A possible framework of the Lipkin model obeying the su(n)-algebra in arbitrary fermion number. I --- The su(2)-algebras extended from the conventional fermion-pair and determination of the minimum weight states ---

The minimum weight states of the Lipkin model consisting of n single-particle levels and obeying the su(n)-algebra are investigated systematically. The basic idea is to use the su(2)-algebra which is independent of the su(n)-algebra. This idea has been already presented by the present authors in the case of the conventional Lipkin model consisting of two single-particle levels and obeying the su(2)-algebra. If following this idea, the minimum weight states are determined for any fermion number occupying appropriately n single-particle levels. Naturally, the conventional minimum weight state is included: all fermions occupy energetically the lowest single-particle level in the absence of interaction. The cases n=2, 3, 4 and 5 are discussed in rather detail.


§1. Introduction
In 1965, at the early stage of the studies of nuclear many-body theories, Lipkin, Meshkov and Glick proposed a schematic model for understanding of microscopic structure of nuclear collective vibration. 1) Hereafter, we will call it the Lipkin model. Naturally, it was an up-to-date problem in those days. The Lipkin model treats many-fermion system consisting of two single-particle levels with the same degeneracy as each other. In this paper, the degeneracy is denoted as 2Ω, which is positive even number. For this model, we can construct the su(2)-algebra in terms of certain bilinear forms in single-particle fermion operators under the condition that the total fermion number operator commutes with the su(2)-generators. The Hamiltonian adopted in this model is expressed as a function of these su(2)-generators. Concerning total fermion number N , the simplest case may be the following: In the absence of interaction, all fermions fully occupy energetically lower single-particle level, i.e., N = 2Ω. Following the review article by Klein and Marshalek, 2) we call this case "closed-shell" system. Conventionally, only this case has been investigated. With the aid of this model, we are able to obtain a schematic understanding of collective vibrational states of the "closed-shell" system in terms of superposition of particle-hole pair excitations. In this case, it is easy to define the particle and the hole operators.
As a natural generalization of the Lipkin model, first, Li, Klein and Dreizler 3) and typeset using PTPT E X.cls Ver.0.9 Meshkov 4) investigated the model consisting of three single-particle levels. Needless to say, this model is treated in the frame of the su(3)-algebra. Further, the generalization to the case of n single-particle levels was performed mainly by Okubo 5) and Klein. 6) The degeneracy of each level is also equal to 2Ω. Mathematical framework in this case is given by the su(n)-algebra with the condition that the total fermion number operator commutes with the su(n)-generators. Needless to say, Hamiltonian should be expressed as a function of the su(n)-generators. Hereafter, we will call it the su(n)-Lipkin model. Including the case n ≥ 3, also only the "closed-shell" system, i.e., N = 2Ω has been investigated. We guess that there exist two reasons why only the case N = 2Ω has been investigated. One of the reasons may be the following: The Lipkin model aims at describing the particle-hole pair type collective vibration and its ideal form may be expected to be realized in this case. If excessively speaking, any case except the "closed-shell" system may be not necessary to investigate. The second is related to the minimum weight state. The Lipkin model is a kind of the algebraic model. Therefore, in order to complete the description of the model, the first task is to determine the minimum weight states. The "closed-shell" system corresponds to the simplest minimum weight state which enables us to formulate various results of the Lipkin model quite easily. However, in the case of the su(2)-Lipkin model, recently, the present authors proposed an idea. 7) Under this idea, the minimum weight states can be determined in the concrete form for the case of any fermion number. The prototype of new boson realization of the su(2)-algebra in the Lipkin model used in 7) can be found in 8). This idea suggests us that we may know the concrete forms of the minimum weight states of the su(n)-Lipkin model for any fermion number. This problem will be discussed in this paper (I). However, even if the minimum weight state can be determined, we have still a problem to be solved. In the su(2)-Lipkin model, the orthogonal set built on a chosen minimum weight state can be easily obtained by operating the raising operator successively on the minimum weight state. In the case of the su(n)-Lipkin model, formally, there exist too many generators which play a role similar to that of the raising operator in the su(2)-Lipkin model. Therefore, in order to make the su(n)-Lipkin model workable, we must present any idea for the operators, the role of which is similar to that of the su(2)-Lipkin model, i.e., the raising operator. This problem will be discussed in next paper (II).
