A possible trial beyond the conventional random phase approximation for the su(2)-Lipkin model including the case of non-closed shell system

Concerning the su(2)-Lipkin model, the calculation of the excitation energy to the 1st excited-state gives rise to the following fact: The two results based on the exact treatment and the conventional random phase approximation (RPA) are in unexpected disagreement. In order to remove this discrepancy, the conventional RPA is renewed. With the aim of this renewal, the terms, which cannot be dealt with in the conventional one, are estimated. The basic framework of this new form is not reorganized from that of the conventional one. In addition, the su(2)-Lipkin model itself is also modified so as to be applicable to the case with non-closed shell system. To this modification, one more su(2)-algebra, which has been already proposed by the present authors, is applied. Through the use of this algebra, the su(2)-Lipkin model is applicable to the case with any total fermion number permitted in the model.

1 Introduction governed by the condition N = 2Ω = 2s. In this paper, we are interested in the case where the condition N = 2Ω = 2s is broken. This paper aims mainly at discussing two subjects. First is to clarify the relationship among the three quantities N, 2Ω and 2s appearing in the su(2)-Lipkin model. For this purpose, we prepare an auxiliary su(2)-algebra, three generators of which commute with the three of the original. This su(2)-algebra has been already taken up for various purposes by the present authors [5]. With the aid of this algebra, we can modify the su(2)-Lipkin model for treating the case where the condition N = 2Ω = 2s is broken, i.e., the non-closed shell case. It should be noted that one of the generators is related to the total fermion number operator which the Lipkin model does not contain. Combining these two su(2)-algebra, we obtain, for example, the relation 2s ≤ N ≤ 4Ω − 2s. Therefore, in the case with s = Ω, automatically, we have N = 2Ω. It suggests us that we can treat the cases with N < 2Ω and N > 2Ω.
The above is the first subject. Second is to develop a possible trial beyond the conventional RPA for the Lipkin model. Figure 1 depicts the frequency ω/(2ǫ) given by the conventional RPA and the exactly calculated excitation energy from the ground-to the 1st excited-state as functions of G(= G/ǫ · 2Ω) for Ω = 5. Here, G denotes the strength of the interaction defined in the relation (79). We are forced to see that the discrepancy is quite conspicuous and cannot see any phase transition [6]. For the above situation, we must develop a new idea. In this paper, we will present a possible form for making up the above discrepancy without breaking down the basic framework of the conventional RPA. For this aim, we focus on the parts replaced with the expectation values given by the free vacuum. Our idea is to replace these parts with certain forms obtained through the relation of the Casimir invariant. With the help of this procedure, we can take up the effects let out in the conventional RPA. As an additional remark, we note that the problem discussed in this paper may be old-fashioned, but, until the present, any clear-cut solutions for them have been not obtained. This is our motivation for the present work. Concerning this point, we are interested in the paper [7].
Of course, there are many attempts to extend the RPA itself to include the higher effects such as an anharmonicity from before. For example, in order to extend the HF and RPA methods, the variational approach to the many-body problem was devised by introducing an appropriate trial state and the second RPA was derived in a useful form for obtaining approximate solutions [8]. Also, by using the Green function formalism, the correction to the RPA was derived as a microscopic theory dealing with an anharmonic effects [9]. In the words of the Beliaev-Zelevinsky boson expansion, the generalized RPA was considered by taking into account the boson-boson correlations [10]. Further, it was shown that, if the variational parameters a, appearing in the generator-coordinate method, Ψ(x) = φ(x; a)f (a)da, depends on the time, the variational parameters oscillate with the same frequency as that derived by the RPA [11], while the relation between the generator-coordinate method and the RPA was indicated in several authors [12].
In the next section, a modified form of the su(2)-Lipkin model is presented. Another generators of the su(2)-algebra adding to the original ones in the su(2)-Lipkin model are introduced. In section 3, a renewed random phase approximation is formulated and the non-closed shell system can be treated. The conventional RPA is included in the renewal of the RPA developed in this section. In section 4, various numerical results obtained by the renewed RPA are shown. Especially, the renewed random phase approximation presents a good result for the first-excited energy compared to the conventional RPA. The last section is devoted to concluding remarks.
