On the Logical Structure of Composite Symmetries in Atomic Nuclei

Abstract

This investigation involves composite symmetries, which connect different components of atomic nuclei. In particular, supersymmetry (SUSY) connects boson and fermion sectors, and multiconfigurational symmetry (MUSY) bridges different configurations, such as shell, quartet, or cluster configurations. Both SUSY and MUSY contain a usual dynamical symmetry in each component, and further symmetry connects the components to each other. The varieties of the connecting symmetries and their logical structure are analyzed and compared.

1. Introduction

Some models of nuclear structure consider the nucleus as a system of two or more components. For example, it is believed to consist of protons and neutrons, bosons and fermions, or different configurations. To be sure, symmetry considerations are also very helpful for these composite models. Their symmetries can be divided into two categories, depending on whether or not they preserve the particle number in each component, separately, or they allow transformations between them. The latter ones are more general, they are the deeper symmetries, and we call them composite symmetries.

A historical example of composite symmetry is the U${}^{ST}$(4) symmetry of the supermultiplet theory [1,2], which contains both the U${}^S$(2) spin and the U${}^T$(2) isospin groups.

Much attention has been focused on nuclear supersymmetry [3,4,5,6,7], in which the nucleons (as fermions), and the excitation quanta of the collective motion (as bosons) are allowed to transfer into each other, i.e., supertransformations are taken into account.

Recently another composite symmetry was investigated as well, which is called multiconfigurational dynamical symmetry [8]. This symmetry connects different configurations (e.g., shell, quartet, or various cluster configurations) to each other. In fact, this symmetry is the common intersection of these fundamental models for the many major-shell excitation problems.

The logical structure of supersymmetry and multiconfigurational symmetry shows a remarkable similarity. Both of them are composite symmetries of composite systems. Each component of the system (boson and fermion sectors of SUSY, or different configurations in MUSY) carries a usual dynamical symmetry, and further symmetry connects the components to each other (supertransformations in SUSY and symmetries in the particle-index space in MUSY).

In addition to the basic similarity, however, there are also differences between the two schemes. Furthermore, the connecting symmetry can be different even within the different SUSY models, as well as within those of the MUSY. Dynamical symmetries are known from the early days of algebraic modeling [9,10], and they are very well studied [11]. Connecting symmetries, however, are not so well explored. Especially insufficient is the analysis of the transformations between the different configurations in MUSY.

In this paper, we intend to survey the logical structure of composite symmetries, with special attention to connecting transformations. MUSY is much less studied so far, and we mainly concentrate on its nature. Nevertheless, we also briefly review the varieties of SUSY from this viewpoint and make some comparisons, when possible. The structure of the paper is as follows: Section 2 presents a discussion on MUSY, first considering its dynamical symmetries and then the symmetries between the different configurations. Section 3 contains a brief review of the dynamical supersymmetry of nuclei from the special viewpoint of the connecting transformations. Finally, a summary is given, and some conclusions are drawn in Section 4. Here, we do not distinguish between Lie (or graded) algebras and their corresponding groups and denote both by capital letters. In the present applications (as well as in many other applications in nuclear physics), only the algebras play important roles.

2. Multiconfigurational Dynamical Symmetry

The multiconfigurational dynamical symmetry connects the fundamental models of nuclear structure to each other. In particular, it provides us with the common intersection of the shell, collective, and cluster models for the many major-shell problems. The MUSY is based on two pillars; on the one side, there is a unified, multiplet structure for the shell, collective, and cluster model states, i.e., there is a similar dynamical symmetry in each configuration:

$$\begin{array}{cc}\hfill {\mathrm{U}}_s\left(3\right)& \hfill \otimes {\mathrm{U}}_e\left(3\right)\supset \mathrm{U}\left(3\right)\supset \mathrm{SU}\left(3\right)\supset \mathrm{SO}\left(3\right)\\ \hfill \left|\right[n_1^s,n_2^s,n_3^s]& \hfill ,[n_1^e,n_2^e,n_3^e],\rho ,[n_1,n_2,n_3],(\lambda ,\mu ),K,L\rangle .\end{array}$$

(1)

On the other side, there are transformations that connect the different configurations and their dynamical symmetries to each other.

2.1. Dynamical Symmetries

Evidently, in order to be able to reveal their connection, models need to be formulated within a symmetry-governed algebraic manner, and the model spaces have to be treated microscopically. Therefore, the symplectic shell model [12,13], the contracted symplectic collective model [14,15], and the semi-microscopic algebraic cluster model [16,17] are the appropriate approaches.

The multi-shell connection is a straightforward extension of the relation between the fundamental models for the single-shell problem found earlier.

2.1.1. Single-Shell Problem

In 1958, Elliott [9,10] showed that the quadrupole deformation and the collective rotation can be deduced from the spherical shell model by selecting a well-defined SU(3) symmetry of the many-nucleon system. Wildermuth and Kanellopoulos established the connection between the shell and cluster models by finding that the model Hamiltonians can be rewritten into each other in the harmonic oscillator approximation [18]. It indicates that their wave functions can be expanded in the other basis of the same eigenvalue. Furthermore, Bayman and Bohr formulated this cluster–shell connection in terms of SU(3) symmetry [19]; thus, the connection between the three basic models was found to be a dynamical symmetry

$$\begin{gathered} \begin{array}{ccc}\mathrm{U}\left(3\right)& \supset & \mathrm{SU}\left(3\right)\supset \mathrm{SO}\left(3\right)\supset \mathrm{SO}\left(2\right)\end{array} \\ [n_1,n_2,n_3],(\lambda ,\mu ),K,L,M. \end{gathered}$$

(2)

Here, we also indicate the representation labels of the groups.

