A possible framework of the Lipkin model obeying the su(n)-algebra in arbitrary fermion number. II --- Two subalgebras in the su(n)-Lipkin model and an approach to the construction of linearly independent basis ---

Standing on the results for the minimum weight states obtained in the previous paper (I), an idea how to construct the linearly independent basis is proposed for the su(n)-Lipkin model. This idea starts in setting up m independent su(2)-subalgebras in the cases with n=2m and n=2m+1 (m=2,3,4,...). The original representation is re-formed in terms of the spherical tensors for the su(n)-generators built under the su(2)-subalgebras. Through this re-formation, the su(m)-subalgebra can be found. For constructing the linearly independent basis, not only the su(2)-algebras but also the su(m)-subalgebra play a central role. Some concrete results in the cases with n=2, 3, 4 and 5 are presented.


§1. Introduction
This paper, (II), is a continuation of (I) 1) and we will mainly discuss an idea, with the aid of which the linearly independent basis is constructed for the su(n)-Lipkin model. As is well known, the su(n)-Lipkin model is a classical model for describing certain phenomenon observing in nuclei, 2) . 3) It is a kind of the Lie algebraic model. Therefore, first task, which is inevitable for investigating this model, is construction of the minimum weight states. By the definition of the minimum weight states, this task may be performed independently of any Hamiltonian expressed as a function of the generators. In (I), we proposed an idea how to construct the minimum weight states for the case with any total fermion number in n single-particle levels. A certain su(2)-algebra, which we called the auxiliary su(2)-algebra in (I), plays a central role for the minimum weight states. The raising operator in this auxiliary algebra is expressed in the form with n-th degree for the fermion creation operators and the Cliford numbers.
After determining the minimum weight states, it may be natural to proceed to construct the linearly independent basis by operating various operators on the minimum weight states such as the raising operator in the su(2)-algebra. However, the following should be noted: Generally, it is impossible for determining the orthogonal sets to prepare sufficient number of the hermitian operators, which commute with one another. Therefore, in this paper, we will aim at constructing the linearly independent basis, and, then, for example, with the aid of the Schmidt method, we are able to obtain the orthogonal sets. In order to realize this aim, first, we pay attention to the point that, in the su(n)-Lipkin model, we can define m independent su(2)subalgebras. Here, m is an integer related to n under the relation n = 2m (n:even) and n = 2m + 1 (n:odd), where m = 2, 3, · · · . Then, by adding these su(2)-algebras, we get the su(2)-algebra. With the use of this su(2)-algebra, we can classify all the generators into the scalars, the vectors and the spinors. And, we can prove that the set of the scalars gives us the su(m)-algebra. Therefore, the remaining vectors and spinors may be treated in the space given under the su(2)-and the su(m)-algebra. The above may be nothing but the re-formation of the su(n)-Lipkin model.
For a given value of the integer m (m = 2, 3, · · · ), we have two Lipkin models: the su(2m)-and the su(2m + 1)-algebra. As a general argument, we know that the minimum weight states of the su(n)-algebra are specified by (n − 1) quantum numbers. In these numbers, m quantum numbers are related to m su(2)-subalgebras and, then, the remaining ones are (m − 1) for n = 2m and m for n = 2m + 1, respectively. In these numbers, (m − 1) quantum numbers may be related to the minimum weight states of the su(m)-algebra. Therefore, in the case of the su(2m)algebra, the number (n − 1) is decomposed into m + (m − 1). In the case of the su(2m+1)-algebra, the number (n−1) is decomposed into m+(m−1)+1, where one quantum number is excess. We will show that this extra number is closely related to the spinor operators, because the spinors appear only in the case with n = 2m + 1. However, these features may be almost impossible to show in general case, and, then, we will discuss them through the cases with n = 2, 3, 4 and 5. The minimum weight states of these four cases were presented in (I) under the re-formed version. The cases with n = 4 and 5 correspond to m = 2 and they can be treated quite easily, because of m = 2, i.e., the su(2)-algebra.