2 A modified form of the su(2)-Lipkin model The su(2)-Lipkin model, which may be the simplest many-fermion model, consists of two single-particle levels with the degeneracies 2Ω. Here, 2Ω denotes an integer and we specify the two levels by the index l with l = 0 and 1. The single-particle states are specified by an appropriate labeling. In this paper, we adopt the labeling µ = 1, 2, · · · , 2Ω for (0, µ) and (1, µ). The basic operators in this model are as follows: Here,c * lµ andc lµ denote the creation and the annihilation fermion operators in the state (l, µ), respectively. Needless to say, the hermitian conjugate are given as The set ( S 0,± ) obeys the su(2)-algebra: The Casimir operator S 2 is given in the form Naturally, ( S 0,± ) commutes with S 2 .
The Hamiltonian treated in this paper is of the form which has been adopted conventionally: Here, 2ǫ and G denote the energy difference between the two single-particle levels and the interaction strength, respectively. We regard the l = 0 level as the lower than the l = 1. In this paper, we will treat the case with First task in the algebraic approach for the Hamiltonian (5) is to give the minimum weight state. Without loss of generality, 2s fermion can occupy the single-particle states µ = 1, 2, · · · , 2s in the level l = 0, because the single-particle energy does not depend on µ. This state, which denotes |s , satisfies the condition Certainly, the state |s is a minimum weight state of the present su(2)-model. Then, we have the state |s, s 0 in the form Conventionally, this model has been investigated in the case with 2s = N = 2Ω (N; total fermion number) under the name of the study of collective motion observed in the closedshell system. With the use of the state |s , we can investigate the case with N = 2s < 2Ω. Then, the next task is to present the minimum weight state in the condition N > 2s. If this task realizes, the basic part of the study of the su(2)-Lipkin model may be completed.
With the aim of obtaining the minimum weight state in N > 2s, we introduce another su(2)-algebra under the same single-particle level scheme as that in the present model: The set ( Λ 0 , Λ ± ) obeys also the su(2)-algebra : As was already mentioned, the algebra ( Λ ±,0 ) has been taken up for the various purposes by the present authors [5]. The Casimir operator Λ 2 is given as The noticeable relation which controls these two algebras is as follows: any of Λ 0,± , any of S 0,± = 0 .
The definition of Λ 0 shown in the relation (9) and its property (16) give us that the total fermion number N in the state (15) is of the form In such a way, we obtained the minimum weight state in the condition N > 2s and the orthogonal set is given as Here, we used the relation (18). If paying attention only to the fact that the set { S ±,0 } obeys the su(2)-algebra, we are able to obtain the eigenvalues of the Hamiltonian (5)  With the aim of making a connection between ( S 0,± , Λ 0,± ) and the minimum weight state |s more clearly, we reform our algebra. First, we decompose S 0,± into the following two parts: Here, the sums (1) µ and (2) µ are defined as They obey The operation of S 0,± on |s leads us to Then, we have For |ss 0 , the relations (24) and (25) give us The above tells us that the eigenvalue problem for ( S 0,± ) in our present model can be treated in the frame of ( S 8 Here,ã * µ andb * µ are defined as Needless to say, ( S 0,± ) obeys the su(2)-algebra: We can see that (ã * µ ,ã µ ) and (b * µ ,b µ ) are nothing but the particle and the hole operators used in the case of the closed-shell system and if Ω used in the closed-shell system is replaced with s, the results are available in the case with s < Ω.
We also decompose ( Λ 0,± ) into the following two parts : These decomposed algebras satisfy 0,± and S (2) 0,± , any of Λ (1) 0,± and Λ Then, by combining the above relations with the form (20), we have The relation (35) tells us that, in the case where the Hamiltonian (5) is investigated, at the first step, we specify the single-particle states µ = 1, 2, · · · , 2s which give us the minimum weight state |s . At the next step, we investigate the Hamiltonian (5) under appropriate method. Then, at the final step, we operate ( Λ (2) + ) 1 2 (N −2s) on the eigenstates obtained at the second step and the eigenstates in the fermion number N are derived. The above is our modified form of the su(2)-Lipkin model and, in 3, we formulate the random phase approximation (RPA) for the Hamiltonian.