The spin–isospin degrees of freedom are described in the Elliot model (as well as in the cluster model) by Wigner’s U${}^{ST}$(4) group, and its spin and isospin subgroups: ${\mathrm{U}}^{ST}\left(4\right)\supset {\mathrm{SU}}^S\left(2\right)\otimes {\mathrm{SU}}^T\left(2\right)$ [1]. When there are m single-particle space orbitals in the shell, the group structure of the single shell problem is provided by the chain as follows:

$$\{{\mathrm{U}}^{\mathrm{ST}}\left(4\right)\supset {\mathrm{SU}}^{\mathrm{S}}\left(2\right)\otimes {\mathrm{SU}}^{\mathrm{T}}\left(2\right)\}\otimes \{\mathrm{U}\left(\mathrm{m}\right)\supset \mathrm{U}\left(3\right)\supset \mathrm{SU}\left(3\right)\supset \mathrm{SO}\left(3\right)\}.$$

(3)

The relevant U(3) (and SU(3)) representations are determined by those of the U(m) group [20]. The total antisymmetry of the many-body wave function requires that U(m) and U${}^{ST}$(4) have associate representations, i.e., their Young patterns are obtained from each other by reflection: interchanging the rows and columns, while the permutation group of S(N) of the N nucleons shares the same Young pattern as the representation of U(m) [21].

A dynamically symmetric Hamiltonian of the single-shell problem is easily obtained:

$$H=C_{\mathrm{U}3}^{\left(1\right)}+\alpha C_{\mathrm{SU}3}^{\left(2\right)}+\delta C_{\mathrm{SO}3}^{\left(2\right)},$$

(4)

where $C^{\left(i\right)}$ stands for the Casimir invariant of degree i of the algebra indicated as a subscript:

$$C_{\mathrm{U}3}^{\left(1\right)}=n,C_{\mathrm{SU}3}^{\left(2\right)}=\frac{3}{4}LL+\frac{1}{4}{Q}^a{Q}^a,C_{\mathrm{SO}3}^{\left(2\right)}=LL,$$

(5)

where $n=n_1+n_2+n_3$ is the number of oscillator quanta. This operator is diagonal in the U(3) basis. Therefore, it splits up the degeneracy of the oscillator spectrum, i.e., deforms it to a realistic one but does not mix its basis states. The original energy operator of the Elliott model [9,10] has also this form (with a fixed ratio of $\alpha$ and $\delta$ to give the quadrupole force).

2.1.2. Multi-Shell Problem

In order to include major-shell excitations, the U(3) symmetry algebra can be extended by incorporating raising and lowering operators of two oscillator quanta, which connect the shells of the same parity, resulting in a symplectic shell model. The ladder operators are: $B_m^{†\left(l\right)}={[{\pi }^{†}\times {\pi }^{†}]}_m^{\left(l\right)},$ and $B_m^{\left(l\right)}={[\pi \times \pi ]}_m^{\left(l\right)},l=0,2$, where ${\pi }^{†}$ and $\pi$ stand for the creation and annihilation operators of the oscillator quanta. The creation operators are $[2,0,0]$ U(3) tensors; therefore, their products also carry U(3) labels: $[n_1^e,n_2^e,n_3^e]$. The relevant group structure of this model is

$$\begin{array}{c}\hfill \mathrm{Sp}(6,\mathrm{R})\supset \mathrm{U}\left(3\right)\supset \mathrm{SU}\left(3\right)\supset \mathrm{SO}\left(3\right)\\ \hfill |[n_1^s,n_2^s,n_3^s],[n_1^e,n_2^e,n_3^e],\rho ,[n_1,n_2,n_3],(\lambda ,\mu ),K,L\rangle .\end{array}$$

(6)

Here, $[n_1^s,n_2^s,n_3^s]$ denotes the symplectic bandhead, which is a U(3) irreducible representation (irrep), being a lowest-weight Sp(6,R) state. Note that the Sp(6,R) basis is given by the outer product of U(3) basis states $[n_1^e,n_2^e,n_3^e]\otimes [n_1^s,n_2^s,n_3^s]$, and $\rho$ distinguishes multiple occurence of $[n_1,n_2,n_3]$ in the product representation. In collective terms, the symplectic model includes monopole and quadrupole vibrations as well as the vorticity degrees of freedom for the description of the rotational dynamics in a continuous range from the irrotational flow to a rigid rotor.