After giving m su(2)-algebras, in §2, the spherical tensor operators (the scalar, the vectors and the spinors) are defined. In §3, the Casimir operator as a quadratic form for the generators is diagonalized and through this procedure, the operators specifying the minimum weight states are introduced. Of course, the Hamiltonian given in (I) is rewritten in terms of the spherical tensors. In §4, our scheme for obtaining the linearly independent basis is presented. Following our scheme, in §5, the cases with n = 2, 3, 4 and 5 are treated. In §6, as a concluding remark, the connection between the su(2m)-and the su(2m + 1)-Lipkin model is discussed. In Appendix A, the su(m)-subalgebra in the su(n)-Lipkin model is presented in detail. Including the results given in (I), various formal aspects of the Lipkin model is also stressed in (II). §2. The su(n)-generators of the Lipkin model in spherical tensor form In §6 of Part I, we gave some examples of the re-formed generators of the Lipkin model. Following these examples, we will treat the present model under two cases: (i) n; even and (ii) n; odd. In the case (i), we introduce the integer m through n = 2m ; n = 2, 4, · · · , i.e., m = 1, 2, · · · . (2.1) We also treat the case (ii) by the integer m, which is given by n = 2m + 1 ; n = 3, 5, · · · , i.e., m = 1, 2, · · · . (2.2) In (I), we learned that the case with m = 1 corresponds the su(2)-and the su(3)-Lipkin model. The former is the su(2)-algebra itself and the latter contains one su(2)-subalgebra. On the other hand, the su(4)-and the su(5)-Lipkin models corresponding to the case with m = 2 contain two su(2)-subalgebras. Extending this aspect, we define m independent su(2)-subalgebras. In the case with n = 2m, the following is given: The case with n = 2m + 1 gives us the form We can see that the cases with (m = 1, r = 1) and (m = 2, r = 1, 2) reduce to the form treated in (I) for the cases with n = 2, 3, 4 and 5. Of course, we have the relation [ S 0 (m; r) , S ± (m; r ′ ) ] = δ rr ′ · ± S ± (m; r) . (2.5) In terms of total sum of S ±,0 (m; r) for r, we obtain the su(2)-algebra: In our re-formation, the su(2)-algebra (2.6) will play a role for classifying all generators in terms of the spherical tensor representation. Main task of §2 is to express all the generators except the forms (2.3) and (2.4) in spherical tensor forms. For this task, it may be convenient to introduce two integers r and k: (i) n ; even (n = 2m) , r = 1, 2, · · · , m − 1, k = r + 1, r + 2, · · · , m, (2.7) (ii) n ; odd (n = 2m + 1) , r = 1, 2, · · · , m − 1, m, k = r + 1, r + 2, · · · , m. (2.8) In the case (ii), if r = m, k becomes meaningless. In the case with n = 2m, S p and S p q (p > q) can be re-formed to scalars and vectors: (α)scalars, In the case with n = 2m + 1, spinors appear: (γ)scalars, (2.16) R 1,−1 (m; r, k) = S 2m−2r+1 2m−2k+2 . r = 1, 2, · · · , m − 1, k = r + 1, r + 2, · · · , m (2.17) The forms (2.13)∼ (2.17) are also for S p and S p q (p > q). The expressions (β) and (ǫ) give us scalars in the form Hermitian conjugate of R µ,ν (m; r, k) is denoted as For the present, the type S p p is re-formed as follows: (i) n = 2m; The operators P 0 (m; r) should be compared with S 0 (m; r). It is noted that we have the relation P 0 (m; m) = S 0 (m; m) in the case (i) and P 0 (m; r) (r = 1, 2, · · · , m − 1, m) are not scalars, but in the case (ii), P 0 (m; r) (r = 1, 2, · · · , m − 1, m) are scalars. Later, we will contact with this point. Total number of S p (n) (p = 1, 2, · · · , n−1) and S p q (n) (p > q = 1, 2, · · · , n−2) is equal to On the other hand, total number of S + (m; r), R 0,0 (m; r, k) and R 1,ν (m; r, k) (ν = ±1, 0) in the case with n = 2m is equal to In the case with n = 2m + 1, total number of S + (m; r), R 0,0 (m; r, k), R Total number of S p p (n) (p = 1, 2, · · · , n − 1) is equal to (n − 1). In the case with n = 2m, total number of S 0 (m; r) and P 0 (m; r) is equal to m + m = 2m = n . (2.25) We have the difference n − (n − 1) = 1. The reason why we have this difference comes from the double counting for (1/2) S 1 1 . Later, we will show that this double counting does not give us any trouble. In the case with n = 2m + 1, total number of S 0 (m; r) and P 0 (m; r) is equal to m + m = 2m = n − 1 . (2.26) Thus, we re-formed the su(n)-generators