A note on the Lipkin model in arbitrary fermion number

TL;DR

This paper introduces a generalized form of the Lipkin model based on the su(6)-algebra, extending previous work on su(4), and details its algebraic structure and basis parameters.

Contribution

It presents a new su(6)-algebra-based Lipkin model and provides a complete algebraic framework and basis parameterization for arbitrary fermion numbers.

Findings

Formulation of the su(6)-algebra Lipkin model

Expression of relations using spherical tensors in su(2)-algebras

Parameterization of the basis with twenty parameters

Abstract

A possible form of the Lipkin model obeying the su(6)-algebra is presented. It is a natural generalization from the idea for the su(4)-algebra recently proposed by the present authors. All the relation appearing in the present form can be expressed in terms of the spherical tensors in the su(2)-algebras. For specifying the linearly independent basis completely, twenty parameters are introduced. It is concluded that, in these parameters, the ten denote the quantum numbers coming from the eigenvalues of some hermitian operators. The five in these ten determine the minimum weight state.

Equations

$$n-1=m+(m-1).(n=2m)$$

$$\frac{1}{2}\left(\left(n^2-1\right)-\left(n-1\right)\right)=\frac{1}{2}n(n-1)=m(2m-1).(n=2m)$$

$$m(2m-1)-\left(m+\frac{1}{2}m(m-1)\right)=3\cdot \frac{1}{2}m(m-1).$$

$$3\cdot \frac{1}{2}m(m-1)=\frac{1}{2}m(m-1)+2\cdot \frac{1}{2}m(m-1).$$

$${\tilde{S}}_{+}(1)={\tilde{S}}_2^3,{\tilde{S}}_{-}(1)={\tilde{S}}_3^2,{\tilde{S}}_0(1)=\frac{1}{2}\left({\tilde{S}}_3^3-{\tilde{S}}_2^2\right),$$

$${\tilde{S}}_{+}(2)={\tilde{S}}^1,{\tilde{S}}_{-}(2)={\tilde{S}}_1,{\tilde{S}}_0(2)=\frac{1}{2}{\tilde{S}}_1^1.$$

$${\tilde{S}}_{\pm ,0}={\tilde{S}}_{\pm ,0}(1)+{\tilde{S}}_{\pm ,0}(2).$$

$${\tilde{R}}_{+}={\tilde{S}}_1^3+{\tilde{S}}^2,{\tilde{R}}_{-}={\tilde{S}}_3^1+{\tilde{S}}_2,{\tilde{R}}_0=\frac{1}{2}\left({\tilde{S}}_3^3+{\tilde{S}}_2^2-{\tilde{S}}_1^1\right).$$

$$\left[\mathrm{any}\mathrm{of}{\tilde{S}}_{\pm ,0},\mathrm{any}\mathrm{of}{\tilde{R}}_{\pm ,0}\right]=0.$$

$${\tilde{R}}^{1,+1}=-{\tilde{S}}^3,{\tilde{R}}^{1,0}=\frac{1}{\sqrt{2}}\left({\tilde{S}}_1^3-{\tilde{S}}^2\right),{\tilde{R}}^{1,-1}={\tilde{S}}_1^2,$$

$${\tilde{R}}_{1,+1}=-{\tilde{S}}_3,{\tilde{R}}_{1,0}=\frac{1}{\sqrt{2}}\left({\tilde{S}}_3^1-{\tilde{S}}_2\right),{\tilde{R}}_{1,-1}={\tilde{S}}_2^1,$$

$$\left[{\tilde{R}}_{+},{\tilde{R}}^{1,\nu }\right]=0,\left[{\tilde{R}}_0,{\tilde{R}}^{1,\nu }\right]={\tilde{R}}^{1,\nu }.(\nu =\pm ,0)$$

$$|{\rho }_0,{\sigma }_0^1,{\sigma }_0^2;\rho ,{\sigma }^1,{\sigma }^2⟩=f_{\rho {\rho }_0}{f}_{{\sigma }^1{\sigma }_0^1}{f}_{{\sigma }^2{\sigma }_0^2}{\tilde{P}}^{\rho {\rho }_0,{\sigma }^1{\sigma }_0^1,{\sigma }^2{\sigma }_0^2}|\rho ,{\sigma }^1,{\sigma }^2⟩,$$