In the large n limit, the symplectic shell model reduces to a collective model with a simpler structure. Here, n is the number of oscillator quanta, including the zero point values. The dynamical group of the model simplifies to U${}_e$(6)⊗U${}_s$(3), i.e., to a compact group, as opposed to the noncompact Sp(6,R) of the shell model. Technically, it is achieved by replacing the raising and lowering operators with boson creation and annihilation operators: $b_m^{†\left(l\right)}=(1/ϵ)B_m^{†\left(l\right)},b_m^{\left(l\right)}=(1/ϵ)B_m^{\left(l\right)}$, where $ϵ$ denotes ${\left[\frac{4}{3}{n}_s\right]}^{\frac{1}{2}};n_s=n_1^s+n_2^s+n_3^s$. This model is called the U(3) boson model [14], or the contracted symplectic model [15]. Mathematical justification for the simplifying assumptions is provided through the application of the group deformation mechanism. This model is more easily applicable, e.g., it has an orthonormal set of basis states defined by the group chain:

$${\mathrm{U}}_{\mathrm{s}}\left(3\right)\otimes {\mathrm{U}}_{\mathrm{e}}\left(6\right)\supset {\mathrm{U}}_{\mathrm{s}}\left(3\right)\otimes {\mathrm{U}}_{\mathrm{e}}\left(3\right)\supset \mathrm{U}\left(3\right)\supset \mathrm{SU}\left(3\right)\supset \mathrm{SO}\left(3\right).$$

(7)

The united U(3) group is generated by the sum of the operators corresponding to the subgroups U${}_s$(3)⊗U${}_e$(3):

$$n=n_s+2n_e,Q=Q_s+Q_e,L=L_s+L_e.$$

(8)

The U${}_s$(3) is Elliott’s shell model symmetry of the $0\hslash \omega$ shell, and U${}_e$(6) is the group of the six-dimensional oscillator, generated by the bilinear products of the ($l=0$ and 2) boson creation and annihilation operators. It is realized in a similar way as the U(6) group of the interacting boson model (IBM) [22]; nevertheless, physically, it is different because, in the case of the contracted symplectic model, the bosons are associated with intershell excitations, not to intrashell ones.

The spin–isospin degrees of freedom of the symplectic and contracted symplectic models can be described in the same way as those of the Elliott model, i.e., using Wigner’s symmetry of ${\mathrm{U}}^{\mathrm{ST}}\left(4\right)\supset {\mathrm{SU}}^{\mathrm{S}}\left(2\right)\otimes {\mathrm{SU}}^{\mathrm{T}}\left(2\right)$. The same procedure is also applied in the semi-microscopic algebraic cluster model.

The semi-microscopic algebraic cluster model (SACM) combines a microscopic model space with a fully algebraic description, i.e., not only the basis states but also the operators carry group symmetry [16]. The internal structure of the clusters is treated here using the Elliott model [9,10]; therefore, this part of the wave function has a U${}_C^{ST}$(4)⊗U${}_C$(3) symmetry. The relative motion of the clusters is accounted for by the modified (U${}_R$(4)) vibron model [23,24].

The coupling between the relative motion and internal cluster degrees of freedom for a binary cluster system results in a group structure:

$${\mathrm{U}}_{{\mathrm{C}}_1}^{\mathrm{ST}}\left(4\right)\otimes {\mathrm{U}}_{{\mathrm{C}}_1}\left(3\right)\otimes {\mathrm{U}}_{{\mathrm{C}}_2}^{\mathrm{ST}}\left(4\right)\otimes {\mathrm{U}}_{{\mathrm{C}}_2}\left(3\right)\otimes {\mathrm{U}}_{\mathrm{R}}\left(4\right).$$

(9)

The spin and isospin degrees of freedom are essential from the viewpoint of the construction of the model space. However, if one is interested only in a single supermultiplet [U${}_C^{ST}$(4)] symmetry, which is typical in cluster problems, then the relevant group structure simplifies to that of the space part. It is characterized by the group chain:

$${\mathrm{U}}_{{\mathrm{C}}_1}\left(3\right)\otimes {\mathrm{U}}_{{\mathrm{C}}_2}\left(3\right)\otimes {\mathrm{U}}_{\mathrm{R}}\left(3\right)\supset {\mathrm{U}}_{\mathrm{C}}\left(3\right)\otimes {\mathrm{U}}_{\mathrm{R}}\left(3\right)\supset \mathrm{U}\left(3\right)\supset \mathrm{SU}\left(3\right)\supset \mathrm{SO}\left(3\right),$$

(10)

Here, U${}_C$(3) stands for the coupled space symmetry of the two clusters.

The model space is free from the Pauli-forbidden states and the spurious excitations of the center of mass, and it is identical to those of the fully microscopic cluster models (when they apply U${}^{ST}$(4) ⊗ U(3) basis).

As for the product representations of $[n_1^s,n_2^s,n_3^s]\otimes [n_1^e,n_2^e,n_3^e]$ appearing in each approach, their physical content is as follows: For the shell and collective models, s stands for the band-head (valence shell), while for the cluster model, it refers to the internal cluster structure. The superscript e indicates in each case the major shell excitations; in the shell and collective model cases, they occur in steps of $2\hslash \omega$, connecting oscillator shells of the same parity, while in the cluster case, they represent the relative motion and can change in steps of $1\hslash \omega$, incorporating all the major shells. For the cluster model, it has only completely symmetric (single-row Young-tableaux) irreps: $[n,0,0]$, while in the case of the shell and collective models, it can be more general.

2.2. Connecting Transformations

The MUSY wave function is determined by the representation indices of the group chain; (1) let us denote it in short by $|u3\rangle$. This is the common part of the (symplectic) shell, (contracted symplectic) collective, and (semi-microscopic algebraic) cluster models. The complete classification of the states requires a further quantum number (or set of quantum numbers) that we denote by Greek letters, which refer to the larger groups and allow differentiation between the configurations, e.g., $|\alpha ,u3\rangle$.

