Multiple SU(3) algebras in shell model and \\ interacting boson model
aa r X i v : . [ nu c l - t h ] M a r Multiple SU (3) algebras in shell model andinteracting boson model V.K.B. Kota ∗ Physical Research Laboratory, Ahmedabad 380 009, India
R. Sahu
National Institute of Science and Technology,Palur Hills, Berhampur-761008, Odisha, India
P.C. Srivastava
Department of Physics, Indian Institute of Technology,Roorkee 247 667, India
Abstract
Rotational SU (3) algebraic symmetry continues to generate new results in the shell model (SM).Interestingly, it is possible to have multiple SU (3) algebras for nucleons occupying an oscillatorshell η . Several different aspects of the multiple SU (3) algebras are investigated using shell modeland also deformed shell model based on Hartree-Fock single particle states with nucleons in sdg orbits giving four SU (3) algebras. Results show that one of the SU (3) algebra generates prolateshapes, one oblate shape and the other two also generate prolate shape but one of them givesquiet small quadrupole moments for low-lying levels. These are inferred by using the standardform for the electric quadrupole transition operator and using quadrupole moments and B ( E K = 0 + band in three different examples. Multiple SU (3) algebras extendto interacting boson model and using sdg IBM, the structure of the four SU (3) algebras in thismodel are studied by coherent state analysis and asymptotic formulas for E sdg IBM further support the conclusions from the sdg shell model examples. ∗ corresponding author, [email protected] . INTRODUCTION Elliott has recognized way back in 1958 that shell model (SM) admits SU (3) ⊃ SO (3)algebra and this will generate rotational spectra in nuclei starting with the interacting par-ticle picture [1, 2]. Following this, SU (3) algebra was developed in considerable detail byvarious groups and this includes methods to obtain SU (3) irreducible representations (ir-reps) and SU (3) Wigner-Racah algebra with codes for calculating SU (3) ⊃ SO (3) and SU (3) ⊃ SU (2) × U (1) reduced Wigner coefficients, SU (3) Racah coefficients, SU (3) co-efficients of fractional parentage and so on [3–10]. By mid 60’s it was recognized that the SU (3) symmetry is good for 1 p and 2 s d shell nuclei but due to the strong spin-orbit force itwill be a badly broken symmetry for 1 p f shell nuclei and beyond. Hecht, Draayer and oth-ers later recognized [11–15] that for heavy deformed nuclei, pseudo- SU (3) based on pseudospin and pseudo Nilsson orbits will be a useful symmetry and it gave rise to many newresults. Very recently, a proxy- SU (3) scheme by Bonatsos, Casten and others [16–18] hasappeared within SM with definite prediction for prolate dominance over oblate shape inheavy deformed nuclei. This SU (3) model is currently being investigated in more detail.In addition, in the multishell situation again SU (3) appears within the Sp (6 , R ) model ofRowe and Rosensteel [19–21] and this has given rise to the no-core-sympletic shell model[22, 23]. Going beyond SM, a major basis for the interacting boson model (IBM) of atomicnuclei is that with s and d bosons the spectrum generating algebra (SGA) is U (6) and it has SU (3) as a subalgebra generating rotational spectrum [24, 25]. Similarly, sdg IBM [26, 27], sdpf
IBM [28, 29] and also IBM-3 with isospin ( T ) and IBM-4 with spin-isospin ( ST ) degreesof freedom [25, 30] all contain SU (3) symmetry generating rotational spectra. In addition, inIBM-3 and IBM-4 models, SU (3) also appears for isospin ( T ) and spin-isospin ( ST ) degreesof freedom respectively. Similarly, for odd-A nuclei we have SU BF (3) × SU F (2) symmetryin IBFM model with Nilsson correspondence [31]. This extends to SU (3) in IBFFM forodd-odd nuclei [32, 33] and SU (3) in IBF M for two quasi-particle excitations [34]. With SU (3) generating rotational spectra within both SM and IBM, it is natural to look for newperspectives for SU (3) symmetry in nuclei.One curious aspect of SU (3) in nuclei is that in a given oscillator shell η , there will bemultiple SU (3) algebras. Very early it is recognized that in SM with s and d orbits there willbe two SU (3) algebras [35] but its consequences are not explored in any detail. Similarly,2n sd IBM there are two SU (3) algebras [25] and they are applied in phase transition studies[36]. Finally, it was also recognized that there will be four SU (3) algebras in sdg IBM [27].Except for the sd IBM, properties of multiple SU (3) algebras are not investigated in anydetail in the past. As we will show, for a given oscillator shell with major shell number η ,there will be 2[ η ] number of SU (3) algebras where (cid:2) η (cid:3) is the integer part of η/