in the Lipkin model in the form of the spherical tensor representation for the su(2)-algebra ( S ±,0 (m)). Finally, we give a comment. We presented a possible re-formation of the original representation of the su(n)-Lipkin model. Concerning the su(2)-subalgebra, it starts in the expressions (2.3) and (2.4). However, it must be noted that these forms are not unique. If the indices i = 1, 2, · · · , n − 1 are changed from the forms (2.3) and (2.4). we obtain various forms. For example, in the case with n = 2m = 4, S ±,0 (m; r) in the relation (2.3) can be expressed as For the form (2.27), we exchange the indices i = 1 and 2 and, then, they becomes The form (2.28) also obeys the su(2)-algebra. §3. The Casimir operator in the present representation and its diagonalization In this section, mainly we will discuss the Casimir operator, Γ su(n) , shown in the relation (I.2.4): We can see in the form (3.1) that Γ su(n) , which is of the quadratic form with respect to the su(n)-generators, is not diagonal. Then, we will consider the diagonalization. The operator Γ su(n) can be rewritten as The part S(m; r) 2 denotes the Casimir operator of the su(2)-subalgebra ( S ±,0 (m; r)) and Γ m is given separately in the cases with n = 2m and 2m + 1: The part Γ (2) m is expressed as The operators Γ m and Γ (2) m are of the quadratic. However, Γ m is diagonalized, but Γ (2) m is not diagonalized. The above is our re-formation of Γ su(n) .
The operator Γ (2) m can be diagonalized in the following: i.e., It should be noted that the second term on the right-hand side of the relation (3.6) disappears in the case with n = 2m. The operator Q 0 (m; r) is obtained from P 0 (m; r) by the orthogonal transformation Here, (C r ′ ,r (m)) denotes an orthogonal matrix: In this paper, we will adopt the following form for C r ′ ,r (m): Here, C r ′ ,r (m) except for the phase factor (−) r−r ′ denotes the Clebsch-Gordan coefficient of the type j 1 , m 1 , j 1 , −m|j 2 , 0 and it is given as Later, F r ′ ,r (m) will be used. For example, we have The relations (3.8) and (3.9) lead us to the expression (3.6). We note that Q 0 (m; r) for r = 1, 2, · · · , m−1 are scalar in the case with n = 2m. Later, it will be shown. In the case with n = 2m, Q 0 (m; m) disappears in Γ (2) m , because of the factor (n−2m) in the relation (3.6). In this case, in spite of the relation P 0 (m; m) = S 0 (m; m), we have treated these two as if they are independent of each other. As was already mentioned in the relation (2.25), total number of P 0 (m; r), i.e., Q 0 (m; r) becomes equal to that of S 0 (m; r). Originally, total number of S p p (n) is equal to n−1, i.e., 2m−1 and in this number, the integer m is fixed as total number of S 0 (m; r). Therefore, total number of Q 0 (m; r) should be reduced to m − 1. We can see in the relation (3.7a) that Q 0 (m; m) disappears, in other word, only Q 0 (m; r) (r = 1, 2, · · · , m − 1) appear. The case with n = 2m + 1 does not contain such a problem, because of n = 2m + 1, i.e., n − 1 = 2m. Now, let us derive the formulae for obtaining the eigenvalues of Γ su(n) . For this aim, we must rewrite (1/2) · [ S + (m; r) , S − (m; r) ] + and other terms of the anticommutators to the forms S + (m; r) S − (m; r) and others. As the result, the linear terms for the generators appear. The part m r=1 S(m; r) 2 shown in the relation (3.3) is simple: (3.12) The linear terms coming from the relation (3.4), which we denote as Γ (1) m can be expressed in the form m can be expressed as We can see that in the case with n = 2m, Q 0 (m; m) disappears also in Γ m ) can be expressed in the following form: By changing the scale of Q 0 (m; r), we can eliminate the irrational numbers contained in Q 0 (m; r): The linear terms indicate the quantum fluctuations. We see that the fluctuations appear only at the points r = m and m − 1. If the orthogonal matrix (3.10) is not adopted, they may spread over various points of r. In the case with n = 2m, P 0 (m; r) (r = 1, 2, · · · , m − 1, m) is not scalar. But, it can be easily proved that Q 0 (m; r), i.e., R 0 (m; r) (r = 1, 2, · · · , m − 1) is scalar and we see that, in Γ (0) m , R 0 (m; m) in the case with n = 2m disappears. With the use of the results in (I), we can calculate the eigenvalues of R 0 (m; r) for a given minimum weight state.