Let us consider a (unitary) transformation $T(\beta ,\alpha )\equiv T$, which takes a state to another one: $T(\beta ,\alpha )|\alpha ,u3\rangle =|\beta ,u3\rangle$. An operator O is transformed as $O^{\prime }=TO{T}^{-1}$. As for the connection between the different configurations, different approaches were considered, which are explained in what follows.

2.2.1. Invariant Operators

This method applies operators that are invariant with respect to the transformations from one configuration to the other [8]. This is a mathematically well-defined and physically simple method. Let the Hamiltonian H be invariant (with respect to the transformations from one configuration to another), and consider its diagonal matrix elements: $\langle \alpha ,u3\left|H\right|\alpha ,u3\rangle$. (Note that for a dynamical symmetry, the Hamiltonian is diagonal.) These matrix elements also remain invariant:

$$\langle \beta ,u3|H^{\prime }|\beta ,u3\rangle =\langle \alpha ,u3|T^{-1}\left\{TH{T}^{-1}\right\}T|\alpha ,u3\rangle =\langle \alpha ,u3\left|H\right|\alpha ,u3\rangle .$$

(11)

i.e., the energy spectra of the two configurations (which share the same $u3$ labels) are identical.

The way to obtain the invariant operators can be summarized as follows: Let us consider a classification scheme different from that of chain (1) that we discussed so far. The scheme of chain (1) is a shell scheme: the symmetries of the nucleons are treated in each shell and then combined. The other scheme, called the particle scheme, is based on the study by Kramer and Moshinsky [25]. They considered the problem of N nucleons in the harmonic oscillator potential.

The spatial part of the total wave function has a U(3N) symmetry, while the spin–isospin part has U(4N). We focus on the space part. The U(3N) group is generated by the 3N×3N number conserving bilinear products of the creation and annihilation operators of oscillator quanta. When they are contracted (summed up) with respect to the particle index, then we have the U(3) space group, while the contraction with respect to the space index results in U(N). Their physically relevant subgroups are the O(3) rotation, and the S(N) permutation, respectively. Similarly, in the spin–isospin model, the important subgroups are the spin and isospin groups U${}^S$(2)⊗U${}^T$(2), and the permutation S${}^{ST}$(N). The antisymmetry is guaranteed when S(N) and S${}^{ST}$(N) have conjugate representations.

The transformation from one configuration to the other manifests itself in redistributing the N particles, and when the operators are insensitive to these changes, they are invariant. This is the case when they are expressed in terms of the contracted operators of U(3) [8]. For instance, the interactions written in terms of these operators do not have any configuration-dependent part.

A simple Hamiltonian that proved to be useful in several applications is

$$H=ϵn+a{C}_{\mathrm{SU}3}^{\left(2\right)}+b{C}_{\mathrm{SU}3}^{\left(3\right)}+d\frac{1}{2\theta }{L}^2,$$

(12)

Here, $\theta$ is the moment of inertia calculated classically for the rigid shape determined by the U(3) quantum numbers (for a rotor with axial symmetry). The parameters $ϵ,a,b$ and d can be fitted to the experimental data.

This Hamiltonian is invariant with respect to the transformations from one configuration to the other; therefore, it results in the identical spectra (of the common U(3) sectors) of different configurations. This kind of Hamiltonian was applied for the unified description of different (quartet, alpha-cluster, exotic cluster) configurations in a large range of excitation energy and deformation [8,26,27]. The gross features of the spectra could be reproduced with reasonable approximation, and in some cases, even predictions could be made for the detailed high-lying cluster spectra from the description of low-lying quartet states [26].

Nevertheless, we should note here that the invariance of the operators is a sufficient condition but not a necessary one. Identical spectra may result from non-invariant Hamiltonians as well.

2.2.2. Eigenvalue Connection

This is a simple and widely applicable method, based on the relation between the wave functions of different configurations. They may be identical when the multiplicity of the U(3) basis is 1 in the shell model space. In such a case, the shell-model expansion of any (cluster) state contains only a single term; therefore, the wave functions of different configurations are identical as a consequence of the antisymmetrization. Then, it is straightforward to require that their energy eigenvalues should be the same [28]. The equations obtained this way result in unique relations between the parameters of the energy functionals.

Let us illustrate the situation with a simple example of two different binary clusterizations (called c, and d) of a nucleus. For the sake of simplicity, let each cluster be spin–isospin scalar, and let one cluster in both configurations be SU(3) scalar (having a closed shell structure). An example is the (c:)${}^{20}$Ne+${}^4$He and (d:)${}^{16}$O+${}^8$Be configurations of the ${}^{24}$Mg nucleus. What is the relation between the energy spectra of the two configurations, if a simple dynamical symmetry holds for them? In [28,29], the following energy functional was applied:

$$E=ϵ+\gamma n_R+\beta L(L+1)+\theta n_{R}L(L+1)+F(\lambda ,\mu ,L),$$

(13)

where $n_R$ is the number of quanta in the relative motion. It is obviously configuration-dependent because the $n_R$-dependent parts of the Hamiltonian are not invariant. The requirement of having the same energy for those states of the two configurations that have identical wave functions puts the following constraints: ${\gamma }_c={\gamma }_d=\gamma$, ${ϵ}_c={ϵ}_d-\gamma n_0$, ${\theta }_c={\theta }_d=\theta$, ${\beta }_c={\beta }_d-\theta n_0$, $F_c(\lambda ,\mu ,L)=F_d(\lambda ,\mu ,L)$, and $n_{cR}=n_{dR}+n_0$, where $n_0$ is the difference of the relative motion quantum number in describing the same U(3) state.