2. In thepresent paper, following the recent investigation of multiple pairing algebras in SM and IBM[37], several different aspects of multiple SU (3)’s in SM and IBM are investigated. Now, wewill give a preview.In Section 2, multiple SU (3) algebras in SM generated by angular momentum operator L q and quadrupole moment operator Q q with different signs for the ℓ → ℓ ± Q · Q operators are given. Usingthese, correlations between different Q · Q operators are studied. In Section 3, Spectra andelectric quadrupole ( E
2) properties of these algebras are studied using shell model codesand also deformed shell model based on Hartree-Fock single particle states (called DSM[30]). Used here are examples with 6 protons, 6 protons plus 2 neutrons and 6 protonsplus 6 neutrons systems. In Section 4, results for multiple SU (3) algebras in IBM’s (withno internal degrees of freedom for the the bosons) are presented. Finally, Section 5 givesconclusions. II. PHASE CHOICE AND MULTIPLE SU (3) ALGEBRAS IN SHELL MODEL
Let us consider the situation where valence nucleons in a nucleus occupying an oscillatorshell with major shell number η . With the spin-isospin degrees of freedom for the nucleons,the spectrum generating algebra (SGA) is U (4 N ) and decomposing the space into orbitaland spin-isospin ( ST ) parts, we have U (4 N ) ⊃ U ( N ) × SU (4). Here, N = ( η + 1)( η + 2) / SU (4) is the Wigner’s spin-isospin SU (4) algebra; see for example [30, 38–41]. Also,for a given η , the the single particle (sp) orbital angular momentum ℓ takes values ℓ = η , η − . . . , 0 or 1. Note that, for nuclei with only valence protons or neutrons SU (4) changesto SU (2) generating spin S . As Elliott has established, the orbital U ( N ) algebra admits SU (3) subalgebra with U ( N ) ⊃ SU (3) ⊃ SO (3) where SO (3) generates orbital angularmomentum. The eight generators of SU (3) are the orbital angular momentum operators L q and quadrupole moment operators Q q . In LST coupling and using fermion creation ( a † )3nd annihilation ( a ) operators, L q = 2 X ℓ r ℓ ( ℓ + 1)(2 ℓ + 1)3 (cid:16) a † ℓ
12 12 ˜ a ℓ
12 12 (cid:17) , , q . (1)Note that ˜ a ℓ − m, − m s , − m t = ( − ℓ − m + − m s + − m t a ℓm, m s , m t where m s and m t are the S z and T z quantum numbers for a single nucleon. Similarly, the quadrupole operator is Q q = 2 X ℓ f ,ℓ i h η, ℓ f || Q || η, ℓ i i√ (cid:16) a † ℓ f
12 12 ˜ a ℓ i
12 12 (cid:17) , , . (2)Closure examination of the reduced matrix element h η, ℓ f || Q || η, ℓ i i of the quadrupoleoperator in the orbital space allows us to recognize that there will be multiple SU (3) sub-algebras in U ( N ). We will turn to this now.As Elliott considered [1], the quadrupole operator is Q q = q π (cid:2) r Y q ( θ, φ ) + p Y q ( θ p , φ p ) (cid:3) with oscillator length parameter b = 1. For a single shell. this is equivalent to using Q q = q π r Y q ( θ, φ ). Therefore, the reduced matrix elements of Q decompose into theradial part and angular part, (cid:10) η, ℓ f || Q || η, ℓ i (cid:11) = * η, ℓ f || r π Y ( θ, φ ) || η, ℓ i + (cid:10) η, ℓ f || r || η, ℓ i (cid:11) , (3)with the angular part given by [42], * η, ℓ || r π Y ( θ, φ ) || η, ℓ + = − s ℓ ( ℓ + 1)(2 ℓ + 1)(2 ℓ + 3)(2 ℓ − , * η, ℓ || r π Y ( θ, φ ) || η, ℓ + 2 + = * η, ℓ + 2 || r π Y ( θ, φ ) || η, ℓ + = s ℓ + 1)( ℓ + 2)(2 ℓ + 3) . (4)Similarly, the radial matrix elements are h η, ℓ || r || η, ℓ i = 2 η + 32 , h η, ℓ || r || η, ℓ + 2 i = h η, ℓ + 2 || r || η, ℓ i = α ℓ,ℓ +2 p ( η − ℓ )( η + ℓ + 3) ; α ℓ,ℓ +2 = α ℓ +2 ,ℓ = ± . (5)The phase factor α ℓ,ℓ +2 arises as there is freedom in choosing the phases of the radial wave-functions of a 3D oscillator. In SM studies, the standard convention is to use α ℓ,ℓ +2 = − ℓ [41–43]. However, Elliott in his SU (3) introductory paper [1] and in sd as well as4 dg IBM and IBFM the choice made is α ℓ,ℓ +2 = +1 for all ℓ [25, 26, 31, 44]. Thus, in generalwe have, L q = 2 X ℓ r ℓ ( ℓ + 1)(2 ℓ + 1)3 (cid:16) a † ℓ
12 12 ˜ a ℓ
12 12 (cid:17) , , q ,Q q ( α ) = − η + 3) X ℓ s ℓ ( ℓ + 1)(2 ℓ + 1)5(2 ℓ + 3)(2 ℓ − (cid:16) a † ℓ