Main aim of this paper is to present concrete forms of the minimum weight states for any fermion number in the su(n)-Lipkin model including the "closed-shell" system. Preliminary argument was performed in the recent paper by the present authors for the su(2)-Lipkin model. 7) In this argument, a certain su(2)-algebra which is independent of the su(2)-algebra in the Lipkin model plays a central role. We called it as the auxiliary su(2)-algebra. The orthogonal sets obtained under this algebra give us the minimum weight states of the su(2)-Lipkin model. We extend this idea to the su(n)-Lipkin model. Condition that the auxiliary su(2)algebra is independent of the su(n)-algebra in the Lipkin model is formulated in the commutation relation [ any auxiliary su(2)−generator , any su(n)−generator in the Lipkin model ] = 0 . (1.1) For construction of this auxiliary algebra, the raising operator in the su(2)-algebra can be expressed in certain form with n-th degree for the fermion creation operators and the Clifford numbers, unfamiliar to nuclear theory. The minimum weight states of the su(n)-Lipkin model are given in terms of the orthogonal sets of the auxiliary su(2)-algebra. In this paper, the terminology, the "closed-shell" system was used for the case in which, in the absence of interaction, all fermions occupy fully energetically the lowest single-particle level, i.e., N = 2Ω. However, in order to formulate the "closed-shell" system rigorously, not only the condition N = 2Ω but also other conditions, for example, in the case of the su(2)-Lipkin model, s = Ω (s : the magnitude of the su(2)-spin for this model) are necessary. In next section, the su(n)-Lipkin model is recapitulated and the condition governing the minimum weight states is given. In §3, the su(2)-algebra auxiliary to the su(n)-Lipkin model is formulated under the condition that any of the su(2)generators commutes with any of the su(n)-generators. The three generators are expressed as functions of single-particle fermion operators. For obtaining the expressions, the Clifford number may be necessary. In §4, formal aspects of the minimum weight states of the su(2)-and the su(3)-Lipkin model are discussed. Section 5 is devoted to presenting the general forms of the minimum weight states concretely in the case of the su(n)-Lipkin model. Finally, in §6, mainly, the minimum weight states for the su(n)-Lipkin model in the cases n = 2, 3, 4 and 5 are given in the form slightly different from that presented in §5 and it will be useful for the discussion in (II). §2. The su(n)-algebra in the Lipkin model Many-fermion model discussed in this paper consists of n single-particle levels, the degeneracies of which are equal to 2Ω = 2j + 1 (j; half-integer). The singleparticle states are specified by the quantum numbers (p, jm). Here, p and m are given by p = 0, 1, 2, · · · , n − 1 and m = −j, −j + 1, · · · , j − 1, j, respectively. Hereafter, we omit the quantum number j. Following the order p = 0 < p = 1 < · · · < p = n−1, the levels becomes higher. The level p = 0 is the lowest. The fermion operators are denoted by (c p,m ,c * p,m ) and, then, the total fermion number operator N(n) for the case n can be expressed as The commutation relations are given in the form In the relation (2.3), we can see that the operators (2.2) obey the su(n)-algebra.
The simplest Casimir operator, Γ su(n) , is given as The operators Γ su(n) and N (n) satisfy [ Γ su(n) and N (n) , any of the operators (2. 2) ] = 0 . (2.5) Further, it should be noted that N (n) can not be expressed in terms of the above su(n)-generators.
For the above su(n)-algebra, we can select a Hamiltonian Here, H 0 (n) is the Hamiltonian of individual levels with energies ε p , for which we set up Then, H 0 (n) can be expressed as The part H 1 (n) is an interaction term which choose, for illustration only, in the form (2.9) Here, G(> 0) denotes the coupling constant. The above Hamiltonian can be found in Ref.