The method of the eigenvalue connection is a general one and is applicable also in the case when there are configuration-dependent interactions. It does not reveal, however, the real transformations from one configuration to the other.

2.2.3. Projection Method

This method has been applied thus far only for the case of two binary clusterizations [29]. It consists of defining an underlying configuration, from which the configurations in question can be obtained through projection. For the two binary clusterizations, the underlying one is a ternary cluster configuration. For the example presented in the previous subsection, namely the ${}^{24}$Mg nucleus with the binary configurations of (c:)${}^{20}$Ne+${}^4$He and (d:)${}^{16}$O+${}^8$Be, the underlying ternary configuration is ${}^{16}$O+${}^4$He+${}^4$He.

A ternary configuration has two independent relative motions, i.e., two sets of oscillator quanta, which can be transformed into each other. Their algebraic description is provided using a model with the group chain $\mathrm{U}\left(6\right)\supset {\mathrm{U}}_{\mathrm{v}}\left(3\right)\otimes {\mathrm{U}}_{\mathrm{t}}\left(3\right)$, and by including a scalar boson for generating the spectrum, one obtains the U(7) dynamical group [30]. U(7) has several chains of subgroups, but the one that is relevant for the MUSY is

$$\begin{array}{ccc}\hfill \mathrm{U}\left(7\right)& \supset & \mathrm{U}\left(1\right)\otimes \mathrm{U}\left(6\right)\supset \mathrm{U}\left(6\right)\hfill \\ & \supset & \{\mathrm{U}\left(3\right)\supset \mathrm{SU}\left(3\right)\supset \mathrm{SO}\left(3\right)\supset \mathrm{SO}\left(2\right)\}\otimes \{{\mathrm{U}}_{\mathrm{p}}\left(2\right)\supset {\mathrm{SU}}_{\mathrm{p}}\left(2\right)\supset {\mathrm{SO}}_{\mathrm{p}}\left(2\right)\}.\hfill \end{array}$$

(14)

Here, U(3) (generated by the sums of the generators of U${}_v$(3) and U${}_t$(3)) and its subgroups act in real space, while U${}_p$(2) and its subgroups act in the pseudo-space of the particle number. A simple dynamically symmetric Hamiltonian of the three-cluster system can be written as

$$H=ϵ+\gamma N_6+\delta C_{\mathrm{SU}\left(3\right)}^{\left(2\right)}+\beta L^2+\theta N_6{L}^2,$$

(15)

which is diagonal in the basis above. Of course, more complicated functional forms of the invariant operators of U(3) and its subgroups can be applied, too, still having the dynamical symmetry. These Hamiltonians have an exact pseudo-spin symmetry.

One can define two different sets of Jacobi coordinates in the ternary configuration. Let $r_O,r_{H{e}_1}$, and $r_{H{e}_2}$ denote the space vectors of the three clusters, and $M_O$ and $M_{He}$ their masses, respectively. Then the sets of Jacobi coordinates are $t_c=r_O-r_{H{e}_1}$, $v_c=r_{H{e}_2}-r_N$, where $r_N=(M_O{r}_O+M_{He}{r}_{H{e}_1})/(M_O+M_{He})$, and $t_d=r_{H{e}_2}-r_{H{e}_1}$, $v_d=r_O-r_B$, where $r_B=(M_{He}{r}_{H{e}_1}+M_{He}{r}_{H{e}_2})/(M_{He}+M_{He})$.

The two binary configurations are obtained through projection. Each of them corresponds to one coordinate set with some constraint on one of the relative motions. The relative motion quantum number $n_{v_c}$ or $n_{v_d}$ of the binary configurations is obtained from the $N_6=n_v+n_t$ ternary quantum numbers, by fixing $n_t$ for the configurations c and d, respectively, (in our example, $n_{t_c}$ = 8, $n_{t_d}$ = 4). Then, the ternary Hamiltonians for the Jacobi coordinates c and d are

$$H_c=(ϵ+8\gamma )+\gamma n_{v_c}+\delta C_{\mathrm{SU}\left(3\right)}^{\left(2\right)}+(\beta +8\theta )L^2+\theta n_{v_c}{L}^2,$$

(16)

$$H_d=(ϵ+4\gamma )+\gamma n_{v_d}+\delta C_{\mathrm{SU}\left(3\right)}^{\left(2\right)}+(\beta +4\theta )L^2+\theta n_{v_d}{L}^2.$$

(17)

Note that these equations have exactly the form found from the eigenvalue condition of the previous subsection (Equation (13) and the constraints on its parameters). Therefore, both the model spaces and the Hamiltonians of the two binary configurations are obtained from the underlying ternary configuration by simple projections.

The projection method discussed here results in identical spectra (of the common U(3) states) of different configurations by applying configuration-dependent interactions.

2.2.4. Transformation of Operators and States

The logic of MUSY with configuration-dependent interaction, as discussed above, shows some similarity to that of the local gauge invariance. In particular, the invariance emerges from the simultaneous transformations of the state vectors and operators.

In order to illustrate the similarity of the logical structure we recall here the basic idea of the local gauge invariance [31]. The gauge transformation is defined as $\psi \to U\psi$, $U=exp\left(\frac{iq}{\hslash c}{\chi }^{\prime }\right)$. When ${\chi }^{\prime }$ is a constant, then the Schrödinger Equation (and the Hamiltonian) is invariant with respect to the gauge transformation because the constant gauge factor ($\chi =\frac{iq}{\hslash c}{\chi }^{\prime }$) "slips through" the differentiation: $\partial \left(e^{i\chi }\psi \right)=e^{i\chi }(\partial \psi )$, $e^{i\chi }H{e}^{-i\chi }=e^{i\chi }{e}^{-i\chi }H=H$. This is the global gauge invariance.