12 12 ˜ a ℓ
12 12 (cid:17) , , q + X ℓ<η α ℓ,ℓ +2 s ℓ + 1)( ℓ + 2)( η − ℓ )( η + ℓ + 3)5(2 ℓ + 3) (cid:20)(cid:16) a † ℓ
12 12 ˜ a ℓ +2 ,
12 12 (cid:17) , , q + (cid:16) a † ℓ +2 ,
12 12 ˜ a ℓ
12 12 (cid:17) , , q (cid:21) ; α = ( α , , α , , . . . , α η − ,η ) for η even , α = ( α , , α , , . . . , α η − ,η ) for η odd , α = ( ± , ± , . . . ) . (6)Now, the most important result that can be proved by using the tedious but straight forwardangular momentum algebra is that the eight operators ( L q , Q q ′ ( α )) generate SU (3) algebraindependent of the choice of the α ’s and they satisfy the commutation relations [1, 41], (cid:2) L q , L q ′ (cid:3) = −√ h q q ′ | q + q ′ i L q + q ′ , (cid:2) L q , Q q ′ ( α ) (cid:3) = −√ h q q ′ | q + q ′ i Q q + q ′ ( α ) , (cid:2) Q q ( α ) , Q q ′ ( α ) (cid:3) = 3 √ h q q ′ | q + q ′ i L q + q ′ . (7)Thus, we have multiple SU (3) algebras SU α (3) in SM spaces generated by the operators inEq. (6). Clearly for a given η , there will be 2[ η ] number of SU (3) algebras; (cid:2) η (cid:3) is the integerpart of η/
2. Then, we have two SU (3) algebras in sd ( η = 2) and pf ( η = 3) shells, four SU (3) algebras in sdg ( η = 4) and pf h ( η = 5) shells, eight SU (3) algebras in ( sdgi ) ( η = 6)and ( pf hj ) ( η = 7) shells and so on. Thus, the first non-trivial situation that is not discussedin literature before is sdg or η = 4 shell with four SU (3) algebras SU ( − , − ) (3), SU (+ , − ) (3), SU ( − , +) (3) and SU (+ , +) (3). Here, α = ( α sd , α dg ) and ( − , − ) means ( α sd , α dg ) = ( − , − α sd , α dg ). In the reminder of this paper, we will use theexample of η = 4 shell to present some results from multiple SU (3) algebras. Before this,we will first consider the quadrupole-quadrupole interaction generated by Q q ( α ).5 . Matrix elements of Quadrupole-quadrupole interaction from multiple SU (3) algebras Investigation of multiple SU (3) algebras in shell model spaces needs firstly the singleparticle energies (spe) and two-body matrix elements (TBME) of the quadrupole-quadrupoleinteraction operator Q ( α ) · Q ( α ) for all phase choices α (also the spe and TBME for thesimpler L · L operator). The methods for obtaining these are well known [42] and we willgive only the final formulas. In order to derive formulas for the spe and TBME generatedby Q ( α ) · Q ( α ) operators, firstly notice that the Q q operator can be written as, Q q ( α ) = 2 X ℓ f ,ℓ i C α ℓ f ,ℓ i (cid:16) a † ℓ f
12 12 ˜ a ℓ i
12 12 (cid:17) , , q . (8)The C α ℓ f ,ℓ i follow easily from Eq. (6). From now on we will drop ’2’ and α in Q q ( α ) whenthere is no confusion. For a many particle system, Q · Q = m X i =1 Q ( i ) · Q ( i ) + 2 m X i 2) and (0 , , / Q ( α ) · Q ( α )operators with α = ( α sd , α dg ) = (+ , +) , (+ , − ) , ( − , +) and ( − , − ) using the results in SectionIIA.Given an operator O acting in m particle spaces ( O is assumed to be real), its traceover the m particle space is hhOii m = P γ h m, γ | O | m, γ i . Note that | m, γ i are m -particle states. Similarly, the m -particle average is hOi m = [ d ( m )] − hhOii m where d ( m )is m -particle space dimension. Using the spectral distribution method of French [45, 46],a geometry can be defined [46] with norm (or size or length) of an operator O given by || O || m = rD ˜ O ˜ O E m ; ˜ O is the traceless part of O . Following this, given any two operators O and O , the correlation coefficient ζ ( O , O ) = D f O f O E m || O || m || O || m , (15)gives the cosine of the angle between the two operators. Thus, O and O are same withina normalization constant if ζ = 1 and they are orthogonal to each other if ζ = 0 [45]. Most7 ABLE I. Correlation coefficient ζ between Q · Q operators with different values for the phases( α sd , α dg ) in sdg shell model m -particle spaces ( m is number of nucleons). Note that the totalnumber of single particle states (with spin and isospin) is 60. The ζ values in column 3 are for m = 4, 8, 12, 20, 30, 40, 50 and 56. See text for other details.