2) with the notations different from the present. We call the above many-fermion system as the su(n)-Lipkin model. The Hamiltonian H(n) obeys Γ su(n) and N (n) , H(n) = 0 . (2.10) The cases n = 2 and 3 reduce to the Hamiltonians of the su(2)-and the su(3)-Lipkin model which have been discussed in various problems. 2) For studies of any many-fermion system, implicitly or explicitly, we must prepare orthogonal sets for the system under investigation. Standard idea for treating the present model may be, first, to prepare an orthogonal set related a chosen minimum weight state. The set may be constructed by operating the generators S p (n) (p = 1, 2, · · · , n − 1) and S p q (n) (p > q = 1, 2, · · · , n − 2) appropriately on the minimum weight state, which we denote |min(n) . The state |min(n) obeys the conditions N (n)|min(n) = N n−1 |min(n) , (2.11) Conventionally, for |min(n) , a "closed-shell" system has been investigated: (p = 1, 2, · · · , n − 1) (2.14) The above teaches us that the level p = 0 is fully occupied and the levels p = 1, 2, · · · , n − 1 are vacant. However, even if the treatment is restricted to the "closed-shell" system, there exist many "closed-shell" systems in the case n ≥ 4, for example, the levels p = 0 and 1 are fully occupied and the other vacant: Including such "closed-shell" systems, it may be interesting to investigate the case with arbitrary fermion number, i.e., 0 ≤ N n−1 ≤ 2nΩ. Further, for constructing the orthogonal set built on |min(n) , appropriate choice of the operators as functions of S p (n) (p = 1, 2, · · · , n − 1) and S p q (n) (p > q = 1, 2, · · · , n − 2) is inevitable. The simplest examples are given by S p (n)|min(n) and S p q (n)|min(n) . However, S p q (n) S q (n)|min(n) and S q (n) S p q (n)|min(n) are not independent of each other, because of the relation [ S p q (n) , S q (n) ] = S p (n). We call the operators appropriately chosen as the building blocks. The above argument tells us that, as was mentioned in §1, we have two tasks for formulate the present model: (1) One is to determine the minimum weight state and (2) the other is to construct the building blocks. Although these two are interrelated with each other, the concrete contents are completely independent of each other. Therefore, after discussing the task (1) in (I), we will treat the task (2) in (II).
Main aim of this paper is to present an idea, under which the minimum weight states of the su(n)-Lipkin model are systematically constructed. In order to make our idea understandable, we show the single-particle level scheme in denote a possible candidate of the minimum weight state of the su(ν)-Lipkin model for 2 ≤ ν ≤ n. We set up the following relations for |min(ν) : The total fermion number N ν−1 is expressed as It may be necessary to give some comment on the relations (2.18)∼(2.20). The definitions of N (ν) and S p p (ν) shown in the relations (2.1) and (2.2b), respectively, for the case n = ν are rewritten in the form The relation (2.21) tells us the following: Since the state |min(ν) is regarded as the eigenstate of N (ν) and S p p (ν), |min(ν) should be also the eigenstate of mc * 0,mc 0,m and mc * p,mc p,m . Then, the relation (2.19) may be permitted to set up and the relations (2.18) and (2.20) are obtained.
The relations (2.16)∼(2.20) are set up for the range 2 ≤ ν ≤ n. However, it may be convenient for later arguments to add the point ν = 1 to 2 ≤ ν ≤ n. Judging from Fig.1, it may be natural to consider that the case ν = 1 may be restricted only to p = 0. Then, in this case, the relations (2.17) and (2.18) are meaningless and the relations (2.16) and (2.19) may be meaningful: Here, N (1) is given by the relation (2.1) for n = 1 in the form Let |min(ν) be obtained. Then, we can show that |min(ν) for ν = 2, 3, · · · , n satisfies the relation The reason is very simple. Since any fermion does not occupy the single-particle levels p = ν, ν + 1, · · · , n − 1, we havẽ The relations (2.23)∼(2.25) teach us that |min(ν) as the solution of Eqs.(2.16)∼(2.18) is also the minimum weight state of the su(n)-Lipkin model. In next section, we will discuss the su(2)-algebra ( Λ ±,0 (n)), which plays a central role for obtaining the state |min(ν) . §3. The su(2)-algebra auxiliary to the su(n)-Lipkin model As was mentioned in 1, an idea preliminary to the present one has been already shown in our recent paper for the case of the su(2)-Lipkin model. 