$${\tilde{P}}^{\rho {\rho }_0,{\sigma }^1{\sigma }_0^1,{\sigma }^2{\sigma }_0^2}={\left({\tilde{R}}_{+}\right)}^{\rho +{\rho }_0}{\left({\tilde{S}}_{+}(1)\right)}^{{\sigma }^1+{\sigma }_0^1}{\left({\tilde{S}}_{+}(2)\right)}^{{\sigma }^2+{\sigma }_0^2},$$

$$f_{\tau {\tau }_0}=\sqrt{\frac{(\tau -{\tau }_0)!}{(2\tau )!(\tau +{\tau }_0)!}}.(\tau =\rho ,{\sigma }^1,{\sigma }^2)$$

$$|\rho {\rho }_0,{\sigma }^1,{\sigma }^2,\sigma {\sigma }_0⟩=\sum _{{\sigma }_0^1{\sigma }_0^2}⟨{\sigma }^1{\sigma }_0^1{\sigma }^2{\sigma }_0^2|\sigma {\sigma }_0⟩|{\rho }_0,{\sigma }_0^1,{\sigma }_0^2;\rho ,{\sigma }^1,{\sigma }^2⟩.$$

$${\tilde{R}}^{l,l_0;\text{l},{\lambda }_0}={\left({\stackrel{\rightharpoonup }{R}}_{-}\right)}^{l-{\lambda }_0}{\left({\stackrel{\rightharpoonup }{S}}_{-}\right)}^{l-l_0}{\left(-{\tilde{S}}^3\right)}^l.$$

$$\displaystyle\left({\stackrel{{\scriptstyle\rightharpoonup}}{{O}}}\right)^{n}{\widetilde{A}}=\underbrace{\left[{\widetilde{O}},\cdots,\left[{\widetilde{O}},\left[{\widetilde{O}}\right.\right.\right.}_{n},{\widetilde{A}}\Bigl{]}\Bigl{]}\cdots\Bigl{]}\ .$$

$$|l,r{r}_0,s{s}_0,\sigma ;\rho {\sigma }^1{\sigma }^2⟩=\sum _{{\lambda }_0{l}_0}⟨l{\lambda }_0\rho {\rho }_0|r{r}_0⟩⟨l{l}_0\sigma {\sigma }_0|s{s}_0⟩{\tilde{R}}^{l,l_0;l,{\lambda }_0}|\rho {\rho }_0,{\sigma }^1,{\sigma }^2,\sigma {\sigma }_0⟩.$$

$${\tilde{R}}^{l,l_0;l,{\lambda }_0}\cdot {\tilde{P}}^{\rho {\rho }_0,{\sigma }^1{\sigma }_0^1,{\sigma }^2{\sigma }_0^2}$$

$${\left({\stackrel{\rightharpoonup }{S}}_{-}\right)}^{l-l_0}{\left(-{\tilde{S}}^3\right)}^l={\left({\stackrel{\rightharpoonup }{S}}_{-}\right)}^{l-l_0}{\left({\tilde{R}}^{1,+1}\right)}^l={\tilde{Z}}^{l,l_0}.$$

$${\tilde{Z}}^{l,l_0}=\sqrt{\frac{l!}{(2l-1)!!}}\sum _{\lambda }{\left(\frac{1}{\sqrt{2}}\right)}^{l-\lambda }\frac{\sqrt{(l+l_0)!(l-l_0)!}}{\left(\frac{l+l_0-\lambda }{2}\right)!\lambda !\left(\frac{l-l_0-\lambda }{2}\right)!}$$

$$\times {\left({\tilde{R}}^{1,+1}\right)}^{\frac{l+l_0-\lambda }{2}}{\left({\tilde{R}}^{1,0}\right)}^{\lambda }{\left({\tilde{R}}^{1,-1}\right)}^{\frac{l-l_0-\lambda }{2}}.$$

$${\stackrel{\rightharpoonup }{S}}_{\pm }{\tilde{Z}}^{l,l_0}=\sqrt{(l\mp l_0)(l\pm l_0+1)}{\tilde{Z}}^{l,l_0\pm 1},$$

$${\stackrel{\rightharpoonup }{S}}_0{\tilde{Z}}^{l,l_0}=l_0{\tilde{Z}}^{l,l_0}.$$