Under the $U=exp\left(\frac{iq}{\hslash c}{\chi }^{\prime }\left(x\right)\right)$ local gauge transformation, however, the Schrödinger equation is not invariant because the phase factor can no longer escape differentiation. In order to make it invariant, we replace the derivation ∂ with covariant derivation $D=\partial +\frac{iq}{\hslash c}A$. In other words, the invariance is obtained when the transformation of the wave function is completed using a proper transformation of the operator.

When MUSY is realized with transformed operators, it has a similar scenario: the transformation of the wave function is equipped with a corresponding transformation of the operators.

3. Supersymmetry

Supersymmetry connects the properties of boson and fermion systems. SUSY finds applications in nuclear physics in different ways. The most extensive studies have been carried out in relation to collective motion, when the bosons are excitation quanta, and the fermions are nucleons. The concept of dynamical supersymmetry in nuclear structure was introduced in relation to quadrupole collectivity [3,4,5], and most of its applications have taken place in this area. Therefore, it is straightforward to introduce the basic concept in this language.

Quadrupole collectivity is known to dominate the low-energy spectrum of nuclei for a long time [32,33]. A successful algebraic model of the quadrupole motion is the interacting boson model [22], in which the building blocks are the quadrupole and monopole bosons. Microscopically, they represent nucleon pairs in the valence shell with $L=2$, and $L=0$ angular momenta, respectively. The (6 × 6 = 36) particle-number conserving bilinear products of their creation and annihilation operators generate the U(6) group, called the dynamical group of IBM. A single irreducible representation of this group generates the complete collective spectrum of a nucleus and provides us with the symmetry classification of the states. The classification schemes and their quantum numbers are defined by the subgroup chains: $\mathrm{U}\left(6\right)\supset \mathrm{G}\supset \mathrm{SO}\left(3\right),$ where G stands for $\mathrm{U}\left(5\right)\supset \mathrm{O}\left(5\right),$ or SU(3), or $\mathrm{O}\left(6\right)\supset \mathrm{O}\left(5\right).$ When the Hamiltonian is expressed in terms of the invariant operators of a single chain, we can speak of dynamical symmetry, and the energy eigenvalue problem has an analytical solution. The three dynamical symmetries correspond to a vibrator (U(5)), rigid rotor (SU(3)), and a gamma-soft (O(6)) rotor, respectively.

If fermions (nucleons) can occupy m single-particle states, then the ($m\times m$) number-conserving bilinear products of their creation and annihilation operators generate the U(m) group. Therefore, the interacting boson fermion model (IBFM) of the quadrupole motion has an algebraic structure of $\mathrm{U}\left(6\right)\otimes \mathrm{U}\left(\mathrm{m}\right)$, as long as the number of bosons and fermions are unchanged [4]. By allowing the transformations between bosons and fermions, one achieves supersymmetry, which is obviously a deeper symmetry and has an algebraic structure of graded (or super) algebra U(6/m): $\mathrm{U}(6/\mathrm{m})\supset {\mathrm{U}}^{\mathrm{B}}\left(6\right)\otimes {\mathrm{U}}^{\mathrm{F}}\left(\mathrm{m}\right)$. In the supersymmetric model, only the total number of particles (i.e., the sum of boson and fermion numbers) is conserved.

More recently, a similar scheme has been proposed in relation to clusterization, i.e., dipole collective motion [6]. In this case, the bosonic sector contains dipole and monopole bosons for the description of the relative motion of the clusters. They are the building blocks of the vibron model of the U(4) group structure [23,24]. The fermionic sector is provided by the nucleons occupying m single-particle states. Thus, the 1$\hslash \omega$ shell ($0p$ shell), e.g., has $m=12$. When the bosons and fermions are conserved separately, then we have a model with U(4)⊗U(m) group structure, while the supertransformations enlarge it to U(4/m).

The overwhelming studies in relation to SUSY in the nuclear structure were carried out for the quadrupole collectivity, based on IBFM, and much less application is available concerning clusterization. However, since the logic and mathematical structure of the two applications are exactly the same, in what follows, we use the general notation, referring to both quadrupole and dipole collectivity, when appropriate.

3.1. Lie Subalgebras

The usual SUSY in nuclear structure is based on the group chain, starting with

$$\mathrm{U}(\mathrm{n}/\mathrm{m})\supset {\mathrm{U}}^{\mathrm{B}}\left(\mathrm{n}\right)\otimes {\mathrm{U}}^{\mathrm{F}}\left(\mathrm{m}\right),$$

(18)

where $n=6$ for quadrupole and $n=4$ for dipole collectivity. As it is seen here already in the first step of the subgroup chain classical Lie groups appear. The quantum numbers that distinguish the states of a supersymmetric multiplet are provided using the representation labels of the subgroups of the group chains

$${\mathrm{U}}^{\mathrm{B}}\left(\mathrm{n}\right)\supset {\mathrm{G}}_{\mathrm{n}}^{\mathrm{B}}\supset {\mathrm{G}}_{\mathrm{n}}^{\prime \mathrm{B}}\supset ...,{\mathrm{U}}^{\mathrm{F}}\left(\mathrm{m}\right)\supset {\mathrm{G}}_{\mathrm{m}}^{\mathrm{F}}\supset {\mathrm{G}}_{\mathrm{m}}^{\prime \mathrm{F}}\supset \dots ,$$

(19)

and by those of the combined boson–fermion groups G${}^{B+F}$.