7) The basic idea is to introduce the su(2)-algebra ( Λ ±,0 (2)), which is characterized by the commutation relation any of Λ ±,0 (2) , any of the su(2)−generators S 1 (2), S 1 (2), S 1 1 (2) = 0 . (3.1) The explicit forms are as follows: It is easily verified that the expression (3.2) satisfies the condition (3.1) and obeys the su(2)-algebra: In our idea, ( Λ ±,0 (2)) plays a central role in deriving the minimum weight state with arbitrary fermion number in the su(2)-Lipkin model. Conventionally, only the case of the fermion number 2Ω has been treated, i.e., the "closed-shell" system. In §4, for illustration of our idea, we will discuss how ( Λ ±,0 (2)) is used in our present problem including the case of the su(3)-Lipkin model. In the form similar to the relation (3.2), we can give the su(2)-algebra ( Λ ±,0 (n)) which is independent of the su(n)-Lipkin model: For constructing ( Λ ±,0 (n)), first, we must have a preliminary argument. We know that system composed of one kind of fermion is regarded as single su(2)-spin system with the magnitude 1/2. Through the following commutation relation, we can understand this point: Here, (c * ,c) denotes fermion operator obeying the anti-commutation relation The anti-commutation relation (3.6) leads us to the relation (3.5). The fermion operatorsc * andc play a role of the raising and the lowering operator, respectively. However, the form (3.5) cannot be straightforwardly translated into the case of many-fermion system, for example, the system specified by p = 0 in this paper: The first of the relation (3.7) is rewritten to The form (3.8) suggests us that it may be impossible to regardc * 0,m as the raising operator of many su(2)-spin system as it stands.
Let us discuss a possible idea for the above problem. Under this idea, the present many-fermion system can be regarded as that composed of independent 2Ω su(2)spins. Each is specified by m and its magnitude is equal to 1/2. This idea is realized through introducing the Clifford numbers e m (m = −j, −j + 1, · · · , j − 1, j) which obey the condition Of course, e m commutes with the fermion operators. With the use of e m , we define the following operators:d * With the aid of the anti-commutation relation (3.7) and the property of the Clifford number (3.9), we can derive the following relation * ) for (d * 0,m ,d 0,m ): In contrast to the form (3.8), we can see that the symbol δ mµ is attached to both of the two terms on the right-hand side of the relation (3.11b). Therefore, the relation (3.11) suggests us that the present many-fermion system consists of 2Ω su(2)-spins which are independent of one other and the generators of the m-th spin are given by The total spin of the present system, ( Λ ±,0 (1)), can be expressed in the form Of course, they obey the su(2)-algebra: 11b) can be derived through the following process: We can treat the eigenvalue problem of ( Λ ±,0 (1)), which will be discussed in 4 in reference to the su(2)-and the su(3)-Lipkin model. The su(2)-algebra (Λ ±,0 (2)) given in the relation (3.2) is expressed as Here, we used (e m ) 2 = 1 and (d * Properties of the above operators are summarized as follows * ) : We are now possible to give explicit forms Λ ±,0 (n). First, we define the following operators:d * Judging from the expressions (3.12) and (3.14), it may be natural to set up the following form for ( Λ ±,0 (n)): The second of the relation (3.15a) can be derived through the following process: It should be noted that the su(2)-algebra ( Λ ±,0 (n)) is extended from the fermionpair for p = 0 and 1 ( Λ ±,0 (2)). With the use of the relations (3.18) and (3.19), we can show that Λ ±,0 (n) obey the su(2)-algebra: Next, we will give the proof of the commutation relation (3.4). For this aim, we express the su(n)-generators (2.2) in the unified form  In this way, we could show that the expression (3.20) satisfies the relation (3.4). In next section, the expressions of Λ ±,0 (2) shown in the relation (3.14) and Λ ±,0 (3) shown in the following play a central role:  In order to illustrate our idea, let us start with the su(2)-Lipkin model. We denote one of the states in which only the single-particle level p = 0 is occupied by N 0 fermions as |N 0 : Here, we omitted any quantum number which does not connect with the algebras under consideration. It is easily verified that |N 0 is a possible candidate of the minimum weight states of the su(2)-Lipkin model: Comparison of the relations (4.1) and (4.2) with (2.24), (2.15a) and (2.26) gives us, for the case (n = 2, ν = 1, p = 1): An example of |N 0 is presented in Appendix.