$${\tilde{Z}}^{l,l_0}⟶\sqrt{\frac{4\pi l!}{(2l+1)!!}}{\mathcal{Y}}_{l,l_0},\left({\mathcal{Y}}_{l,l_0}=r^l{Y}_{l{l}_0}(\theta \varphi )\right).$$

$${\stackrel{\rightharpoonup }{R}}_{+}{\tilde{Z}}^{l,l_0}=0,{\stackrel{\rightharpoonup }{R}}_0{\tilde{Z}}^{l,l_0}=l{\tilde{Z}}^{l,l_0}.$$

$${\tilde{S}}_{+}(1)={\tilde{S}}_4^5,{\tilde{S}}_{-}(1)={\tilde{S}}_5^4,{\tilde{S}}_0(1)=\frac{1}{2}\left({\tilde{S}}_5^5-{\tilde{S}}_4^4\right),$$

$${\tilde{S}}_{+}(2)={\tilde{S}}_2^3,{\tilde{S}}_{-}(2)={\tilde{S}}_3^2,{\tilde{S}}_0(2)=\frac{1}{2}\left({\tilde{S}}_3^3-{\tilde{S}}_2^2\right),$$

$${\tilde{S}}_{+}(3)={\tilde{S}}^1,{\tilde{S}}_{-}(3)={\tilde{S}}_1,{\tilde{S}}_0(3)=\frac{1}{2}{\tilde{S}}_1^1.$$

$${\tilde{S}}_{\pm ,0}={\tilde{S}}_{\pm ,0}(1)+{\tilde{S}}_{\pm ,0}(2)+{\tilde{S}}_{\pm ,0}(3).$$

$${\tilde{R}}_{+}={\tilde{S}}_3^5+{\tilde{S}}_2^4,{\tilde{R}}_{-}={\tilde{S}}_5^3+{\tilde{S}}_4^2,{\tilde{R}}_0=\frac{1}{2}\left({\tilde{S}}_5^5+{\tilde{S}}_4^4-{\tilde{S}}_3^3-{\tilde{S}}_2^2\right),$$

$${\tilde{R}}^{\frac{1}{2},\frac{1}{2}}={\tilde{S}}_1^5+{\tilde{S}}^4,{\tilde{R}}_{\frac{1}{2},\frac{1}{2}}={\tilde{S}}_5^1+{\tilde{S}}_4,$$

$${\tilde{R}}^{\frac{1}{2},-\frac{1}{2}}={\tilde{S}}_1^3+{\tilde{S}}^2,{\tilde{R}}_{\frac{1}{2},-\frac{1}{2}}={\tilde{S}}_3^1+{\tilde{S}}_2,$$

$$\tilde{R}=\frac{1}{2}\left({\tilde{S}}_5^5+{\tilde{S}}_4^4+{\tilde{S}}_3^3+{\tilde{S}}_2^2\right)-{\tilde{S}}_1^1.$$

Full text

1 Introduction

This paper is a continuation of two papers, recently, published by the present authors [1, 2]. Hereafter, these two will be referred to as (I) and (II), respectively. In these papers, we treated the Lipkin model [3, 4] with arbitrary single-particle levels and fermion number. In (I), mainly an idea of how to construct the minimum weight state, which is the starting point of the algebraic approach, was proposed. In (II), we discussed how to express the linearly independent basis built on a chosen minimum weight state. Present paper aims mainly at supplementing the results of (II) with the discussion on the $su(6)$-algebra, which we promised in (II).

First, let us consider the Lipkin model obeying the $su(n)$-algebra for the case with $n=2m$ in rather general framework. Here, $m$ denotes integer. Since the total fermion number $N$ is a constant of motion, we omit the discussion on $N$. Our present argument is restricted to the case with even integer $n$ with $n=4,\ 6,\cdots$, i.e., $n=2m$ with $m=2,\ 3,\cdots$. If the generators in the $su(n)$-algebra are expressed appropriately, we can show that this model includes $m$ $su(2)$-subalgebras. This point has been shown in (II) and the generators are given in the relation (II.2.3). Therefore, as the total sum, we can define the $su(2)$-algebra $({\widetilde{S}}_{\pm,0})$ which will play a central role in our approach. Of course, the generators in each $su(2)$-algebra form a vector. Further, in (II), we showed that the $su(n)$-Lipkin model includes one $su(m)$-subalgebra, all generators in which are scalar for $({\widetilde{S}}_{\pm,0})$ and the remains form $m(m-1)/2$ vectors. They are given in the relations (II.2.9) and (II.2.10) $\sim$ (II.2.12), respectively. Generally, the minimum weight states in the $su(n)$-algebra are specified by $(n-1)$ quantum numbers. In the present case, we can decompose the number $(n-1)$ into two parts:

[TABLE]

As was already mentioned, the case with $n=2m$ includes $m$ $su(2)$-algebras and one $su(m)$-algebra. The first and the second term, $m$ and $(m-1)$, represent, respectively, the numbers of the quantum numbers related with $m$ $su(2)$-algebras and of one $su(m)$-algebra for the minimum weight state.

Next, let us consider the orthogonal set constructed by operating “certain operators” on any minimum weight state. We will call them as the excited state generating operators. They should be expressed in terms of $n(n-1)/2$ quantum numbers coming from the relation

[TABLE]

The number $n(n-1)/2$ is equal to that of the $su(n)$-generators with the types ${\widetilde{S}}^{p}(n)$ $(p=1,\ 2,\cdots,\ n-1)$ and ${\widetilde{S}}_{q}^{p}(n)$ $(p>q=1,\ 2,\cdots,\ n-2)$: $(n-1)+(n-1)(n-2)/2=n(n-1)/2$. We will call them as the raising operators and their hermitian conjugates as the lowering operators. The definition of ${\widetilde{S}}^{p}(n)$ and ${\widetilde{S}}_{q}^{p}(n)$ has been given in the relation (I.2.2). We know that there exist $m$ $su(2)$-subalgebras and one $su(m)$-algebra and, then, in the excited state generating operators, $m$ and $m(m-1)/2$ operators are related to these subalgebras, respectively. Therefore, we must investigate the remain, the number of which is given by

[TABLE]

It should be noted that the number $m(m-1)/2$ is just equal to that of the vector operators for $({\widetilde{S}}_{\pm,0})$, which are presented in (II). With the use of these three types of the operators, we can expect to obtain a possible idea for constructing the excited state generating operators. Further, we notice that the relation (3) can be decomposed into

[TABLE]

First term corresponds to the number of the raising or the lowering operators in the $su(m)$-subalgebra. It is well known that we can construct a tensor operator with two parameters in terms of a vector, for example, the solid harmonics ${\mathcal{Y}}_{l,l_{0}}=r^{l}Y_{ll_{0}}(\theta\phi)$ is constructed in terms of the position vector ($x=r\sin\theta\cos\phi,\ y=r\sin\theta\sin\phi,\ z=r\cos\phi)$. In our case, there exist $m(m-1)/2$ vectors and each gives us a tensor operator in the $su(2)$-subalgebras, which is specified by two parameters. This argument leads us to the following: Second term represents the total number of the parameters contained in the operators which are built by $m(m-1)/2$ vectors. On the other hand, first term $m(m-1)/2$ corresponds to the number of the lowering operators in the $su(m)$-subalgebra. A merit of our idea may be as follows: All generators of the $su(n)$-Lipkin model are expressed in terms of the spherical tensors for $({\widetilde{S}}_{\pm,0})$ and, then, we can apply the technique of the angular momentum coupling rule.

As a simple example of the Lipkin model, we will give a brief summary for the case with $n=4$, i.e., $m=2$, which has been discussed in (II), but the form in this paper is a little bit different from that of (II). The generators in the present two $su(2)$-subalgebras are copied from the relation (II.5.9):

[TABLE]

The sum is given by

[TABLE]

The generators in the $su(m=2)$-subalgebra are expressed as

[TABLE]

Since ${\widetilde{R}}_{\pm,0}$ are scalars for $({\widetilde{S}}_{\pm,0})$, we have the relation

[TABLE]

The vector operators for $({\widetilde{S}}_{\pm,0})$ are given as

[TABLE]

The relations (6)$\sim$ (8) are copied from the relations (II.5.11)$\sim$ (II.5.14). The vectors ${\widetilde{R}}_{\pm,0}$ satisfy

[TABLE]

