The first excitation energy in the two-level pairing model is investigated in terms of the equilibrium and the small fluctuation around it. The basic idea is an extension of results presented in a previous paper by the present authors. In this investigation, the result obtained in the previous paper plays a central role: At the limit of the weak and the strong interaction strength, the results are in good agreement with the exact one. The former and the latter are calculated in the framework of the su(2) ⊗ su(1, 1)- and the su(2) ⊗ su(2)-coherent state, respectively. On the basis of the above conclusion, the intermediate region for the interaction strength is described in terms of the idea of the interpolation and it is shown that the agreement of the result with the exact one is quite good. 1 typeset using PTPTEX.cls 〈Ver.0.9 〉 The description of the first excited state in the two-level pairing model for the closed shell system has been regarded as a basic, but a modest problem in many-body physics. For example, the recent investigations can be found in Ref. 1). Following this quiet stream, recently, the present authors have considered this problem from tow slightly different viewpoints. One
The first excitation energy in the two-level pairing model is investigated in terms of the equilibrium and the small fluctuation around it. The basic idea is an extension of results presented in a previous paper by the present authors. In this investigation, the result obtained in the previous paper plays a central role: At the limit of the weak and the strong interaction strength, the results are in good agreement with the exact one. The former and the latter are calculated in the framework of the su(2) ⊗ su(1, 1)- and the su(2) ⊗ su(2)-coherent state, respectively. On the basis of the above conclusion, the intermediate region for the interaction strength is described in terms of the idea of the interpolation and it is shown that the agreement of the result with the exact one is quite good. 1 typeset using PTPTEX.cls 〈Ver.0.9 〉 The description of the first excited state in the two-level pairing model for the closed shell system has been regarded as a basic, but a modest problem in many-body physics. For example, the recent investigations can be found in Ref. 1). Following this quiet stream, recently, the present authors have considered this problem from tow slightly different viewpoints. One
The first excitation energy in the two-level pairing model is investigated in terms of the equilibrium and the small fluctuation around it. The basic idea is an extension of results presented in a previous paper by the present authors. In this investigation, the result obtained in the previous paper plays a central role: At the limit of the weak and the strong interaction strength, the results are in good agreement with the exact one. The former and the latter are calculated in the framework of the su(2)⊗su(1,1)- and the su(2)⊗su(2)-coherent state, respectively. On the basis of the above conclusion, the intermediate region for the interaction strength is described in terms of the idea of the interpolation and it is shown that the agreement of the result with the exact one is quite good.
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).
A simple model of a boson system is considered for which the exact Schr\"odinger equation reduces to a difference equation which can be solved numerically. Application of the Bogoliubov method and comparison of its results with the numerical results then yield some insight into the validity of the Bogoliubov approximations. The simple model considered has, in the absence of interactions, only a zero energy state and two states of unit energy available to each boson. Initially there is assumed to be no interaction between bosons in the degenerate excited states and all existing interactions are assumed to be repulsive. The ground-state energy is calculated in the Bogoliubov approximation, with and without the depletion effect considered, and in another numerical approximation. These results and the results of the numerical solution of the exact Schr\"odinger equation converge to the Bogoliubov result with depletion ignored when the number of particles in the system approaches infinity. It is surprising to note that, for the interaction strengths considered, the Bogoliubov result is within 3% of the numerical result for as few as 32 particles in the system, and within 10% for only 4 particles. A modification of the above system is considered in which there exists an additional two-body interaction between particles in the degenerate excited states which may be attractive or repulsive. It is shown that the ground-state energy, with this additional interaction present, deviates from the Bogoliubov value linearly with the strength of the added interaction, i.e., $W={W}_{\mathrm{Bog}}+\ensuremath{\alpha}F$ (with F the strength parameter of the interaction). The factor $\ensuremath{\alpha}$ is found to be small and constant over a wide range of F and in this range the deviation of W from the Bogoliubov value is small. There is, however, a transition point FT, beyond which $\ensuremath{\alpha}$ is constant and large and the usual Bogoliubov approximation is invalid for interactions more attractive than that characterized by FT. A Bogoliubov-like approximation is shown to be quite accurate in this region. It is then shown that $F_{T}^{\ensuremath{-}1}$ is a linear function of the number of particles in the system.