The state |N 0 is also the minimum weight state of the su(2)-algebra ( Λ ±,0 (2)): Therefore, by operating Λ + (2) successively on |N 0 , we are able to obtain the states orthogonal to |N 0 in the form The state |N 1 , N 0 satisfies The state |N 1 , N 0 is also the minimum weight state of the su(2)-Lipkin model with the same property as that shown in the relation (4.2). But, it is not the minimum weight state of the su(2)-algebra ( Λ ±,0 (2)). For |min(2) = |N 1 , N 0 , we obtain the following: Inversely, we have The above relations lead us to the inequalities Since Λ ±,0 (2) obey the su(2)-algebra, the relations (4.4b) and (4.8b) give us the following inequalities: Fermion numbers in the single-particle levels p = 0 and p = 1 are given in the relation (4.10) and, then, we have 14) For the relation (4.14), we should note that S 1 (3) = S 1 (2) and, further, |N 1 , N 0 does not contain any fermion in the level p = 2 and S 2 (3) and S 1 2 (3) contain the annihilation operator in p = 2. Although |N 1 , N 0 is not minimum weight state of ( Λ ±,0 (2)), it is the minimum weight state of ( Λ ±,0 (3)): is also the minimum weight state of the su(3)-Lipkin model. It may be clear from the relations (4.13)∼(4.16). Then, we introduce the state |N 2 , N 1 , N 0 in the form The state |N 2 , N 1 , N 0 satisfies The state |N 2 , N 1 , N 0 is also the minimum weight state of the su(3)-Lipkin model with the same property as that shown in the relation (4.14) and (4.15). But, it is not the minimum weight state of the su(2)-algebra ( Λ ±,0 (3)). For |min(3) = |N 2 , N 1 , N 0 , we obtain the following: Inversely, we have The above relations lead us to The operators Λ ±,0 (3) obey the su(2)-algebra and, then, the relations (4.15) and (4.21b) lead us to the following inequalities: The relations (4.24a) and (4.24b), together with the inequality in the relation (4.23), are rewritten as The relation (4.23) gives us In last section, we discussed the cases of the su(2)-and the su(3)-Lipkin model. As are given in the relations (4.4) and (4.16), |N 0 and |N 1 , N 0 are the minimum weight states of ( Λ ±,0 (n)) for n = 2 and 3, respectively. The example of |N 0 and the explicit form of |N 1 , N 0 are shown in the relation (A . 7) and (4.5), respectively. These two forms suggest us the following form: If we adopt the form (A . 7), the state (5.1a) can be expressed as Hereafter, we will use only the form (5.1a). Therefore, our treatment is valid for n ≥ 3. If the form (5.1) is accepted, the minimum weight state of the su(n)-Lipkin model may be given as First, let us prove the relation For this aim, some preliminary argument is necessary. For the case ν < n, the operatord m (n) introduced in the relation (3.15) can be factorized into the form The operator (δ * m (n, ν),δ m (n, ν)) satisfies The relation (5.6) may be self-evident, because (δ * m (n, ν),δ m (n, ν)) andd * µ are composed from the operators different of each other. It can be seen in the relation (5.5). The operator Λ − (n) is expressed as Then, with the use of the relations (3.19) and (5.6), we have Successive use of the relation (5.8) and the condition Λ − (n)|N 0 = 0 lead us to the relation (5.3). Next, we consider that the state |N n−2 , N n−3 , · · · , N 1 , N 0 is the eigenstate of Λ 0 (n) and its eigenvalue should be obtained. The relations (3.20b), (5.4) and (5.6) lead us to Λ 0 (n) in the following form: In order to calculate [ Λ 0 (n) , Λ + (ν) ], we must use the relation For the derivation of the relation (5.11), we used the relations (3.18) and (3.19).
With the use of the relations (5.8b) and (5.11), we obtain the following: Successive use of the relation (5.12) gives us the relation Here, we used the relation (5.8b) and Thus, we learned that |N n−2 , N n−3 , · · · , N 1 , N 0 is the minimum weight state of ( Λ ±,0 (n)). The (N n−1 − N n−2 )/n-time operation of Λ + (n) on this minimum weight state, we have the form (5.2): |N n−1 , N n−2 , N n−3 , · · · , N 1 , N 0 = Λ + (n) N n−1 −N n−2 n |N n−2 , N n−3 , · · · , N 1 , N 0 Next, we will show that the state (5.15) is the minimum weight state of the su(n)-Lipkin model. First, the following relations are derived from the relation (3.15): With the use of the relation (5.17), we have Noting the relations S p (n)|N 0 = 0, S q p (n)|N 0 = 0 and S p p (n)|N 0 = −N 0 |N 0 , we can show that the state (5.15) is the minimum weight state of the su(n)-Lipkin model: Thus, we could find the minimum weight state for the general case.
