Some algebraic structures for the bosonic three-level systems

A simple systematic procedure to construct the three-level unitary and orthogonal algebras which appear in the bosonic three -level pairing models and for arbitrary choice of level degeneracies is presented. We draw attention to the existence of the three types of dynamical symmetry. The new analytical expressions of the energies are derived. The results presented show very clearly the duality between orthogonal algebras and the quasi-spin algebras for the three-level system with pairing interactions. The seniority selection rules of the transition operators in three level boson system are given. Illustrative calculations are carried out for U21 algebra.


Introduction
Since the birth of quantum mechanics, symmetry has acquired a central role in all branches of physics and group theory provides the mathematical tool to formulate symmetry principles. Modern developments in symmetry are putting more emphasis on the concept of dynamical symmetry (DS) [1][2][3]. The DS is a type of symmetry in which the Hamiltonian is expanded in elements of a Lie algebra, (G 0 ), called the spectrum generating algebra (SGA). The DS occurs if the Hamiltonian can be written in terms of the Casimir operators (COs) of a chain of nested algebras, The DS has been widely exploited in many different fields ranging from hadronic [4,5] and molecular physics [6,7] to nuclear physics [8][9][10][11]. On the other hand, in some applications [12,13], the predictions of an exact DS are not fulfilled and one is compelled to break it. The required symmetry breaking is achieved by including terms associated with (two or more) different subalgebra chains of the parent SGA in the Hamiltonian.
The notable application of DS and SGA in nuclear physics is the study of collective states for even-even nuclei using the interacting boson model (IBM) [14,15]. In the original version of the IBM, the identical bosons with angular momentum l is referred to as the l-bosons. Hence, the nucleus is described in terms of interacting s −(l=0) and d−(l=2) bosons. The SGA of the IBM is the unitary algebra U 6 . One obvious extension is gIBM [16][17][18][19][20][21] in which the next even angular momentum is considered hence to include a g−(l=4) boson. The U 15 is the SGA of the gIBM. In addition to those mentioned above, more important versions of the IBM [22][23][24] consists in the introduction of additional angular momenta (p, f with l=1, 3, respectively) together with the s and d boson. In the extensions of IBM, the bosons with p, f and g angular momentum are useful mostly as supplements of dipole and octupole degrees of freedom, which are generated by the system of the s and d boson.
Over the last few years, more attention is paid to the development of the IBM in a more general form [25][26][27]. The two-level boson model, or s−a boson model, is described in terms of interacting s− and a− boson where a is the positive integer angular momentum. The SGA of the s−a IBM is + U , n 1 a where = + n a 2 1.
a Within this framework, the purpose of this paper is to extend the work of the two-level pairing model to the generic three-level pairing model. Consequently, we have a more explicit insight into the algebraic structure of the IBM. Clearly, such an algebraic approach is preferable not only because it gives the opportunity to expand the model to other cases without having to develop a completely new model, but also because it provides the analytical expressions of physical observations.
The generic three-level boson model ors ab boson model can be defined in terms of the s, a and bbosons. The SGA of thes ab IBM is + + U ,

General symmetries in IBM
By studying the algebraic structure of the spdIBM, sdgIBM, spdfIBM and sdgpfIBM, it becomes clear that there is a general algebra structure. It is as follows where the indices α, β and γ take the values a, b and c in cyclic order, n α = 2α + 1, n βγ = n β + n γ and n abc = n a + n b + n c . For instance, figure 1 shows lattice of algebras of U 21,pfh , using (1). It is important to note that if a, b or c equals zero then the corresponding unitary and orthogonal algebra will be º É a U U O. n 1 1 Therefore, O 3,α will be neglected in (2). Later, figure 2 illustrates this case for the algebra U 21 .
In the subalgebra chains, (1), seniority quantum number is the most important. Seniority refers to the number of particles that are not in pairs coupled to angular momentum = J 0. Seniority can be given a grouptheoretical definition starting from the Lie subalgebra chain  In general, this classification is not complete as indicated by the dots in the equations (3a)-(3c). The reduction is not multiplicity free and generally requires a multiplicity label α. The multiplicity of the angular momentum a in the decomposition of the IR [v a ] may be calculated by means of Littlewood rules [33].

Three level unitary and orthogonal algebras
Beginning, in order to simplify the notations which will appear later, A, B and C are used to designate spherical tensors (STs) with integer angular momenta a, b and c respectively (i.e., A a ≡A and A aα ≡A α ). An irreducible tensor product Ä [ ] A B e of two STs A and B is defined as the tensor of rank e whose components  Ä [ ] A B e can be expressed in terms of A α and B β according to, are the Clebsch-Gordan coefficients. The tensor commutator, [34,35], with definite angular momentum e and z component  is defined as , .
The tensor commutator [26] of two coupled operators is Consider, the unitary algebra U n abc which appears in the problem of many bosons placed in a three-level system, of possibly unequal degeneracies, with integer angular momenta a, b and c and we call it three level unitary algebra. The algebra U n abc is spanned by the generators The two level subalgebra U n bc is spanned by the generators ( n abc The number of COs of the U n abc is identical with the rank of the algebra. So, there are n abc COs of the U . n abc Since the Hamiltonian of IBM includes one and two body interactions, in most applications, there will be no need for COs beyond quadratic. The following expressions for the linear and quadratic COs of three level unitary algebra U n abc are derived. We obtain Table 1. The IRs of all subalgebras in the chains (1).