Concerning the energy of harmonic oscillator, a prescription is proposed for making the original form unchanged even after q-deformation. Applicability of the prescription is limited, but, it can be applied to various cases which are well known. 1 typeset using PTPTEX.sty <ver.1.0>An idea of q-deformation serves us an interesting viewpoint for the understanding of dynamics of many-body systems described by boson operators. For the q-deformation, the present authors recently gave a possible scheme in relation to the generalized boson coherent state. 1)-3) It is, in some sense, an extension of the idea given by Penson and Solomon. 4) Before entering the central part of this paper, first, we recapitulate the basic idea of Ref. 1). We treat a boson system described by one kind of boson (ĉ, ĉ ∗). As the deformation of (ĉ, ĉ ∗), the operator (ˆγ, ˆγ ∗ ) is introduced in the definition ˆγ = F ( ˆ N)ĉ, ˆγ ∗ = ĉ ∗ F ( ˆ N). (1) Here, F ( ˆ N) denotes a function of the boson number operator, ˆ N, defined by ˆN = ĉ ∗ ĉ. (2) The function F ( ˆ N) obeys a certain condition which is proper to each case under investigation. The present deformation is characterized by the function F ( ˆ N). The commutation relations [ĉ, ĉ ∗ ] and [ˆγ, ˆγ ∗ ] are as follows: [ĉ, ĉ ∗ ] = 1, [ˆγ, ˆγ ∗ ] = ( ˆ N + 1)F ( ˆ N) 2 − ˆ NF ( ˆ N − 1) 2. (3a) (3b) Later, we discuss the case in which the relation (3b) does not hold. If following our deformed boson scheme, any operator which is expressed in terms of a function (ĉ, ĉ ∗), Ô = O(ĉ, ĉ ∗), is deformed by replacing (ĉ, ĉ ∗ ) with (ˆγ, ˆγ ∗ ): Ôq = O(ˆγ, ˆγ ∗ ). (4) However, in the case of the Hamiltonian, a special consideration is necessary for the replacement
The treatment for the multiboson coherent state is extended for the completeness of the formulation. The basic idea for the extension is presented in terms of expanding the boson space. The MYT boson mapping plays a central role. The resultant multiboson coherent state includes the state obtained in our previous treatment and the form suggests various multiboson coherent states are possible in the deformed boson scheme. 1 typeset using PTPTEX.sty <ver.1.0>Recently, the present authors published four papers 1), 2), 3) concerning the deformed boson scheme including conventional q-deformation in time-dependent variational method. The series of these papers consists of (I), 1) (II), 2) (III) 2) and the short note 3) and, in this series, we discussed various aspects of the q-deformation appearing in the boson coherent state and its generalized form. For example, in (I), 1) a possible form of the multiboson coherent state and its generalization were described. Its treatment is, in some sense, in close connection with that given by Penson and Solomon. 4) However, the treatment in (I) is limited to a certain special boson subspace. In this sense, it is unsatisfactory and to make it complete is the aim of this paper. Further, the present paper gives a generalization of the treatment proposed by the present authors, which is related to a new boson realization in the Lipkin 5), 6) model. In (I), a possible form of the multiboson coherent state, which we denote as |c0〉〉, was defined in the boson space ˜ B composed by the boson operator (˜c, ˜c ∗ ) in the following form:
Pair transfer matrix elements between the ground and excited states of a two-level system with monopole pairing interaction as a function of the pairing strength are calculated. The exact energies of excited states are compared with predictions of the quasiparticle random phase approximation. It is shown that at relatively low values of the pairing strength, two-particle transfer to the first and possibly second excited states must be taken into account for a more accurate description of pairing properties.