In the relations (4.11), (4.12), (4.25) and (4.26), we showed the inequalities, which the fermion numbers N ν−1 and γ ν−1 (p) in the cases of the su(2)-and the su(3)-Lipkin model should satisfy. As final remark of this section, we will give the inequalities for the general case. First, the relation between N n−1 and γ n−1 (p) for the su(n)-Lipkin model must be discussed. The minimum weight state |min(n) = |N n−1 , N n−2 , · · · , N 1 , N 0 shown in the relation (5.16) gives us the following relation: The relation (5.22) is written compactly as The relation (5.23) is inversely expressed as γ n−1 (p) − (ν + 1)γ n−1 (ν + 1) , (ν = 0, 1, · · · , n − 2) (5.24a) The form (5.24b) is nothing but the relation (2.20). We can rewrite (5.22) to the following: The right-hand side of the relation (5.25) should be zero or positive and, then, we have At the present, the upper limit cannot be determined. For the determination of the upper limit, we note that ( Λ ±,0 (n)) obeys the su(2)-algebra and the relations (5.13) and (5.16) give us the following inequalities: .
The relation (5.28) combined with the relation (5.26) lead us to , (n = 2, 3, · · · )(5.31) For the relations (5.29) and (5.30) for the cases n = 2 and 3 reduce to the relations (4.11) and (4.25), respectively. The inequality (5.29a) leads us to the following: For the derivation, we used the relation (5.23). Then, we have Thus, we could present the minimum weight state of the general case. It should be noted that all relations given in this section are available for n ≥ 3. §6. Discussions Until the present stage, we developed a possible idea how to give concrete expressions of the minimum weight states for the su(n)-Lipkin model in arbitrary fermion number. In this section, we will treat some simple examples of the minimum weight states from a viewpoint slightly different from that in last section. This argument is also in preparation for next paper (II). Our discussion starts in to mention that the su(n)-Lipkin model contains the su(2)-subalgebras. Its number depends on the number n. In this section, we will discuss the cases n = 2, 3, 4 and 5. The case n = 2 is the su(2)-algebra itself and the case n = 3 has one su(2)-subalgebra. On the other hand, the cases n = 4 and 5 contain two su(2)-algebras. One by one, we will show this point.
In the case n = 2, S 1 (= S + ), S 1 (= S − ) and S 1 1 /2(= S 0 ) form the su(2)-algebra and Γ su(2) is given as (6.1) The above is nothing but the original Lipkin model. The minimum weight state |min(2) is specified by two quantum numbers N and s, the eigenvalues of N and − S 0 : |min(2) = |N ; s . Of course, these two are related to the algebra. Then, for the orthogonal set, we have In this case, we obtain the relation Then, with the use of the inequality (4.12), we can show that the relation (6.3) holds in the following domains: Fig. 2. The relation between s and N is shown in the inequality (6.4).
The above domains can be illustrated in Fig.2. A "closed-shell" system appears in the case (N = 2Ω, s = Ω), where, in the absence of interactions, the level p = 0 is occupied fully by the fermions and the level p = 1 is vacant. The point C in Fig.2 corresponds to the "closed-shell" system. But, s can decrease from s = Ω to s = 0, where the level p = 0 and p = 1 are occupied in equal fermion number Ω. Next, we treat the case n = 3. The operators S 2 1 (= S + ), S 1 2 (= S − ) and ( S 2 2 − S 1 1 )/2(= S 0 ) form the su(2)-subalgebra and, further, we have the scalar R 0 with respect to ( S ±,0 ) in the form The Casimir operator Γ su(3) is expressed as In addition to N , |min(3) can be specified by the eigenvalues of S 0 and R 0 , −σ and −ρ, respectively: |min(3) = |N ; ρ, σ . Then, we have Therefore, for constructing the orthogonal sets, we, further, must take account of ( S 2 , S 1 ). It will be discussed in (II). In the present, we have the relation The inequality (4.23) leads us to the following domains for the relation (6.8).