The minimum weight state is expressed as $|{\rho,\sigma^{1},\sigma^{2}}\rangle$, where the eigenvalues of ${\widetilde{R}}_{0}$, ${\widetilde{S}}_{0}(1)$ and ${\widetilde{S}}_{0}(2)$ are denoted as $-\rho$, $-\sigma^{1}$ and $-\sigma^{2}$, respectively. The explicit form of $|{\rho,\sigma^{1},\sigma^{2}}\rangle$ is given in (I). Then, the eigenstate of $({\widetilde{\mib R}}^{2},{\widetilde{R}}_{0})$, $({\widetilde{\mib S}}(1)^{2},{\widetilde{S}}_{0}(1))$ and $({\widetilde{\mib S}}(2)^{2},{\widetilde{S}}_{0}(2))$ with the eigenvalues $(\rho(\rho+1),\rho_{0})$, $(\sigma^{1}(\sigma^{1}+1),\sigma_{0}^{1})$ and $(\sigma^{2}(\sigma^{2}+1),\sigma_{0}^{2})$ can be expressed as

[TABLE]

[TABLE]

The eigenstate of $({\widetilde{\mib S}}^{2},{\widetilde{S}}_{0})$ with the eigenvalue $(\sigma(\sigma+1),\sigma_{0})$ is given by

[TABLE]

We are investigating the case with $n=4$, i.e., $m=2$. Therefore, as can be seen in the relation (3), further, we must search the operator characterized by three quantum numbers for obtaining the excited state generating operators. For this task, in (II), the following form is adopted:

[TABLE]

Here, we adopted the notation for ${\widetilde{O}}$ and ${\widetilde{A}}$ in the form

[TABLE]

Since ${\stackrel{{\scriptstyle\rightharpoonup}}{{S}}}_{+}\left(-{\widetilde{S}}^{3}\right)^{l}={\stackrel{{\scriptstyle\rightharpoonup}}{{R}}}_{+}\left(-{\widetilde{S}}^{3}\right)^{l}=0$ and ${\stackrel{{\scriptstyle\rightharpoonup}}{{S}}}_{0}\left(-{\widetilde{S}}^{3}\right)^{l}={\stackrel{{\scriptstyle\rightharpoonup}}{{R}}}_{0}\left(-{\widetilde{S}}^{3}\right)^{l}=l\left(-{\widetilde{S}}^{3}\right)^{l}$, ${\widetilde{R}}^{l,l_{0};l,\lambda_{0}}$ can be regarded as spherical tensor operator for the $su(2)$-algebras $({\widetilde{S}}_{\pm,0})$ and $({\widetilde{R}}_{\pm,0})$ with rank $l$. Operating ${\widetilde{R}}^{l,l_{0};l,\lambda_{0}}$ on the state (12) and applying the angular momentum coupling rule, we have

[TABLE]

However, the set composed by the states (15) cannot be regarded as orthogonal. The reasons are as follows: (1) the symbol $l$ is not a quantum number, but a parameter, the value of which is given from the outside and (2) the operator ${\widetilde{R}}^{l,l_{0};l,\lambda_{0}}$ contains the degrees of freedom which do not contain in the states $|{\rho\rho_{0},\sigma^{1},\sigma^{2},\sigma\sigma_{0}}\rangle$, for example, such as $-{\widetilde{S}}^{3}$. Therefore, they form the linearly independent basis and by appropriate method, for example, by the Schmidt method, we must construct the orthogonal set. In the sense mentioned above, the excited state generating operator is given by

[TABLE]

The above is the summary of (II) for the case with $n=4$, i.e., $m=2$. In the preparation for the discussion on the case with $n=6$, i.e., $m=3$, we rewrite the relation (13) in the form slightly different from the original. The form $({\stackrel{{\scriptstyle\rightharpoonup}}{{S}}}_{-})^{l-l_{0}}(-{\widetilde{S}}^{3})^{l}$ can be rewritten as

[TABLE]

Here, ${\widetilde{Z}}^{l,l_{0}}$ is of the form

[TABLE]

The sum for $\lambda$ obeys the following condition : if $|l-l_{0}|=$even or odd, $\lambda$ cannot be odd or even, respectively. The operator ${\widetilde{Z}}^{l,l_{0}}$ is a tensor with rank $l$ for $({\widetilde{S}}_{\pm,0})$:

[TABLE]

In the case that $({\widetilde{R}}^{1,\pm 1},{\widetilde{R}}^{1,0})$ is a position vector $(r_{\pm 1}=\mp(x\pm iy)/\sqrt{2},r_{0}=z)$, ${\widetilde{Z}}^{l,l_{0}}$ is reduced to the solid harmonics:

[TABLE]

An important property of ${\widetilde{Z}}^{l,l_{0}}$ is as follows:

[TABLE]

With the use of the relation (9), we are able to obtain the property (21).