As an illustrative example, let us consider here the U(6/12) supersymmetry. This is an especially interesting case. The nucleons can occupy the orbitals with angular momenta $j=1/2,3/2$, and $5/2$, resulting in the fermion dynamical algebra U${}^F$(12). By introducing a pseudo-spin $\tilde{s}=1/2$ and pseudo-orbital angular momenta, $\tilde{l}=0,2$ one obtains ${\mathrm{U}}^{\mathrm{F}}\left(12\right)\supset {\mathrm{U}}^{\mathrm{F}}\left(6\right)\otimes {\mathrm{U}}^{\mathrm{F}}\left(2\right)$. Due to the isomorphism between the U${}^B$(6), and U${}^F$(6) algebras, one can establish supersymmetries corresponding to each dynamical symmetry of IBM (see [5] and references therein). For instance, the SO(6) dynamical supersymmetry is defined by the following group chain:

$$\begin{array}{c}\hfill \mathrm{U}(6/12)\supset {\mathrm{U}}^{\mathrm{B}}\left(6\right)\otimes {\mathrm{U}}^{\mathrm{F}}\left(12\right)\supset {\mathrm{U}}^{\mathrm{B}}\left(6\right)\otimes {\mathrm{U}}^{\mathrm{F}}\left(6\right)\otimes {\mathrm{U}}^{\mathrm{F}}\left(2\right)\supset {\mathrm{U}}^{\mathrm{B}+\mathrm{F}}\left(6\right)\otimes {\mathrm{SU}}^{\mathrm{F}}\left(2\right)\supset \\ \hfill {\mathrm{SO}}^{\mathrm{B}+\mathrm{F}}\left(6\right)\otimes {\mathrm{SU}}^{\mathrm{F}}\left(2\right)\supset {\mathrm{SO}}^{\mathrm{B}+\mathrm{F}}\left(5\right)\otimes {\mathrm{SU}}^{\mathrm{F}}\left(2\right)\supset {\mathrm{SO}}^{\mathrm{B}+\mathrm{F}}\left(3\right)\otimes {\mathrm{SU}}^{\mathrm{F}}\left(2\right)\supset \mathrm{Spin}\left(3\right)\end{array}$$

(20)

A dynamically symmetric Hamiltonian (without the terms which contribute to the binding energy only) is:

$$\begin{array}{ccc}\hfill H& =& {\kappa }_0{C}_2\left[{\mathrm{U}}^{\mathrm{B}+\mathrm{F}}\left(6\right)\right]+{\kappa }_3{C}_2\left[{\mathrm{SO}}^{\mathrm{B}+\mathrm{F}}\left(6\right)\right]+{\kappa }_4{C}_2\left[{\mathrm{SO}}^{\mathrm{B}+\mathrm{F}}\left(5\right)\right]\hfill \\ & +& {\kappa }_5{C}_2\left[{\mathrm{SO}}^{\mathrm{B}+\mathrm{F}}\left(3\right)\right]+{\kappa }_5^{\prime }{C}_2\left[\mathrm{SU}\left(2\right)\right].\hfill \end{array}$$

(21)

This supersymmetry is realized in the ${}^{194,195}$Pt nuclei.

This kind of dynamical supersymmetry, which applies Lie algebras as the first subalgebras, is the most widely used type of nuclear supersymmetry. The Hamiltonian H of the system is expressed in terms of the invariant operators of the Lie subgroups; therefore, it does not show invariance with respect to the supergroup. Depending on whether or not it contains contributions from the Casimir operators of the boson and fermion groups separately or only from the invariants of the combined boson–fermion groups, it produces different, or similar, spectra in the boson and fermion sectors.

3.2. Graded Subalgebras

There are (exceptional) cases when superalgebras appear as subalgebras as well (see [34] and references therein). Such a dynamical symmetry may give rise to a Hamiltonian, that is invariant with respect to some supertransformations. As an example, we cite here the U(6/2) model of [34]. It contains quadrupole and monopole bosons and single-nucleon states with $j=1/2$, and the relevant group chain is

$$\mathrm{U}(6/2)\supset \mathrm{U}\left(5\right)\otimes \mathrm{U}(1/2).$$

(22)

In fact, the supertransformations are acting only between the monopole boson and the fermion states. This supersymmetry is realized in the ${}^{102}$Ru-${}^{103}$Rh-${}^{104}$Pd region. The U(1/2) supersymmetry has a similar formalism to that of the supersymmetry model of the quantum field theory, and in [34] a detailed comparison is presented. The Hamiltonian contains the number operators of the s and d bosons and the fermions:

$$H=ϵ(N_s+N_F)+N_d.$$

(23)

This Hamiltonian is invariant with respect to the supertransformations between the s bosons and fermions. It results in the degenerate states of different boson and fermion numbers. One can realize the similarity with the case of the invariant operators of MUSY.

