons from which we can derive the Hamiltonian. Such a procedure respects all the fermion symmetries.

DOI
10.48550/arxiv.nucl-th/0412042

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We derive a boson Hamiltonian from a Nuclear Hamiltonian whose potential is expanded in pairing multipoles and determine the fermion-boson mapping of operators. We use a new method of bosonization based on the evaluation of the partition function restricted to the bosonic composites of interest. By rewriting the partition function so obtained in functional form we get the euclidean action of the composite bosons from which we can derive the Hamiltonian. Such a procedure respects all the fermion symmetries.
We test a new approach to bosonization in relativistic field theories and many-body systems, based on the use of fermionic composites as integration variables in the Berezin integral defining the partition function of the system. The method appears promising since at zeroth order it correctly describes the propagators of the composites, which can be evaluated in a number of significant cases. Still to be established is a general procedure for deriving the free action of the composites starting from the one of the constituents. To shed light on this problem and to explore further features of the method we study a simplified version of the BCS model. In this simple case the action of the composites can indeed be obtained: whether this result can be generalized it remains hewever to be seen. Yet an interesting property of the wave operators appearing in the free actions of bilinear composites already emerges from the simple problem we have treated: here the wave operators do not describe any time evolution, even though they generate the right propagators. This outcome relates to the basic properties of the integrals over the even elements of a Grassmann algebra where the composites live, which entails that the propagators are no longer the inverse of the wave operators.
The microscopic basis of the Interacting Boson Model for deformed nuclei is discussed. The IBM Hamiltonian is constructed microscopically in the following two steps. In the first step, the collective nucleon pairs of J = 0/sup +/ (S), 2/sup +/ (D), etc. are mapped onto the corresponding bosons. Nucleon-nucleon interactions are also mapped onto boson-boson interactions. This mapping method for deformed nuclei was proposed recently, and it turned out that this method is consistent with the Hartree-Fock-Bogoliubov + angular momentum projection calculation. Low-lying collective states primarily consist of S and D pairs. Consequently, the corresponding boson states mainly consist of s and d bosons, while there are some admixture of g-bosons. In the second step, effects of these g-bosons are included within the s-d boson space by a unitary transformation which transforms a combination of d and g bosons into a new d-boson. By minimizing the coupling between new d and g bosons with an appropriate mixing angle, one can neglect the coupling and obtain the IBM Hamiltonian with s and d bosons. It is demonstrated that the s-d Hamiltonian thus derived indeed reproduces spectra of the original s-d-g Hamiltonian.
I derive in a novel fashion exact formulas for the calculation of general matrix elements, including the overlap (norm) matrix, between states constructed from fermion pairs. Mapping the fermion pairs to bosons, I show how to construct finite and exact (in the sense of preserving matrix elements) boson representations of the norm operator and one- and two-fermion operators. This may lead to a microscopic basis for the Interacting Boson Model, as well as new truncation schemes for the nuclear shell model.