Under the above preparation, we will investigate the case with $n=6$, i.e., $m=3$. This case includes three $su(2)$-subalgebras in the form

[TABLE]

The total sum of the above is denoted as

[TABLE]

The eight generators in the $su(m=3)$-subalgebra, which are scalars for $({\widetilde{S}}_{\pm,0})$, are written down as follows:

[TABLE]

The set (24a) forms the $su(2)$-algebra and $({\widetilde{R}}^{\frac{1}{2},\frac{1}{2}},{\widetilde{R}}^{\frac{1}{2},-\frac{1}{2}})$ and $(-{\widetilde{R}}_{\frac{1}{2},-\frac{1}{2}},{\widetilde{R}}_{\frac{1}{2},\frac{1}{2}})$ are spinors for $({\widetilde{R}}_{\pm,0})$. The present case includes three vectors for $({\widetilde{S}}_{\pm,0})$:

[TABLE]

The hermitian conjugates of the vectors are omitted to give. The expressions (22), (24) and (25) are obtained by putting $m=3$ in the relations (II.2.3) and (II.2.9)$\sim$(II.2.12). For $\nu=\pm 1,\ 0$, the vector $({\widetilde{R}}^{1,\nu}(1))$ satisfies the relation

[TABLE]

For $\nu=\pm 1,\ 0$ and $k=2,\ 3$, two vectors $({\widetilde{R}}^{1,\nu}(k))$ obey the relation

[TABLE]

The above are the relations for our present case $(n=6)$. However, ${\widetilde{R}}^{1,\nu}(2)$ and ${\widetilde{R}}^{1,\nu^{\prime}}(3)$ do not commute mutually, but, they commute with ${\widetilde{R}}^{1,\nu^{\prime\prime}}(1)$. We will treat the case with $n=6$ as a natural generalization from the case with $n=4$.

Now, in parallel with the $su(4)$-Lipkin model, we are possible to give our scheme for obtaining the linearly independent basis for the $su(6)$-Lipkin model. In order to avoid unnecessary complication, we will not apply the angular momentum coupling rule, together with the associating numerical factors, for example, such as the form (11). The minimum weight state is expressed in the form $|{\rho,\rho^{1},\sigma^{1},\sigma^{2},\sigma^{3}}\rangle$, where $\rho$, $\rho^{1}$, $\sigma^{1}$, $\sigma^{2}$ and $\sigma^{3}$ denote the eigenvalues of $-{\widetilde{R}}$, $-{\widetilde{R}}_{0}$, $-{\widetilde{S}}_{0}(1)$, $-{\widetilde{S}}_{0}(2)$ and $-{\widetilde{S}}_{0}(3)$ defined in the relations (24) and (22), respectively. As a possible extension of ${\widetilde{P}}^{\rho\rho_{0},\sigma^{1}\sigma_{0}^{1},\sigma^{2}\sigma_{0}^{2}}$ shown in the relation (10b), we introduce the following operator:

[TABLE]

With the use of the operator (28), we define the state

[TABLE]

The state (29) corresponds to the state (10a) and is expressed in terms of eleven parameters, some of which play the role of quantum numbers.

The above is first step in our approach to the case with $n=6$. The linearly independent basis of our present case should be expressed totally in terms of twenty parameters. Therefore, we must, further, investigate how to consider nine parameters which may be related with three vectors. For this task, we extend the idea adopted in the case with $n=4$. This case starts in the relation (9). The relation (9) for the vector (17) with (1) for the definition of ${\widetilde{Z}}^{l,l_{0}}$ leads us to the relation (21) for the raising operator ${\widetilde{R}}_{+}$ and the hermitian operator ${\widetilde{R}}_{0}$. These two correspond to $({\widetilde{R}}_{+},{\widetilde{R}}^{\frac{1}{2},\frac{1}{2}},{\widetilde{R}}^{\frac{1}{2},-\frac{1}{2}})$ and $({\widetilde{R}}_{0},{\widetilde{R}})$, respectively for the present case. As can be seen in the relation (27a), the operation of ${\stackrel{{\scriptstyle\rightharpoonup}}{{R}}}_{+}$ and ${\stackrel{{\scriptstyle\rightharpoonup}}{{R}}}^{\frac{1}{2},\pm\frac{1}{2}}$ on the vectors labeled $k=2,\ 3$ do not vanish. Then, if we try to use the form extended from the relation (9), we must introduce new vectors. As the candidate of these new, the following are possible to use:

[TABLE]

The vector (30b) is derived by the formula $Z_{\nu}=\sum_{\lambda,\mu}\langle 1\lambda 1\mu|{1\nu}\rangle X_{\lambda}Y_{\mu}$ for two vectors $X$ and $Y$. The vectors (30) satisfy the following relation for $\nu=\pm 1,\ 0$ and $k=1,\ 2,\ 3$:

[TABLE]

With the use of the vectors (31), we can define the operator

[TABLE]

Here, ${\widetilde{Z}}^{l^{k}l_{0}^{k}}(k)$ $(k=1,\ 2,\ 3)$ is obtained by replacing $ll_{0}$ and ${\widetilde{R}}^{1,\nu}$ in the relation (1) with $l^{k}l_{0}^{k}$ and ${\breve{R}}{}^{1,\nu}(k)$. As was already mentioned, some pairs of $({\widetilde{R}}^{1,\nu}(k))$ do not commute and, therefore, the ordering of ${\widetilde{Z}}^{l^{k}l_{0}^{k}}(k)$ in the definition (32) should be fixed beforehand, for example, as is shown in the definition (32). The operator (32) satisfies

[TABLE]

We can see that the above is a natural extension from the relation (21).

The operator (32) is expressed in terms of six parameters $(l^{1},l_{0}^{1})$, $(l^{2},l_{0}^{2})$ and $(l^{3},l_{0}^{3})$. Then, in order to accomplish our task, we must search, further, three parameters. As can be seen in the relation (13), the case with $n=4$ is completed by taking into account the lowering operator ${\widetilde{R}}_{-}$ in the form ${\stackrel{{\scriptstyle\rightharpoonup}}{{R}}}_{-}$. The relation (33a) tells us that the operation of the raising operators ${\widetilde{R}}_{+}$ and ${\widetilde{R}}^{\frac{1}{2},\pm\frac{1}{2}}$ makes the results vanish. It may be enough to consider three lowering operators ${\stackrel{{\scriptstyle\rightharpoonup}}{{R}}}_{-}$ and $\mp{\stackrel{{\scriptstyle\rightharpoonup}}{{R}}}_{\frac{1}{2},\pm\frac{1}{2}}$ on the operator (32). First, we note that the operator (32) is nothing but tensor specified by

[TABLE]

Then, tensor operator specified by $l$ and $l_{0}$ $(l_{0}=-l,\ -l+1,\cdots,\ l-1,\ l)$ can be given in the form

[TABLE]

Next, we consider the operators $\mp{\widetilde{R}}_{\frac{1}{2},\mp\frac{1}{2}}$, which form the spinor for $({\widetilde{R}}_{\pm,0})$. As is well known, tensor operator specified by $\lambda$ and $\lambda_{0}$ $(\lambda_{0}=-\lambda,\ -\lambda+1,\cdots,\ \lambda-1,\ \lambda)$ is constructed in the form

[TABLE]

In the case of applying the angular momentum couping rule, it may be convenient to attach the numerical factor $g_{\lambda\lambda_{0}}$:

[TABLE]

Then, we introduce the operator

[TABLE]

Product of the operators (35) and (38) gives us the operator with nine parameters:

[TABLE]

Thus, multiplying the operator (40) by (28), we obtain the state generating operator with fifteen parameters.

Until the present, we have not contacted with the spherical tensor representation of our model with respect to the $su(2)$-algebras $({\widetilde{R}}_{\pm,0})$ and $({\widetilde{S}}_{\pm,0})$. Its explicit expression is omitted to show, but, we will discuss the basic idea for this problem. The linearly independent basis obtained in this paper is expressed in terms of twenty parameters, which, at least, five are related with the quantum numbers for the minimum weight state, i.e., $\rho$, $\rho^{1}$, $\sigma^{1}$, $\sigma^{2}$ and $\sigma^{3}$. Then, it may be important to investigate which parameters play a role of the quantum numbers or not. It is our final problem of this paper.