( α sd , α dg ) ( α ′ sd , α ′ dg ) ζ ( − , − ) (+ , − ) 0 . , . , . , . , . , . , . , . − , +) 0 . , . , . , . , . , . , . , . , +) 0 . , . , . , . , . , . , . , . , − ) ( − , +) 0 . , . , . , . , . , . , . , . , +) 0 . , . , . , . , . , . , . , . − , +) (+ , +) 0 . , . , . , . , . , . , . , . recent application of norms and correlation coefficients is in understanding the structure ofmultiple pairing algebras in shell model [37].Applying Eq. (15), we have calculated ζ between the operators Q ( α sd , α dg ) · Q ( α sd , α dg )and Q ( α ′ sd , α ′ dg ) · Q ( α ′ sd , α ′ dg ) for all possible combinations of α ’s and ( α ′ )’s. Some resultsfor ζ are given in Table I. It is seen from the table that Q ( − , − ) · Q ( − , − ) is stronglycorrelated with Q (+ , − ) · Q (+ , − ). Similarly, the Q · Q ’s with ( α sd , α dg ) = (+ , +) and( − , +) are strongly correlated. However, the correlations between other pairs of Q · Q arequite small. Thus, SU ( − , − ) (3) and SU (+ , − ) (3) are expected to give similar results but quitedifferent from SU (+ , +) (3) and SU ( − , +) (3). This is seen in the results of detailed calculationspresented in the next section. It is important to stress that all the four SU α (3) algebrasgenerate the same spectrum for H ( α ) = Q ( α sd , α dg ) · Q ( α sd , α dg ) independent of ( α sd , α dg ).We will consider these in more detail in the following.8 II. RESULTS FOR SPECTRA, QUADRUPOLE MOMENTS AND E TRANSI-TION STRENGTHS FROM SM AND DSM With the sdg example, we have four Q · Q Hamiltonians, H ( − , − ) Q = − Q ( − , − ) · Q ( − , − ) ,H (+ , − ) Q = − Q (+ , − ) · Q (+ , − ) ,H ( − , +) Q = − Q ( − , +) · Q ( − , +) ,H (+ , +) Q = − Q (+ , +) · Q (+ , +) . (16)In this section we will present the results generated by these four H ’s for the yrast levels,quadrupole moments Q ( J ) of these levels and the B ( E J upto 10. Used for this purpose are the Antoine shell model code [47] and also the deformedshell model (DSM) based on Hartree-Fock states [30]. DSM is particularly important forbringing out shape information in a transparent manner and also it is useful for largerparticle numbers where SM calculations are impractical. We will test the SM results withanalytical results derived using SU (3) algebra and also test DSM using SM results. We willfirst present some analytical results from SU (3) algebra. A. Analytical results from SU (3) algebra With SU (3) symmetry of the H Q Hamiltonians, the shell model space for a m nucleonsystem decomposes into SU (3) irreducible representations (irreps) due to the equivalencebetween H Q and C ( SU (3)) as given by Eq. (14). If we have identical nucleons (protons orneutrons), the ground band belongs to the leading SU (3) irrep ( λ H , µ H ) with spin S = 0and J = L for even m (similarly with S = 1 / m ). It is easy to write a formula forobtaining ( λ H , µ H ) as given in [48]. The irreps for m identical nucleons in η = 4 shell aregiven in Table II. Similarly, for m nucleons with isospin T , we need to consider the lowestspin-isospin SU (4) irrep allowed for this system [38, 40] and this will then give ( λ H , µ H ) [48].The irreps ( λ H , µ H ) for m nucleons with T = | T z | are given in Table II. The eigenstates of9 ABLE II. Ground state or leading SU (3) irrep ( λ H , µ H ) for a given number m of identical nucleonsand also for a given number m of nucleons with isospin T = | T Z | . Results are given for the oscillatorshell η = 4. The ( λ H , µ H ) are given in the table as ( λ H , µ H ) m for identical nucleons with m ≥ λ H , µ H ) m,T for nucleons with T = | T z | and 3 ≤ m ≤ 15; for odd m values, 2 T value giveninstead of T value. More complete results are available in [48]. η = 4: identical nucleons(8 , ,(10 , ,(12 , ,(15 , ,(18 , ,(18 , , (18 , ,(19 , ,(20 , ,(22 , ,(24 , ,(22 , , (20 , ,(19 , ,(18 , ,(18 , ,(18 , , (19 , ,(20 , ,(16 , ,(12 , ,(9 , ,(6 , , (4 , ,(2 , ,(1 , ,(0 , ,(0 , , (0 , η = 4: even number of nucleons(16 , , ,(14 , , ,(12 , , ,(20 , , , (20 , , ,(19 , , , (18 , , ,(24 , , ,(25 , , ,(26 , , ,(22 , , ,(18 , , , (30 , , ,(30 , , ,(28 , , ,(26 , , , (23 , , ,(20 , , ,(36 , , ,(33 , , ,(30 , , ,(29 , , , (28 , , ,(26 , , , (24 , , ,(36 , , ,(36 , , ,(34 , , ,(32 , , ,(32 , , , (32 , , ,(26 , , ,(20 , , η = 4: odd number of nucleons(12 , , , (10 , , , (18 , , , (16 , , , (15 , , , (22 , , , (23 , , , (22 , , ,(18 , , , (27 , , , (28 , , , (26 , , , (22 , , , (19 , , , (33 , , , (30 , , ,(28 , , , (27 , , , (24 , , , (22 , , , (36 , , , (33 , , , (31 , , , (30 , , ,(30 , , , (28 , , , (22 , , , (36 , , , (37 , , , (35 , , , (34 , , , (34 , , ,(30 , , , (24 , , , (19 , , H Q are | m ; ( λ H µ H ) KL ; S : J T i and the ( λ H µ H ) → L reduction is well known giving,( λµ ) −→ L : K = min ( λ, µ ) , min ( λ, µ ) − , · · · , ,L = K, K + 1 , K + 2 , · · · , K + max ( λ, µ ) f or K = 0 ,L = max ( λ, µ ) , max ( λ, µ ) − , · · · , f or K = 0 , ( λ, µ ) → L ⇐⇒ ( µ, λ ) → L . (17)It is easy to see that the energies of the yrast levels in a even m system (assuming spin S = 0) are given by, E ( J = L ) = − ( λ H + µ H + λ H µ H + 3( λ H + µ H )) + 34 L ( L + 1) . (18)10n the examples presented ahead in the present paper we will only consider even m systemswith ( λ H µ H ) = ( λ 0) and then λ is even. A ( λ, 0) irrep with λ even, as seen from Eq. (17),generates the ground band with J = 0, 2, 4, . . . , λ . The ground state energy E gs = ( λ + 3 λ )and the energies of the J levels with respect to E gs are just 3 J ( J + 1) / 4. In addition, if wechoose the E Q of one of the H Q , then formulas for Q ( J )and B ( E 2) will be simple for the ( λ, 0) irrep of the corresponding SU (3) algebra. Just as itis considered in SM and DSM codes, we will take the E T E for identical nucleonsystems to be T E = Q q ( − , − ) e eff b (19)where b is the oscillator length parameter and e eff is effective charge. Then, analyticalformulas for the quadrupole moments ( Q ( J )) of the yrast levels and B ( E SU (3) algebra for the eigenstates obtained for H ( − , − ) Q as they belongto SU ( − , − ) (3). Using the results in [1, 31], we have for H ( − , − ) Q in Eq. (16) with T E in Eq.(19), Q (( λ, 0) : J = L ) = − L L + 3 (2 λ + 3) e eff b ,B ( E 2; ( λ, J = L → J − L − 2) = 516 π (cid:26) J ( J − λ − J + 2)( λ + J + 1)(2 J − J + 1) (cid:27) ( e eff ) b . (20)However, for systems with valence protons and neutrons, the E T E = (cid:2) e peff Q q ( − , − ; p ) + e neff Q q ( − , − ; n ) (cid:3) b (21)where e peff and e neff are proton and neutron effective charges. Again, using eigenstatesobtained for H ( − , − ) Q as they belong to SU ( − , − ) (3) and the T E in Eq. (21), a simple for-mula is obtained for Q ( J ) and B ( E | ( λ π , λ ν , λ π + λ ν , K = 0 , L, S = 0 , J = L i for a system with protons ( π ) and neu-trons ( ν ). Now, carrying out the SU (3) algebra using the mathematical formulation andanalytical results given in [6, 14, 49, 50] we have, Q (( λ, 0) : J = L ) = − L L + 3 2( λ + 3) X eff b ,B ( E 2; ( λ, J = L → J − L − 2) = 516 π (cid:26) J ( J − λ − J + 2)( λ + J + 1)(2 J − J + 1) (cid:27) ( X eff ) b ; X eff = e peff ( λ π + 3 λ π + λ π λ ν ) + e neff ( λ ν + 3 λ ν + λ π λ ν )( λ + 3 λ ) , λ = λ π + λ ν . (22)11ests of Eqs. (18), (20) and (22) are carried out using SM and DSM in the next threesubsections.It is important to stress that in the event we use the eigenstates of other H α Q , the groundband generated by them will belong to the ( λ 0) irrep of the corresponding SU α (3). However,then the Q ’s in T E in Eqs. (19) and (21) are no longer generators of these SU α (3)’s andhence the formulas in Eqs. (20) and (22) will not apply. In this situation, we have to use Q q ( − , − ) = Q q ( α )+∆ Q and ∆ Q follows easily from Eq. (6). Then, one has to carry out the SU (3) tensorial decomposition of ∆ Q with respect to SU α (3) and use the SU (3) Wigner-Racah algebra as described for example in [6, 14, 49, 50] for obtaining the matrix elementsof ∆ Q in the | ( λ K = 0 , L i states. This exercise is postponed to a future publication andinstead we will present results of full (without any truncation) SM results along with someDSM results in the next two subsections and only DSM results in the third subsection. Inaddition, to gain more insight into the other SU α (3) algebras, we will use the asymptoticformulas for quadrupole moments and B ( E sdg IBM in Section IV. B. SM and DSM results for multiple SU (3) algebras: ( sdg ) p example In our first example, we have analyzed a system of 6 protons in η = 4 shell, i.e. ( sdg ) p system by carrying out SM calculations using the four H Q Hamiltonians in the full SM space(matrix dimension in the m -scheme is ∼ ) using the Antoine code. For this system, theleading SU (3) irrep (see Table II) is (18 , 0) with S = 0. Then, Eq. (18) gives E gs =378 and SM calculations for all four H Q ’s are in agreement with this SU (3) result. Also,in the SM results the excitation energies of the yrast J states or ground band members( J = 0 , , , , , . . . ) are seen to follow for all