A renewal of the conventional random phase approximation
In a way similar to that in the conventional random phase approximation (RPA), we begin our discussion with the analysis of the following equation of motion: In the case with s = Ω, the term ∓2G(2s −ñ) S ± ] can be linearized for S (1) ± . In the case with s < Ω, the term ∓2G(2s −ñ) S (2) ∓ exists as the operator. However, as long as the equation of motion is investigated in the orthogonal set given in the relation (26), this term may be permitted to throw away. If we follow the above argument, our problem is to result in developing the idea how to treatñ. Under the assumption that the change of the occupation of the single-particle states is small, it might be permitted beforehand to set upñ = 0.
However, in the case where the interaction strength becomes large, the effect coming from n is conjectured to be a subject of our investigation even if s = Ω.
In the case with (s = Ω, n = 0), we can see that the relation (46) By adopting the approximation (41), the relation (47) becomes In the case with (s = Ω, n 0 = 0), the relation (48) is written as The above is well-known formula under the name of the relation between the forward-and the backward-amplitude. The relations (45) and (48) (45) and (48) give us the following results : If s = Ω and n 0 = n 1 = 0, ω/(2ǫ) can be expressed as In the case with G · 2Ω/ǫ = 1, ω/(2ǫ) = 0 and at this point, the phase transition occurs. In our case, also the condition G · (2s − n)/ǫ = 1 leads us ω/(2ǫ) = 0. However, we can see that the above condition depends on n. The case (S ± ) 2 is also in the same situation as the above. As was mentioned in the above, our further task is to determine n 0 and n 1 which may be functions of ω/(2ǫ). Our first task is concerned with n 0 (= (0|ñ|0)). The idea comes from the Casimir operator (4). By calculating the expectation value (0| S 2 |0)(= s(s + 1)), we have Here, (0|[ S Since 2s − n 0 ≥ 0, we pick up the positive solution of the above-quadratic equation for (2s − n 0 ): The inverse of the above is expressed as The relation (55) will be taken up again in the relations (72), (73) and (74). We can see that the solution (55) gives us the following boundary values : (ii) If ω/(2ǫ) = 0 , 2s − n 0 = 0 , i.e., n 0 = 2s .
The relation (55) can be expressed as a function of S − : The above expression is approximated in the form n 0 = (S − ) 2 2(2s + 1) . if 2s + 1 ≫ 2(S − ) , i.e., n 0 ≪ s + 1 2 or n 0 ≪ S − , The form (58) may be regarded as the simplest approximate expression for n 0 . As a supplementary remark, we mention the following: About sixty years ago, the effect of n 0 as the quasi-particle number was investigated in the study of the quasi-particle RPA for microscopic description of the first excited state of medium heavy even-even nuclei [13]. If the formula used at that time is translated into the present model, it may be reduced to the form (58). In this case, S − / √ 2s plays a role of the backward amplitude in the quasi-particle RPA. Of course, the formula (58) is not applicable to the case with n 0 ≈ 2s. Next, we discuss our idea for n 1 (= (1|ñ|1)). First, we notice the relation (36), which can be rewritten as follows : We discuss the case with N = 2s and, then, omit the superscript (1) 13 Associating with the relation (60), we take up the following : For arbitrary values of A and B, we have the relations (60) and (61). We will show that, for specific values, they are useful for our purpose. For the both sides of the relation (60), the states |0) and |1) lead us to The left-hand side The right-hand side Here, S ± are shown in the relation (51) and, for the derivation of the relation (62), we adopt the approximation (42) and n is given in the relation (46) (n = (n 1 + n 0 )/2). For the relation (62b), we impose the condition Under this condition, the relation (62b) vanishes and (A, B) is of the form Here, C denotes an arbitrary c-number factor (C = 0). The term (AS + + BS − ) in the relation (62a) is expressed as The above consideration leads us to Since E 1 − E 0 = ω, (n 1 − n 0 ) is determined in the form Combining the relation (67) with (56) for the boundary values of n 0 , we have (i) If ω/(2ǫ) = 1 , n 0 = 0 and, then n 1 = 2 , (ii) If ω/(2ǫ) = 0 , n 0 = 2s and, then n 1 = 2s .
