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Physical Review C

Exactly solvable fermion-boson mapping representations

Osvaldo Civitarese, M. Reboiro

Published Jun 1, 1998• 11 citations
Publisher

Abstract

The Holstein-Primakoff boson mapping technique is applied to transform Hamiltonians defined in terms of the generators of the SU(2) and Sp(4) groups. These solvable models consist of a pair of fermions which interact with hard bosons. It is shown that the mapping to ideal bosons preserves the algebra and that exact matrix elements of the original fermionic Hamiltonian can be reconstructed from the bosonic representation.

DOI
10.1103/physrevc.57.3055

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The Holstein-Primakoff boson mapping technique is applied to transform Hamiltonians defined in terms of the generators of the SU(2) and Sp(4) groups. These solvable models consist of a pair of fermions which interact with hard bosons. It is shown that the mapping to ideal bosons preserves the algebra and that exact matrix elements of the original fermionic Hamiltonian can be reconstructed from the bosonic representation.
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.
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. 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.
The formalism derived for the boson–fermion and the baryon mapping in Paper I of this series is applied to the single-orbit quark shell model developed by Petry and co-workers for the description of nuclear properties. The multiquark space is reduced to the collective subspace of multinucleon states. A collective representation of states and operators in terms of bosons and ideal fermions or in terms of ideal baryons, respectively, is constructed. The results of the original model in the multinucleon space are exactly reproduced. Finally, the two transformations are compared with a related mapping technique recently published by Pittel, Engel, Dukelsky, and Ring.
Abstract An inversion of the Holstein-Primakoff transformation is proposed such that creation and annihilation operators for a bosonic field are rewritten as operators of an SU (2) algebra. In association with more common quadratic combinations for fermionic operators, that inverse transformation sets a quantum Hamiltonian fully in terms of SU (2) operators. A subsequent application of the prescription by Lieb, to obtain the classical limit for spin operators, then allows one to write effciently a classical Hamiltonian for the system. This process is illustrated for a bilayer model undergoing an (electron-hole)-to-exciton quantum phase transition.
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.