the four H Q ’s the 3 J ( J + 1) / SU (3). Thus, it is verified by explicit SM calculations that all the four H Q ’s give SU (3)symmetry. Though the energy spectra are same, the wavefunctions of the yrast J statesare different. This is established by calculating Q ( J ) and B ( E T E given by Eq. (19). In all the calculations, e eff = 1 e and b = A / f m with A = 86 are used. Results from SM for the four H Q ’s are given in Tables III. It iseasy to see that the results for H ( − , − ) Q are in complete agreement with the results the SU (3)formulas given by Eq. (20). This is expected as T E in Eq. (19) is a generator of SU ( − , − ) (3)generated by H ( − , − ) Q . However, the results from the other three H Q ’s are quite different12 ABLE III. Shell model results for quadrupole moments Q ( J ) and B ( E J → J − 2) values forthe ground K = 0 + band members for a system of 6 protons in η = 4 shell. Results are givenfor the four Hamiltonians in Eq. (16). In the table ( − , − ) means we are using the wavefunctionsobtained using H ( − , − ) Q and similarly others. For other details see text. J Q ( J ) ef m ( − , − ) (+ , − ) ( − , +) (+ , +)2 +1 − . − . − . 85 13 . +1 − . − . − . 71 17 . +1 − . − . − . 12 19 . +1 − . − . − . 96 20 . +1 − . − . − . 99 18 . J B ( E J → J − e f m ( − , − ) (+ , − ) ( − , +) (+ , +)2 +1 . 97 291 . 34 0 . 42 42 . +1 . 31 388 . 79 1 . 38 58 . +1 . 68 377 . 33 3 . 90 61 . +1 . 24 325 . 68 9 . 24 58 . +1 . 52 254 . 56 18 . 38 53 . and do not follow the SU (3) results in Eq. (20) as the T E chosen is not a generator ofthe SU (3)’s generated by the three H Q ’s. It is seen from Tables III that the results for Q ( J ) and B ( E H (+ , − ) Q are closer to those from H ( − , − ) Q and this is consistent withthe correlation coefficients shown in Table II. The B ( E H ( − , +) Q are much smaller inmagnitude. Moreover, H ( − , − ) Q generates prolate shape and H (+ , +) Q oblate as seen clearly fromTable III. Quadrupole moments show that H Q (+ , − ) and H Q ( − , +) also generate prolateshapes but the deformation from H Q ( − , +) is quite small for the low-lying levels. To gainmore insight into these results, we have performed DSM calculations using the four H Q ’swith results as follows.Starting with the same model space, sp energies and two-body interaction, in DSMone solves Hartree-Fock (HF) sp equations self-consistently assuming axial symmetry. The13 E n e r g y ,5/2 ,3/2 ,7/2 ,5/2 ,9/2 X X X X X X O OO O O O FIG. 1. Hartree-Fock sp spectrum (it is same for both protons and neutrons) and the lowestintrinsic state for the ( sdg ) p , n system generated by the four H Q operators in Eq. (16). In thefigure, the symbol × denotes neutrons and 0 denotes protons. Shown in the figure are the k valuesof the sp orbits and each orbit is doubly degenerate with | k i and |− k i states. The spectrum is samefor all the four Hamiltonians although the sp wavefunctions are different. The HF energy E HF for the lowest intrinsic state is -1351.73 (note that E is unit less and the unit MeV has to be putback after multiplying with an appropriate scale factor if the results are used for a real nucleus)for all the four Hamiltonians. The intrinsic quadrupole moments (in units of b ), calculated using T E = Q q ( − , − ) b as the quadrupole operator, for H ( − , − ) Q , H (+ , − ) Q , H ( − , +) Q and H (+ , +) Q are 71 . . 43, 5 . 06 and − . 45 respectively. See text for other details. p - h ) exci-tations over the lowest-energy intrinsic state (lowest configuration). Carrying out angularmomentum projection from each intrinsic state and performing band mixing, orthonormal-ized | J K i states are obtained. See [30] for full details and many applications of DSM. Latestapplication of DSM is to dark matter studies [51]. In the present DSM calculations, only thelowest intrinsic state is considered. It is found that the four H Q ’s generate the same HF spspectrum and it is same as shown in Fig. 1 ahead except for some scale factors. The lowestintrinsic state is obtained by putting two protons each in the 1 / , 1 / and 3 / states.The intrinsic quadrupole moments (in units of b ) for H ( − , − ) Q , H (+ , − ) Q , H ( − , +) Q and H (+ , +) Q are+35 . 85, 24 . 4, 1 . 83 and − . 