Let us show that Q 0 (m; r) for r = 1, 2, · · · , m − 1 in the case with n = 2m is scalar. For example, we have Then, the relation (3.18) with (3.9) and (3.11a) leads us to In the case with n = 2m + 1 for r = 1, 2, · · · , m − 1, m, we have Other cases with S − (m) and S 0 (m) are also in the same situation as the above. Further, for r = 1, 2, · · · , m − 1 in the case with n = 2m, the following relation is derived: Here, P 0 (m; r) is defined as Therefore, in place of using Q 0 (m; r), it may be permitted also to use P 0 (m; r). In parallel to the case with n = 2m, we also use P 0 (m; r) in the case with n = 2m + 1: However, the relation (3.21) does not hold in the case with n = 2m + 1. In (I), we showed an example of the Hamiltonian: The term H 0 (n) is the Hamiltonian of individual levels with energies ǫ p : The part H 1 (n) is the interaction term: For the above rewriting, the relation (3.25b) is used. In the case with n = 2m + 1, we have Then, we obtain (3.30) The term H 1 (n) is rather simple. In the case with n = 2m, we have In the case with n = 2m + 1, we obtain In (I), we investigated the structure of the minimum weight states of the su(n)-Lipkin model. In 6 of (I), we sketched the minimum weight states in the representation developed in the present paper (II) for the cases with n = 2, 3, 4 and 5. Although our final aim of this section is to present our idea for constructing the linearly independent basis, the first task is to generalize the above four cases to the case with arbitrary n.
In the notation familiar to the su(2)-algebra, σ 0 (r) is expressed as σ 0 (r) = σ r + σ r 0 . (4.8) Of course, with the use of the angular momentum coupling rule, we are able to obtain the eigenstate of ( S ±,0 (m)). Next, we consider some problems related to the quantum number ρ r . We notice R 0,0 (m; r, k) defined in the relations (2.9) and (2.13), their hermitian conjugates, R 0,0 (m; r, k) and R 0 (m; r) introduced in the relation (3.17a) form the su(m)-algebras for the cases with n = 2m and n = 2m + 1. The detail is given in A. First, we show that the state (4.6) is the minimum weight state of the su(m)-algebra. For this aim, the following relations for the cases with n = 2m and 2m + 1 are useful: The relations (4.10) and (4.12) tell us that the state (4.6) is the minimum weight state of the su(m)-algebra.
