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Progress of Theoretical Physics

On the Schwinger Boson Representation of an Extended (M+1)(N+1)-Dimensional Algebra Containing the su(M+1)- and the su(N, 1)-Algebra

A. Kuriyama, João da Providência, M. Yamamura

Published Feb 1, 2000• 0 citations
Publisher

Abstract

With the aim of describing statistically mixed states in realistic many-body systems, the su(n)- and the su(n, 1)-algebra in the Schwinger boson representation are presented in a form generalized from the su(2)- and the su(1, 1)-algebra. The generators of the two algebras are expressed in terms of certain bilinear forms for boson operators. Some concrete examples are discussed in detail.

DOI
10.1143/ptp.103.285

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With the aim of describing statistically mixed states in realistic many-body systems, the su(n)- and the su(n, 1)-algebra in the Schwinger boson representation are presented in a form generalized from the su(2)- and the su(1, 1)-algebra. The generators of the two algebras are expressed in terms of certain bilinear forms for boson operators. Some concrete examples are discussed in detail.
Following the Schwinger boson representation for the su(M + 1)- and su(N, 1)-algebras presented by two of the present authors (J. da P. and M. Y.) and Kuriyama, a possible counterpart of the Lipkin model in the su(M + 1)-algebra formulated in the fermion space is presented. The free vacuum, which plays a fundamental role in the conventional treatment of the Lipkin model, is generalized in a quite natural way, and further, the excited state generating operators, such as the particle-hole pairs, are also given in a natural scheme. As concrete examples, the cases of the su(2)-, su(3)- and su(4)-algebras are studied. Especially, the case of the su(4)-algebra is investigated in detail in relation to the nucleon pairing correlations and the high-temperature superconductivity.
Considering the present situation of Lie-algebraic studies of high-temperature superconductivity, a Schwinger-type boson realization of the su(4) algebra is investigated. The basic framework comes from a possible form of the su(n) and the su(n,1) algebras in the Schwinger representation, which was recently proposed by two of the present authors (M. Y. & A. K.), together with J. da Providência. The form investigated in this paper is, in some sense, a natural extension of the Schwinger boson representation for the so(5), so(4) and su(2)⊗su(2) algebras, which was also proposed by the present authors. The orthogonal set for the irreducible representation of the su(4) algebra is completely analogous to that expressed in terms of fermion operators for pairing correlation.
With the use of two kinds of boson operators, a new boson representation of the |su(2)su(2)su(2)|-algebra is proposed. The basic idea comes from the pseudo |su(1,1)su(1,1)su(1,1)|-algebra recently given by the present authors [Y. Tsue et al., Prog. Theor. Exp. Phys. 2013, 103D04 (2013)]. It forms a striking contrast to the Schwinger boson representation of the |su(2)su(2)su(2)|-algebra, which is also based on two kinds of bosons. It is proved that this new boson representation obeys the |su(2)su(2)su(2)|-algebra in a certain subspace in the whole boson space constructed by the Schwinger boson representation of the |su(1,1)su(1,1)su(1,1)|-algebra. This representation may be suitable for describing the time dependence of the system interacting with the external environment in the framework of the thermo field dynamics formalism, i.e., phase space doubling. Further, several deformations related to the |su(2)su(2)su(2)|-algebra in this boson representation are discussed. On the basis of these deformed algebras, various types of time evolution of a simple boson system are investigated.
A new method for describing the two-level pairing model in a many-fermion system is presented. The basic idea comes from the Schwinger boson representation for the su(2) ⊗ su(2)-algebra, in which four kinds of boson operators govern the dynamics. It is well known that in the original fermion space, it is almost impossible to introduce any trial state for the variation except the BCS state in the framework of the mean field approximation. However, in the Schwinger boson representation, various trial states can be defined on the condition that the mean field approximation is useful. In these states, the su(2) ⊗ su(1,1)-coherent state is mainly discussed in this paper.