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Canadian Journal of Physics

Boson Method not Violating the Pauli Principle Applied to the Three-Quasiparticle System

P. A. Simard, M. Banville

Published Dec 15, 1971• 6 citations
Publisher

Abstract

The boson method is studied for a system with one and three quasiparticles. A basis is constructed in the ideal space in such a way that the Pauli principle is fully satisfied. The spurious states due to the particle nonconservation are properly eliminated. Numerical calculations are presented and a comparison is made with Kuo's method. The boson method can be easily extended to the study of systems having two phonons.

DOI
10.1139/p71-372

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The boson method is studied for a system with one and three quasiparticles. A basis is constructed in the ideal space in such a way that the Pauli principle is fully satisfied. The spurious states due to the particle nonconservation are properly eliminated. Numerical calculations are presented and a comparison is made with Kuo's method. The boson method can be easily extended to the study of systems having two phonons.
An antisymmetric basis has been set up in the ideal space as a combination of one boson and one ideal quasi-particle state; in this way a three quasi-particle system may be studied. The knowledge of solutions of linearized equations for two quasi-particles is a requisite for the setting up of this basis. The exclusion principle is thus introduced correctly. The spurious states coming from the non-conservation of the number of particles are separated out and can be rejected easily. The method can be used only in QTD approximation. The one-shell case is discussed as a verification of the method.
We propose a new boson expansion theory that does not premise the closed-algebra approximation, indispensable for formulation until now, as an extension of the conventional practical boson expansion methods that have tried to elucidate nuclear collective motion, which is a method that allows the closed-algebra approximation not to be used or to be used appropriately, enables us to obtain the boson expansion easier, and reproduces the fermion subspace onto the boson subspace more faithfully than the conventional practical methods. The two-phonon norm matrix composed of all the phonon excitation modes is investigated in detail, which reveals the mechanism, essential for the boson expansion methods, of how we should construct the fermion subspace to be mapped from the whole fermion space. The conventional practical boson expansion methods have applied the closed-algebra approximation improperly to strengthen the effect of the Pauli principle inappropriately, which should be replaced by those that do not use that approximation or use it properly.
The total number of states of any system is produced by all the possible interactions of the particles in that system. It follows therefore that there is need to obtain the states of the system before a model can be applied to it. Recently we developed a simple method to obtain the electronic states in the two electron lattice systems in all the three dimensions. This method encompasses rules formulated from a detailed study of each dimension. In the current study, the method will be extended to bosons which do not obey the Pauli exclusion principle like the electrons so that their interactions are different from those of the latter. The need for the extension of the method to bosons is also discussed in the work. Keywords: lattices, bosons, fermions, interactions, bosonic states Global Journal of Pure and Applied Sciences Vol. 12(1) 2006: 129-135
The random-phase approximation (RPA) is examined in a boson formalism and special attention is focused on the problem of the violation of the Pauli principle affecting this theory. A mapping technique is discussed and the boson image of a two-body fermion Hamiltonian is constructed. Within the boson space so defined, an extension of RPA is proposed where structure and energy of the ground state, as well as those of the one-phonon states, are analyzed with regard to the role of the spurious components associated with a violation of the Pauli principle. A multistep minimization procedure for the determination of a better ground state is also examined. Numerical tests are performed within the Lipkin-Meshkov-Glick model.