63 respectively. Thus, H ( − , − ) Q generates prolate shape and H (+ , +) Q generates oblate shape in agreement with SM. It is important to emphasize that the intrinsicquadrupole moments are calculated using T E = Q q ( − , − ) b as the quadrupole operator.The ground state energy for the 6 proton system is found to be, for all the four H Q ’s sameas the exact SU (3) values within less than 1% deviation. The energies of the yrast J statesfrom the ground state are also same for four H Q ’s and they follow the 3 J ( J + 1) / Q ( J )’s and B ( E H ( − , − ) Q , the Q ( J ) values (in ef m unit) are − . − . − . − . 07 and − . 29 for J = 2, 4, 6, 8 and 10 respectively. The corresponding B ( E J → J − 2) values(in e f m unit) are 582 . 78, 810 . 44, 848 . 78, 822 . 26 and 755 . 63 respectively. Thus, for largerparticle systems where SM calculations are not possible, one can use with confidence DSMfor further insight into the results from the four H Q ’s, i.e. from multiple SU (3) algebras andthis is used in Section III-D. C. SM results for multiple SU (3) algebras: ( sdg ) (6 p , n ) T =2 example In our second example, we have considered a system of 6 protons and 2 neutrons in η = 4shell, i.e. ( sdg ) p , n system and carried out SM calculations using the four H Q Hamiltoniansin the full SM space (dimension in the m -scheme is ∼ × ) using Antoine code. For thissystem, the leading SU (3) irrep (see Table II) is (26 , 0) with S = 0 and T = 2. Then, Eq.(18) gives E gs = 754 and SM calculations for all four H Q ’s is in agreement with this SU (3)result. Also, in the SM results the excitation energies of the yrast J states or ground band15 ABLE IV. Shell model results for quadrupole moments Q ( J ) and B ( E J → J − 2) values forthe ground K = 0 + band members for a system of 6 protons and 2 neutrons in η = 4 shell. Resultsare given for the four Hamiltonians in Eq. (16). In the table ( − , − ) means we are using thewavefunctions obtained using H ( − , − ) Q and similarly others. For other details see text. J Q ( J ) ef m ( − , − ) (+ , − ) ( − , +) (+ , +)2 +1 − . − . − . 84 23 . +1 − . − . − . 96 30 . +1 − . − . − . 17 32 . +1 − . − . − . 17 32 . +1 − . − . − . 13 32 . J B ( E J → J − e f m ( − , − ) (+ , − ) ( − , +) (+ , +)2 +1 . 70 629 . 26 17 . 14 140 . +1 . 04 876 . 18 25 . 78 198 . +1 . 85 920 . 91 30 . 98 214 . +1 . 68 899 . 62 36 . 37 218 . +1 . 87 841 . 70 42 . 61 215 . members ( J = 0 , , , , , . . . ) are seen to follow for all the four H Q ’s the 3 J ( J + 1) / SU (3). Thus, it is again verified by explicit SM calculations that all the four H Q ’s give SU (3) symmetry. The wavefunctions of the yrast J states are investigated bycalculating Q ( J ) and B ( E T E in Eq. (21). In allthe calculations, e peff = 1 . e , e neff = 0 . e and b = A / f m with A = 88 are used. Note thatthe ground (26 , 0) irrep arises from the strong coupling of the (18 , 0) irrep for the 6 protons(see the previous Section) and the (8 , 0) irrep for the two neutrons. Therefore, formulas inEq. (22) will apply for the states from H Q ( − , − ). Results from SM for the four H Q ’s aregiven in Tables IV. It is easy to see that the results for H ( − , − ) Q are in complete agreementwith the formulas in Eq. (22). This is expected as the proton and neutron parts of T E in Eq. (21) are generators of SU ( − , − ) (3) for protons and neutrons respectively. However,16 ABLE V. Deformed shell model results for quadrupole moments Q ( J ) and B ( E J → J − K = 0 + band members for a system of 6 protons and 6 neutrons (with T = 0)in η = 4 shell. Results are given for the four Hamiltonians in Eq. (16). In the table ( − , − )means we are using the wavefunctions obtained using H ( − , − ) Q and similarly others. Numbers in thebrackets in the second column are exact SU (3) results for H ( − , − ) Q . For other details see text. J Q ( J ) ef m ( − , − ) (+ , − ) ( − , +) (+ , +)2 +1 − . − . − . − . 63 26 . +1 − . − . − . − . 02 33 . +1 − . − . − . − . 66 37 . +1 − . − . − . − . 91 39 . +1 − . − . − . − . 86 40 . J B ( E J → J − e f m ( − , − ) (+ , − ) ( − , +) (+ , +)2 +1 . . 93) 1069 . 13 4 . 61 164 . +1 . . 59) 1503 . 04 7 . 43 233 . +1 . . 13) 1608 9 . 98 254 . +1 . . 99) 1613 . 12 13 . 44 261 . +1 . . 51) 1565 . 64 18 . 