Algebra
IRs Algebra IRs Algebra IRs where the quadratic CO of single level unitary algebra U n a is given by and N a is the number operator, From now on we assume, unless stated explicitly otherwise, that the summation over the pair of indices (i, j) and (I, J) ranges over the elements of the sets In general, the quadratic COs of any multi-level unitary algebra have many interesting properties. For instance, they will be reduced to combination of linear and quadratic COs of single level unitary algebras. For example, the COs of four level unitary algebra can be represented as a combination of linear and quadratic COs of single level unitary algebras Now, let us consider a single level orthogonal algebra. The generators T t (A, A) with t=1, 3, K, 2a−1. form a subalgebra of the U n a which is referred to as the O . n a Moreover, for the single-level algebra, O , n a the operator T commutes with all the generators of the O . n a As a result, the normalized CO of the O n a is given by The O n abc subalgebra is spanned by the generators        Let ω ij =(−1) η with η=0 or 1 for i, j=a, b, c and i≠j. Then, we choice overall arbitrary phases Accordingly, the O n abc subalgebra is spanned by the generators in (15) and the set of generators The two level subalgebra O n bc is spanned by the generators T t (B, B), T t (C, C) and K t (B, C) which are obtained by omitting all terms with A in (15) and (22).
In order to write CO of the three-level orthogonal algebra in terms of multipole operators, the following operators are introduced  (15) and (22), using (5). The quadratic CO of two level orthogonal algebra O n ab is a special case of the (24).
The generators and COs of the three-level unitary and orthogonal algebras in all subalgebra chains of the sdgIBM are established in [17] and they are consistent with our results (6), (8), (15), (22) and (24), with a=0, b=2 and c=4. The main result of this section is that the COs of three level unitary algebras can be expressed in terms of the COs of single level algebras. However, the COs of O n abc can be written in terms of the COs of single level algebras and multipole operators.

The multipole operator and quasi-spin algebra
First, let us consider the QS algebra SU(1, 1) which appears in the problem of many bosons placed in a single level of angular momentum a (a integer) which we denote by SU a (1, 1) and call it single level QS algebra. The SU a (1, 1) algebra is spanned by S a + , S a− and S a0 which obeying the following commutation relations = + = -= - Next, we shall introduce the pair creation operator (S a+ ) and pair annihilation operator (S a− ) as, For the three-level system, the product S + S − is given by Using the recoupling and commutation relations, the mixed term of QS operators, 4(S a + S b− +S b + S a− ), is given by In the same way, we get The one level terms and mixed terms of S + S − may thus be combined to give an expression involving the O n abc and U n abc COs, if and only if the sign ωī j arising in the definition of the sum QS algebra and the sign ω ij entering into the definition of K t are related by   The interesting result is that the three-level pairing Hamiltonian can be expressed in terms of COs of algebras appearing in the subalgebra chains of U n abc (not just at the DS limit). Furthermore, by using results of section 3, the three-level pairing Hamiltonian can be written in terms of COs of single level algebras (N k , [ ] C U n

Three level transition operators
However, the operators become quasispin vectors in the choice The results of the section 4 show very clearly the duality between orthogonal algebras and the QS algebras for the three-level system with pairing interactions. In the language of group theory, the scalar operators + ( )

The dynamical symmetries
The DS plays an essential role in IBM. The main advantage of DS is that, whenever one such symmetry occurs, the following properties are then observed.   c bc abc   j j  abcl  3 , , , , , , In the following part, we briefly discuss the spectra of eigenvalues of the pairing Hamiltonian in the DS bc for the U 21 algebra. There are seven three element sets of unequal odd integers that have a sum of members equal to 21. The U 21 algebra can be constructed using seven different three level systems. Figure 2 represents these systems.
, , is used as the label of orthogonal algebra generated by the three levels a, b and c. Figure 2 shows also the lattice of algebras in the U 21 by taking y = -,   c the results for electromagentic transition strengths will be presented elsewhere -beyond the special cases conventionally considered. The group theoretical problems needed for these are being solved.
In addition, an important finding that emerges from this study is that the same unitary algebra and consequently its orthogonal subalgebra can be constructed using many different three level systems. Therefore, we can compare the spectra of eigenvalues of the pairing operator that produced by the different three level systems.
The relation between the orthogonal algebras and the QS algebras was discussed. The relation between pairing operators and the multipole operators for the three-level boson system is given in a closed form. The results presented here show very clearly the duality between orthogonal algebras and the QS algebras for the three-level system with pairing interactions.
Finally, using F spin and double spherical tensors, the extended pairing algebras with two-species boson will be the topic of future work, for which the quantum phase transitions have not yet been completely studied.