Since we get the minimum weight states of the present su(m)-algebra, we will consider R 0,0 (m; r, k) operated on these states. First, we define the following raising operator: (4.13) λ 0 = λ 0 (1, 2), λ 0 (1, 3), · · · , λ 0 (1, m), λ 0 (2, 3), λ 0 (2, 4), · · · , λ 0 (2, m), · · · , λ 0 (m − 1, m) . (4.14) It should be noted that, since the operators R 0,0 (m; r, k) are not mutually commuted, the ordering of their product is fixed beforehand. Total number of λ 0 is equal to m(m − 1)/2. Operating R 0,0 (m; λ 0 ) on the minimum weight states (4.6), we define the states |N ; λ 0 , ρ, σ 0 = R 0,0 (m; λ 0 )|N ; ρ, σ 0 , (4.15a) |N ; λ 0 , ρ, ρ m , σ 0 = R 0,0 (m; λ 0 )|N ; ρ, ρ m , σ 0 . (4.15b) The states (4.15) are the eigenstates of R 0 (m; r) given in the relation (4.2). After lengthy calculation, we can derive the commutation relation The relation (4.16) is only applicable to the case with n = 2m + 1. With the use of the relation (4.16), we can show that the states (4.15) are the eigenstates of R 0 (m; r), the eigenvalues of which we denote ρ r 0 (r = 1, 2, · · · , m − 1, m): In a form similar to that given in the relation (4.4), we will use the notation ρ 0 : Let us pick up two states specified by λ 0 (1) and λ 0 (2) with λ 0 (1) = λ 0 (2). If these two give us ρ 0 (1) and ρ 0 (2), respectively, with ρ 0 (1) = ρ 0 (2), these two states are orthogonal to each other. But, if ρ 0 (1) = ρ 0 (2), they are not always orthogonal. However, they are linearly independent of each other, because the states (4.15) are specified by λ 0 , the number of which is m(m − 1)/2. It is equal to total of the quantum numbers characterizing the su(m)-algebra except the minimum weight states. In such cases, we must adopt some idea, for example, the Schmidt method for constructing the orthogonal set. We must, further, investigate the role of the vectors and the spinors for constructing the linearly independent basis. These are defined in the relations (2.10)∼(2.12) and (2.14)∼(2.17). A simple idea for this task is to construct the raising operators by appropriate products of the vectors and the spinors, which are similar to the forms (4.5) and (4.13). With the help of this procedure, we can construct the linearly independent basis, which may be justified in the relations (2.23) and (2.24). However, for this task, we have a problem to be reconsidered. As was already mentioned, our present system contains two subalgebras, i.e., the su(2)-and the su(m)-algebra. In our form, the vectors and the spinors are introduced as the tensor operators in the su(2)-algebra. Until the present, we do not have contacted with the argument on the relation between the tensors and the su(m)-algebra. Therefore, not only from the side of the su(2)-algebra, but also from the side of the su(m)-algebra, it should be reconsidered. But, this consideration may be impossible to perform for the general case with arbitrary m. In next section, some concrete results for the cases with n = 2, 3, 4 and 5 will be presented. On that occasion, we will have to contact with this point. §5. Some simple examples and discussion In 6 of (I), we showed the minimum weight states of the cases with n = 2, 3, 4 and 5. In this section, mainly, for the above cases, we will show the results obtained in this paper. First, will treat the su(2)-Lipkin model, i.e., n = 2 and m = 1. In this case, only one case with r = 1 appears and the index k is unnecessary. Three generators are given by The orthogonal set {|ss 0 } is given in the relation Hereafter, we will omit the total fermion number N and the notations used in 6 of (I) will be adopted, for example, S ±,0 in the relation (5.1).
Of course, σ = |σ 1 − σ 2 |, · · · , (σ 1 + σ 2 ). Therefore, the set composed of the states (5.28) may be the linearly independent basis for the su(4)-Lipkin model. Finally, we will discuss the case with n = 5, i.e., m = 2. This case contains also two su(2)-algebra: (2)). (5.29b) Then,the addition of the two is expressed as The scalar operators are as follows: In the present case, two spinors appear: The operators P 0 (2; 1) and P 0 (2; 2) can be expressed as i.e., R 0 (2) = P 0 (2; 1) + P 0 (2; 2) = 1 2 S 4 4 + S 3 3 + S 2 2 + S 1 1 . (5.37b) With the use of the above relations, we will show the results for the case with n = 5 simply in comparative argument with the case with n = 4. In (I), the minimum weight state in the present case is given by |ρ 1 , ρ 2 , σ 1 , σ 2 , where