3.3. SUSY without Dynamical Symmetry

It is not necessary that supersymmetry involves dynamical symmetry, i.e., the Hamiltonian is expressed in terms of invariant operators of a group chain. In [5,35], a different kind of SUSY was investigated, which is not based on the dynamical symmetries of the boson and fermion sectors; rather, the situation is in-between the limiting cases. This means that the Hamiltonian contains contributions from more than one group chain. This scenario is more general than the one we mentioned in the introduction with the common characteristics of SUSY and MUSY, i.e., having dynamical symmetry in each component of the composite system. Therefore, it enlarges the territory of the possible application of supersymmetry.

In [5,35], the U(6/12) SUSY model was applied with the Hamiltonian

$$\begin{array}{cc}\hfill H=& {\kappa }_0{C}_2\left[{\mathrm{U}}^{\mathrm{B}+\mathrm{F}}\left(6\right)\right]+{\kappa }_1{C}_1\left[{\mathrm{U}}^{\mathrm{B}+\mathrm{F}}\left(5\right)\right]+{\kappa }_1^{\prime }{C}_2\left[{\mathrm{U}}^{\mathrm{B}+\mathrm{F}}\left(5\right)\right]\\ \hfill +& {\kappa }_2{C}_2\left[{\mathrm{SU}}^{\mathrm{B}+\mathrm{F}}\left(3\right)\right]\\ \hfill +& {\kappa }_3{C}_2\left[{\mathrm{SO}}^{\mathrm{B}+\mathrm{F}}\left(6\right)\right]+{\kappa }_4{C}_2\left[{\mathrm{SO}}^{\mathrm{B}+\mathrm{F}}\left(5\right)\right]\\ \hfill +& {\kappa }_5{C}_2\left[{\mathrm{SO}}^{\mathrm{B}+\mathrm{F}}\left(3\right)\right]+{\kappa }_5^{\prime }{C}_2\left[\mathrm{SU}\left(2\right)\right].\end{array}$$

(24)

This kind of supersymmetry has been applied to ruthenium and rhodium isotopes [5,35].

Due to the construction of the Hamiltonian operator, the energy of the boson and fermion states are identical in the following sense: The zero fermion case of the SUSY Hamiltonian results in the boson Hamiltonian of IBM. Furthermore, the Hamiltonian (and the spectrum) of an odd-A nucleus can be predicted from those of the neighboring even–even nuclei to a large extent (apart from, e.g., spin–orbit splitting, which does not occur in boson systems but is present in odd-A nuclei).

The Hamiltonian (24) guarantees the similarity of the boson and fermion spectra, though its construction was not governed by the supertransformations. (In fact, the role of these transformations is not discussed in detail in [5,35].) This feature (i.e., the appearance identical spectra without explicit connecting transformations) show some similarity to the eigenvalue method of MUSY.

4. Summary and Conclusions

In this paper, we discussed some composite symmetries of atomic nuclei, which appear when the nucleus is thought to consist of two or more components. Two examples were considered. The multiconfigurational symmetry connects different (shell, quartet, and cluster) configurations of a nucleus. The supersymmetry unifies the description of boson and fermion sectors of nuclei. Bosons and fermions typically correspond to excitation quanta of the collective motion, and to nucleons, respectively. These symmetries are composite ones in the sense that they include a usual dynamical symmetry for each component, and there is a further symmetry that bridges the different components to each other. We paid special attention to the nature of the connecting symmetries.

Both in MUSY and in SUSY studies, one can find examples of exact connecting symmetries, which emerge from the application of invariant Hamiltonians. They produce similar spectra in different components: in the boson and fermion sectors of SUSY, or in different configurations of MUSY. In the case of MUSY, the invariant Hamiltonian is obtained as an operator that is contracted with respect to the particle index; therefore, it is a U(N) scalar, where N is the number of nucleons. In SUSY, it is a Casimir operator of a graded subalgebra in the algebra chain defining the dynamical supersymmetry. As for MUSY, the majority of the applications are carried out with invariant Hamiltonians, while in the case of SUSY, it is only exceptional.

Similar spectra of different components can be obtained with non-invariant operators as well. Suitable Hamiltonians can be found without well-defined connecting symmetries. In such cases, the mathematical structure of the connecting symmetry is not known explicitly. In particular, in the multiconfigurational symmetry, this situation is available via the wave function (or eigenvalue) connection. In this method, configuration-dependent interactions are present as well. The degeneracy of the corresponding states in different configurations is a result of the simultaneous transformations of the states and of the operators. In the supersymmetry, the Casimir operators of the combined boson–fermion system result in this kind of exact symmetry (as opposed to the case when also contributions from boson and fermion subgroups are involved).

For MUSY, a further type of connection was also found, which was mentioned above as the projection method. In this case, the transformation between two (binary cluster) configurations is considered, with configuration-dependent interactions, and it is found as projections of the transformations of an underlying (ternary) configuration, with an invariant Hamiltonian. In this case, the invariance of the operator is not valid with respect to all the transformations in the particle-index space (U(N)); rather, it is restricted to a (U(2)) subgroup of it.

Based on the present discussion, it seems that there is an interesting and rich variety of connecting transformations between the components of atomic nuclei. Further studies along this line can most probably deepen our understanding of these unifying symmetries.

The great advantage of composite symmetries is that they can describe a large amount of spectroscopic data in an integrated way. They may also be able to give predictions, e.g., for the spectrum of an odd-A nucleus from that of the neighboring even–even nucleus in SUSY [4,5], or for the high-lying cluster spectrum extrapolated from the low-lying quartet states in MUSY [36].