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Journal of Physics A Mathematical and General

Schwinger representation approach to the Lipkin model

Masatoshi Yamamura, Constança Providência, João da Providência, Seiya Nishiyama, Flávio Cordeiro, Y. Tsue

Published Sep 19, 2006• 2 citations
Publisher

Abstract

The Schwinger representation and the Marumori–Yamamura–Tokunaga boson expansion are used to describe the Lipkin model in terms of generalized coherent states. The groundstate, first excited state and RPA energies are obtained within several variant types of coherent states. It has been found that generalized coherent states defined in consonance with the parity symmetry of the model describe particularly well the transition from weak to strong coupling, providing a remarkable improvement of the mean-field description of the transition zone.

DOI
10.1088/0305-4470/39/40/012

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The Schwinger representation and the Marumori–Yamamura–Tokunaga boson expansion are used to describe the Lipkin model in terms of generalized coherent states. The groundstate, first excited state and RPA energies are obtained within several variant types of coherent states. It has been found that generalized coherent states defined in consonance with the parity symmetry of the model describe particularly well the transition from weak to strong coupling, providing a remarkable improvement of the mean-field description of the transition zone.
The Schwinger representation and the Marumori–Yamamura–Tokunaga boson expansion are used to describe the Lipkin model in terms of generalized coherent states. The ground-state energy is obtained within several variant types of coherent states. It has been found that two of the kinds considered describe particularly well the transition from weak to strong coupling, providing a remarkable improvement of the mean-field description of the transition zone. The time evolution predicted by generalized coherent states has been investigated numerically by comparing its dynamics with the exact one.
It is shown that the Bloch or angular momentum coherent states furnish a particularly efficacious basis for a discussion of various aspects of the Lipkin model of the ’’nucleus.’’ The Hartree–Fock description (as well as its projected version) is elegantly obtained in this framework. It is demonstrated that the ’’transition probability’’ between the first excited and ground states is proportional to the square of the number of ’’nucleons,’’ representing (in contrast to what obtains in the random phase approximation) a cooperativity of the ’’super-radiant’’ type. The extension of the model through the introduction of bosons permits, with the use of Bloch and Glauber coherent states, a succinct description of the phenomenon of boson condensation.
The two-level pairing model was investigated in the framework of the Schwinger representation. It has been shown that zero-point fluctuations, which are important in the crossover region, are well described by certain coherent states which conserve the constants of motion. These so-called conserving approximations provide excellent descriptions of the ground-state energy and of the first excitation energy over the whole range of coupling constant values. The description of the ground-state energy provided by the Glauber coherent state is also reasonable, but not so good. Although the Glauber coherent state fails to provide the description of the excitation energy around the critical point, its performance in the strong coupling limit is very good both for the description of the ground-state energy and of the first excitation energy. The present results also show that already at the mean-field level, the bosonic description in the independent particle approximation incorporates important correlation corrections to the conventional mean-field description based on the fermionic realization of the su (2) algebra (BCS theory).
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.
Following an idea, recently, proposed by the present authors for the two-level pairing model, the Lipkin model is reexamined. It is a natural generalization of the method already developed by the present authors with Kuriyama. This model is a schematic model for many-fermion systems and obeys the su(2)-algebra. It is shown that the use of the Schwinger boson representation, the model is expressed in terms of the su(1,1) ⊗ su(1,1)-algebra and with the aid of the MYT mapping method, it is disguised from the orginal form in terms of the Holstein-Primakoff representation. Further, under various coherent states, the classical counterparts are derived. It is concluded that the Lipkin model can be treated in the common ring as that of the two-level pairing model.
With the aim of applying to the Lipkin model in the case of open shell system, a possible form of the boson realization for the su(2)-algebra is proposed both in the Schwinger and the Holstein-Primakoff representation. The basic idea is borrowed from the Schwinger boson realization for the su(4)-algebra for many-quark system recently presented by the present authors. As the simplest approximation, a certain approximate form is given and the result is a natural generalization of the RPA result in the case of closed shell system.