ρ 1 , ρ 2 , σ 1 and σ 2 denote the eigenvalues of R 0 (1) (= R 0 ), R 2 (2), S 0 (1) and S 0 (2), respectively. The set ( R ±,0 ) also forms the su(2)-algebra obeying the same relation as that given in the relation (5.16) and there exists also one set of the vector for ( S ±,0 ). Therefore, the treatment for these two may be similar to the case with n = 4. The difference of these two can be found in the appearance of the hermitian operator R 0 (2) and two sets of the spinors. Under the same idea as that in the case with n = 4, we introduce R 1 2 ,µ; 1 2 ,ν : Then, we can define the following scalar operator for ( S ±,0 ) and ( R ±,0 ): The above should be compared with the relation (5.23). Further, we have the operator R j,j 0 ;j,κ 0 : The operators (5.39) and (5.40) satisfy R 0 (2) , R 0,0;0,0 n 0 R j,j 0 ;j,κ 0 = 5(n 0 + j) R 0,0;0,0 n 0 R j,j 0 ;j,κ 0 . (5.41) In this connection, we have R 0 (2) , R l,l 0 ;l,λ 0 = R 0 (2) , R + = R 0 (2) , S + (1) = R 0 (2) , S + (2) = 0 . (5.42) Of course, R l,l 0 ;l,λ 0 is obtained by replacing (− S 3 ) with S 4 in the relation (5.25). The above operators give us the following state: We can see that the state (5.43) is expressed in terms of fourteen parameters including ρ 1 , ρ 2 , σ 1 and σ 2 . By using the angular momentum coupling rule, the state (5.43) can be rewritten to the form with the eigenvalues of S 2 , S 0 , R 2 and R 0 , which is similar to the state (5.28). Of course, the states (5.43) form the linearly independent basis for the su(5)-Lipkin model. The relations (5.41) and (5.42) teach us that the state (5.43) is the eigenstate of R 0 (2) with the eigenvalue 5(n 0 + j) − ρ 2 . The above is an outline of our idea for the su(5)-Lipkin model. Through the above arguments, we can learn some characteristic features of the Lipkin model. Classification of the model into two cases with n = 2m and 2m + 1 may be essential for understanding the features demonstrated by the model. Number of the su(2)-subalgebras is equal to m for both cases and, further, they contain the su(m)-subalgebra. In the case with n = 2m + 1, there exists the degree of freedom related with R 0 (m) and its eigenvalues are influenced by the spinors which do not exist in the case with n = 2m. §6. Concluding remarks In the present two papers, we discussed formal aspects of the su(n)-Lipkin model in rather general form. In (I), we are mainly concerned with the determination of the minimum weight states. In (II), after re-forming the model in the frame of spherical tensor representation, we re-formulated the model by classifying it into two cases: n = 2m and 2m + 1. Especially, in §5, we gave some concrete results for the cases with n = 2, 3, 4 and 5, which, in (I), we discussed only the minimum weight states. As a concluding remark, we will show connection between the su(2m)-and the su(2m + 1)-Lipkin model directly. Concerning the single-particle level schemes, we regard these two as the models illustrated in Fig.1. Let the single-particle level specified by the integer l (l = 0, 1, 2 · · · , 2m − 1) in the su(2m)-Lipkin model be shifted to the integer (l + 1) in the su(2m + 1)-Lipkin model. Therefore, the zero-th level cannot get any counterpart in the su(2m)-Lipkin model. As a possible candidate, the following correspondence may be permitted: , S p −→ S 1 p+1 , S p q −→ S p+1 q+1 − δ pq S 1 1 . (p, q = 1, 2, · · · , 2m − 1) (6.1) The set { S p+1 1 , S 1 p+1 , S p+1 q+1 } forms the su(2m)-algebra and it becomes the u(2m)algebra, if 2m p=1 S p p is included. Further, if the operators S p and S p (p = 1, 2, · · · , 2m) are included, it becomes the su(2m + 1)-algebra. In fact, total number of the generators is given by [(2m) 2 − 1] + 1 + 2 · (2m) = (2m + 1) 2 − 1 . Adding the operators S p , S p and 2m p=1 S p p to the form (6.3) in terms of 2m p=1 [ S p , S p ] + /2 and (1/(2m(2m + 1))( 2m p=1 S p p ) 2 /2, we obtain Γ su(2m+1) . As the su(m)-subalgebra, we treated the simplest case; m = 2. In near future, we will report some results in the cases with m > 2. It may be interesting to investigate how the cases with m = 3 and 4 related with the Lipkin models with n = 3 and 4, which were discussed in this paper. The operator Q 0 (m; r) is essentially equivalent to R 0 (m; r). The operator Q 0 (m; m) plays a role similar to total fermion number N (n) in the su(n)-Lipkin model. Thus, we know that the case with n = 2m + 1 contains the su(m)-algebra in the su(n)-Lipkin model.