In order to examine the validity of the form (67), we apply our idea to the relation (61). With the use of |0) and |1), this relation is expressed in the form The above gives us With the aid of the form (64), we obtain another expression for (n 1 − n 0 ) : Substituting the relation (51) for (S ± ) 2 into (71), we can see that the relation (71)

Discussion with numerical results
The su(2)-Lipkin model is, needless to say, constructed under the su(2)-algebra and it contains two basic parameters, s and G/ǫ, the numerical values of which are given from the outside. The parameter ǫ plays only a role of the measure of the energy. Therefore, the quantity such as, for example, ω/(2ǫ), should be specified in terms of a function of s and G/ǫ. However, as can be seen in the relation (50) together with (55) and (67), G/ǫ is simply expressed as a function of ω/(2ǫ). But, the inverse seems to be too complicated to give the explicit form. In this section, mainly, we will try to search the explicit form by incorporating the numerical results in the discussion.
Let us start with the discussion on n 0 and n 1 . As was already mentioned, G/ǫ can be easily expressed as a function of ω/(2ǫ). We can express (2s − n) in the form The quantity ω/(2ǫ) changes its value in the range 0 ≤ ω/(2ǫ) ≤ 1 and we notice the following inequality : if s ≫ 0 , 4s(s + 1) For example, the case with s = 5 and ω/(2ǫ) = 1/2 gives us 120/( √ 31 + 1) = 18.271 ≫ 1. This indicates that the first term in the bracket [ ] in the relation (72) is much larger than the second, 1. Then, it may be permitted to neglect the second term, 1 and as a possible form of (2s − n), we can adopt the form 2s − n = L 4s(s + 1) .
Here, L is defined in the form The above approximation corresponds to the condition n 1 = n 0 . By substituting the relation (74) into (50), we obtain G/ǫ which is expressed in terms of ω/(2ǫ) : The relation (76) gives us ω/(2ǫ) expressed in terms of G/ǫ and s. If s is finite, it may be easy to see that relation (76) leads us to the following two cases : inversely, In the conventional RPA, the case (i) is familiar to us, but, the case (ii) is ununderstandable. Especially, the case (ii) informs us that we cannot expect the phase transition which is well known in the conventional RPA. Therefore, it may be interesting to investigate the present RPA in parallel with the conventional RPA. For this aim, we introduce an auxiliary parameter g/ǫ which plays a role similar to that of G/ǫ: The relation (76) can be expressed in the following form by introducing the parameter δ : Formally, the relation (79) appears in the conventional RPA described under the interaction strength g/ǫ. Associating with δ, we define δ 0 in the form 16 The relations (75) and (78)∼(80) give us to the following relation: With the aid of the relation (81), δ can be expressed as a function of δ 0 and s. Of course, δ 0 and δ are restricted to the condition Squaring both sides of the relation (81), we have We can see that the relation (83) is a cubic equation for δ 2 . For example, the case with δ 0 = 1 is easily solved in the exact form Of course, the above obeys the condition (82). However, the process for arriving at the exact solutions in more general cases is too troublesome to treat directly the cubic equation (83).
Then, we will try to find a possible approximate solutions. For this task, we use the solution (84).
Then, our problem is reduced to depict δ as a function of δ 0 and it is carried out by exchanging the axis δ 0 for the axis δ in Fig.3. In this way, we get Fig.4. For this task, the derivatives (dδ/dδ 0 ) δ 0 =0 = s/(s + 1) and (dδ/dδ 0 ) δ 0 =1 = (3δ 1 − 1)/(2(δ 1 ) 2 + 1) are used The reason why we set up the from (86) can be given as follows : Figure 4 suggests us that the curve OQP resembles the function sin θ in the range 0 ≤ θ ≤ π. Of course, θ may be very complicated function of δ 0 . Then, let us image a right triangle △ABC shown in As for F (δ 0 ), we adopt the following form : Here, a 0 , a 1 , a 2 and a 3 denote the constants determined under the condition that δ passes the points O, Q and P with the derivatives at the points O and Q already shown. The result is as follows : For the comparison, the approximate curve is shown together with the exact one in Fig.6. The agreement is quite good.