Application of the angular momentum coupling rule to the state (29) leads us to the following:

[TABLE]

[TABLE]

The inequalities (41a) and (41b) are related with the coupling rule for $({\widetilde{R}}_{\pm,0})$ and $({\widetilde{S}}_{\pm,0})$, respectively. If we note that $\rho^{1}$, $\sigma^{1}$, $\sigma^{2}$ and $\sigma^{3}$ denote the quantum numbers coming from the Casimir operator, $\mu$, $\eta$, $\sigma^{12}$ and $\sigma^{123}$ play a role of the parameters obeying the inequality (41). Next, we consider the operator (38). In this case, the coupling rule gives us

[TABLE]

[TABLE]

Under the relation (34), the seven, $\lambda$, $\xi$, $l^{1}$, $l^{2}$, $l^{12}$, $l^{3}$ and $l^{123}$ have to be regarded as parameters obeying the inequality (43). Finally, we consider ${\widetilde{\mib R}}^{2}$, ${\widetilde{R}}_{0}$, ${\widetilde{\mib S}}^{2}$ and ${\widetilde{S}}_{0}$ and, further, ${\widetilde{R}}$. Let the eigenvalues of ${\widetilde{\mib R}}^{2}$, ${\widetilde{R}}_{0}$, ${\widetilde{\mib S}}^{2}$ and ${\widetilde{S}}_{0}$ denote $r(r+1)$, $r_{0}$, $s(s+1)$ and $s_{0}$, respectively. For $({\widetilde{\mib R}}^{2}$, ${\widetilde{R}}_{0})$ and $({\widetilde{\mib S}}^{2}$, ${\widetilde{S}}_{0})$, we have the relations

[TABLE]

[TABLE]

At the present stage, we can summarize our discussion as follows: The eight symbols $\rho^{1}$, $\sigma^{1}$, $\sigma^{2}$, $\sigma^{3}$, $(r,r_{0})$ and $(s,s_{0})$ denote the quantum numbers and the eleven $\lambda$, $\mu$, $\xi$, $\eta$, $\sigma^{12}$, $\sigma^{123}$, $l^{1}$, $l^{2}$, $l^{12}$, $l^{3}$ and $l^{123}$ must be regarded as the parameters, totally the nineteen. However, in the above analysis, there are one point missing. We must take account of $\rho$ and $R$ (the eigenvalue of ${\widetilde{R}}$). After straightforward calculation, we can show that our linearly independent basis is a set of the eigenstates of ${\widetilde{R}}$ and $R$ can be expressed in the form

[TABLE]

Then, we have to add $R$ and $\rho$ to the quantum numbers. Therefore, one of three parameters $\mu$, $\lambda$ and $(l^{1}+l^{2}+2l^{3})/2$, for example, $\mu$, depends on the others if $R$ and $\rho$ are given.

Thus, we have a summary: In our linearly independent basis, $\rho$, $\rho^{1}$, $\sigma^{1}$, $\sigma^{2}$, $\sigma^{3}$, $r$, $r_{0}$, $s$, $s_{0}$ and $R$ denote the quantum numbers and $\sigma^{12}$, $\sigma^{123}$, $l^{1}$, $l^{2}$, $l^{12}$, $l^{3}$, $l^{123}$, $\xi$ and $\eta$ play a role of the parameters. Then, with the use of an appropriate method, for example, the Schmidt method, we can construct the orthogonal set for the $su(6)$-Lipkin model with arbitrary fermion numbers.

Including (I) and (II), this note has been devoted to the discussion on the Lipkin model, which belongs to the classical model in nuclear many-body theories. The authors tried to stress that there exist still some problems which remain unsolved until the present.

Acknowledgment

Two of the authors (Y.T. and M.Y.) would like to express their thanks to Professor J. da Providência and Professor C. Providência, two of co-authors of this paper, for their warm hospitality during their visit to Coimbra in spring of 2015. The author, M.Y., would like to express his sincere thanks to Mrs K. Yoda-Ono for her cordial encouragement. The authors, Y.T., is partially supported by the Grants-in-Aid of the Scientific Research (No.26400277) from the Ministry of Education, Culture, Sports, Science and Technology in Japan.