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Progress of Theoretical Physics

A Boson System Interacting with an External Harmonic Oscillator: The su(1,1)-Spin Like Behavior in the su(2)-Spin System

A. Kuriyama, João da Providência, Y. Tsue, M. Yamamura

Published Oct 1, 1997• 0 citations
Publisher

Abstract

In order to avoid the introduction of a fictitious degree of freedom in the su(1,1)-algebraic approach for describing thermal effects in a boson system, the su(2)-algebraic approach is proposed in terms of a boson system interacting with an external harmonic oscillator. The relation between the two approaches is investigated from two points of view. One is a quantal point of view based on the MYT boson expansion. The other is a classical point of view based on one of three forms of coherent states for the su(2)-spin system proposed recently by the present authors. Solution of Hamilton's equation of motion is discussed in relation to that in the su(1,1)-algebraic approach.

DOI
10.1143/ptp.98.893

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In order to avoid the introduction of a fictitious degree of freedom in the su(1,1)-algebraic approach for describing thermal effects in a boson system, the su(2)-algebraic approach is proposed in terms of a boson system interacting with an external harmonic oscillator. The relation between the two approaches is investigated from two points of view. One is a quantal point of view based on the MYT boson expansion. The other is a classical point of view based on one of three forms of coherent states for the su(2)-spin system proposed recently by the present authors. Solution of Hamilton's equation of motion is discussed in relation to that in the su(1,1)-algebraic approach.
A method for describing a boson system interacting with an external harmonic oscillator, which was proposed recently by the present authors, is extended to the case of a statistically mixed state. The basic idea comes from the su(2, 1)-boson model, the general form of which was recently investigated by the present authors. In terms of the interweaving of the phase space doubling and the su(1, 1)-spin-like behavior in the su(2)-spin system, the extension is performed in the framework of a canonical form of classical mechanics. The results obtained with regard to the present idea are interpreted from various viewpoints, for example, in relation to linear response theory of non-equilibrium statistical mechanics.
A pseudo su(1,1)-algebra is formulated as a possible deformation of the Cooper pair in the su(2)-algebraic many-fermion system. With the aid of this algebra, it is possible to describe the behavior of individual fermions which are generated as the result of interaction with the external environment. The form presented in this paper is a generalization of a certain simple case developed recently by the authors. The basic idea follows the su(1,1) algebra in the Schwinger boson representation for treating energy transfer between the harmonic oscillator and the external environment. The Hamiltonian is given following the idea of phase space doubling in the thermo-field dynamics formalism, and the time-dependent variational method is applied to this Hamiltonian. Its trial state is constructed in the frame deformed from the BCS-Bogoliubov approach to superconductivity. Several numerical results are shown.
With the aim of obtaining the classical counterpart of the su(2)-spin system and its quantal fluctuations in the frame of the boson coherent state, the Holstein-Primakoff and Marumori-Yamamura-Tokunaga boson expansions are investigated. The starting point is found in the Schwinger boson representation for the su(2)-spin system and its reformation. Dirac's canonical theory for constraint systems is also applied to the su(2)-spin system. In conclusion, the latter boson expansion is found to be superior to the former with respect to the use of the boson coherent state.
In order to develop a previously proposed method to investigate thermal effects in quantum many-particle systems, basic ideas are formulated in more general systems obeying su(2)-algebra. It is shown that Hamiltonians of a single orbit shell model with pairing correlation, the Lipkin model, a symmetric rigid rotor model and an asymmetric rigid rotor model have almost the same structure, that is algebraic structure of su(1, 1), by the use of the Holstein-Primakoff boson representation for su(2)-algebra and so-called phase-space doubling introduced to study thermal phenomena. It is also discussed that the interaction term between intrinsic system and external environment approximately represents the interesting physical process in each system at finite temperature.