In Fig.1, the exact and the RPA results for the first-excited energy in the su(2)-Lipkin model are depicted. It is ready to compare our result developed in this paper with the exact and the RPA results. In Fig.7, the first-excited energy from the ground-state energy is  Here, it is interesting to see the value n ≡ (n 1 + n 0 )/2 introduced in 3. As was already mentioned, if n 0 = n 1 = 0, namely n = 0, then Eq. (52) was deduced with s = Ω. This result is nothing but the one derived in the conventional RPA as is depicted in Fig.7. However, in our renewed RPA, n 0 and n 1 are determined within the framework developed in 3, and as a result, n is obtained. In Fig.8, the behavior of n is shown numerically as a function of G. For small G closer to zero, n is nearly equal to zero. Therefore, the conventional RPA results are almost obtained as is shown in Fig.7. However, in the region before and after G ≈ 1, n moves apart from zero. Thus, it may be good to say that our approximate results for the first-excited energy in Fig.7 go away from the conventional RPA results and our renewed RPA results draw closer to the exact ones for G ≫ 1.
In conclusion, it would be said that the renewed random phase approximation developed in this paper is workable beyond the conventional RPA in all the region of the interaction strength G in the su(2)-Lipkin model.

Concluding remarks
In this paper, we have discussed two problems, which connote the su(2)-Lipkin model.
Through the discussion on the first, we gave a relation among the three quantities 2Ω, N and s. The result is depicted in Fig.2. Conventionally, the su(2)-Lipkin model has been applied only to the case with 2Ω = N = 2s, i.e., the closed shell nuclei. However, our idea supports that we can treat this model also in the case satisfying the inequality (19). In the second, preserving the framework of the conventional RPA, we developed an idea how to investigate the term neglected in the conventional treatment. The result is depicted in Fig.7, which suggests us that we must reconsider the terminology, the "phase" used in various situations. For example, the two-level pairing model involves two phases, the nonsuperconductive and the superconductive phase in RPA. Figure 1 suggests us the following : The phase, which is usually interested in and may be called the normal phase, is effective in the range 0 ≤ G/(2ǫ) · 2s ≤ 1. In the range 1 < G/(2ǫ) · 2s, we cannot see anything. This is the result based on the conventional RPA. On the other hand, our RPA gives us the frequency ω/(2ǫ) (> 0) in any range of G/(2ǫ) · 2s and, moreover, the result approaches the exact one. In this sense, we can treat the su(2)-Lipkin model only in the frame of the normal phase. The above is the second problem. If we take these two conclusions into consideration, it may be necessary to investigate another phase, which may be called the deformed phase.
With the aid of the Caylee-Klein parameters, the deformed phase in the su(2)-Lipkin model may be formulated. A new mean field may be constructed by the following canonical transformation from the original fermions (c 1µ ,c * 0µ ) to the new ones (d 1µ ,d * 0µ ) for µ = 1, 2, · · · , 2Ω : Here, (u, v) and their complex conjugate are related to the Caylee-Klein parameters, the set of which is well known in the dynamics of rigid body. They satisfy the relation |u| 2 + |v| 2 = 1 .
The set (u, v) is related with the D-function with j = 1/2 and in terms of the Euler angles (φ, θ, ψ), they can be expressed as The transformation (90) does not violate the basic framework of the model : After transformation, still the model consists of two single-particle levels with the degeneracy 2Ω and, further, the forms of the total fermion number operator, the Casimir operator and Λ ±,0 are unchanged. Of course, the Hamiltonian changes its form. Therefore, it may be an interesting problem to investigate the deformed phase. Finally, we stress that the su(2)-Lipkin model still connotes the problem to give the solution.