32 262 . the results from the other three H Q ’s are quite different as in the previous ( sdg ) p example.Again, it is seen from Tables IV that the results for Q ( J ) and B ( E H (+ , − ) Q are closerto those from H ( − , − ) Q . The B ( E H ( − , +) Q and H (+ , +) Q are much smaller in magnitude.Moreover, H ( − , − ) Q generates prolate shape and H (+ , +) Q oblate as in the previous example.Finally, let us mention that we have also carried out DSM calculations for this example andthey are all in agreement with SM results. 17 . DSM results for multiple SU (3) algebras: ( sdg ) (6 p , n ) T =0 example In our final example we have considered a system of 12 nucleons with T = 0 in η = 4shell, i.e. ( sdg ) (6 p , n ) T =0 system. Here the dimension in the m -scheme in SM is ∼ and therefore SM calculations are not possible with our computational facilities. Thus, inthis example DSM gives the predictions for four H Q ’s and only for H ( − , − ) Q we have exact SU (3) results (they will be same as SM results if performed) from Section III.A. Carryingout DSM calculations for this system, it is found that the four H Q ’s generate the same HFsp spectrum as shown in Fig. 1. Using the lowest intrinsic shown in Fig. 1, it is seenfrom the intrinsic quadrupole moments for the four H ’s that H ( − , − ) Q generates prolate shapeand H (+ , +) Q generates oblate shape in agreement with SM. The ground state energy for thesystem is found to be − . H Q ’s against the exact SU (3) value − SU (3) irrep for the ground band is (36 , 0) and thisgenerated by the irrep (18 , 0) for the 6 protons and (18 , 0) for the 6 neutrons. The energiesof the yrast J states are also same for four H Q ’s and they are also within 1% deviation fromthe 3 J ( J + 1) / Q ( J ) and B ( E e peff = 1 . e , e neff = 0 . e and b = A / f m with A = 92. Note that the ground (36 , 0) irrep arises fromthe strong coupling of the (18 , 0) irreps of the 6 protons and the 6 neutrons. Therefore,formulas in Eq. (22) will apply for the states from H Q ( − , − ). DSM results for H ( − , − ) Q , asshown in Table V are in complete agreement with the formulas in Eq. (22) as expected.However, the results from the other three H Q ’s are quite different as in the previous ( sdg ) p and ( sdg ) p, n examples. Again, it is seen from Tables V that the results for Q ( J ) and B ( E H (+ , − ) Q are closer to those from H ( − , − ) Q . The B ( E H ( − , +) Q and H (+ , +) Q are much smaller in magnitude. Moreover, H ( − , − ) Q generates prolate shape and H (+ , +) Q oblateas in the previous examples. Thus, the results in Tables III-V are generic results for thefour H Q ’s. 18 V. MULTIPLE SU (3) ALGEBRAS IN INTERACTING BOSON MODEL In the interacting boson models with sd ( ℓ = 0 , 2) or sdg ( ℓ = 0 , , 4) bosons (and theirappropriate generalizations to pf , sdgi etc.), the eight operators ( L q , Q q ( α )) are L q = X ℓ r ℓ ( ℓ + 1)(2 ℓ + 1)3 (cid:16) b † ℓ ˜ b ℓ (cid:17) q ,Q q ( α ) = − (2 η + 3) X ℓ s ℓ ( ℓ + 1)(2 ℓ + 1)5(2 ℓ + 3)(2 ℓ − (cid:16) b † ℓ ˜ b ℓ (cid:17) q + X ℓ<η α ℓ,ℓ +2 s ℓ + 1)( ℓ + 2)( η − ℓ )( η + ℓ + 3)5(2 ℓ + 3) (cid:20)(cid:16) b † ℓ ˜ b ℓ +2 (cid:17) q + (cid:16) b † ℓ +2 ˜ b ℓ (cid:17) q (cid:21) ; α ℓ,ℓ +2 = ± . (23)Note that b † and b are boson creation and annihilation operators and ˜ b ℓm = ( − ℓ − m b ℓ − m .Again, after some tedious angular momentum algebra, it is easy to prove that for all choicesof α ℓ,ℓ +2 = ± 1, Eq. (7) is valid and therefore giving a SU (3) algebra for each choice of the α ’s. With α ℓ,ℓ +2 taking +1 or − η there will be 2 [ η/ number of SU (3)algebras in IBM’s just as in SM. It is important to stress that α ℓ,ℓ +1 = +1 for all ℓ valuesis the standard choice in sd IBM and sdg IBM. As an example, in sd IBM with η = 2, the( L q , Q q ) operators generating multiple SU (3) algebra are, L q = √ (cid:16) d † ˜ d (cid:17) q ,Q q ( α sd ) = √ " − √ (cid:16) d † ˜ d (cid:17) q + α sd (cid:16) s † ˜ d + d † ˜ s (cid:17) q ; α sd = ± . (24)giving two SU α (3) algebras. In sd IBM they are discussed in the context of quantumphase transitions (QPT) [36]. The α sd = +1 and − sdg IBM with η = 4 there will be four SU α (3) algebrasgenerated by, L µ = √ (cid:16) d † ˜ d (cid:17) µ + 2 √ (cid:0) g † ˜ g (cid:1) µ ,Q µ ( α sd , α dg ) = r ( − r 221 ( d † ˜ d ) µ − r 337 ( g † ˜ g ) µ + α sd r (cid:16) s † ˜ d + d † ˜ s (cid:17) µ + α dg √ (cid:16) d † ˜ g + g † ˜ d (cid:17) µ ) , (25)with α sd = ± α dg = ± 1. 19 = 0 , ( α sd , α dg ) = (+ , +) γ = 0 , ( α sd , α dg ) = (+ , − ) γ = π/ , ( α sd , α dg ) = ( − , − ) γ = π/ , ( α sd , α dg ) = ( − , +) (a) (b) E FIG. 2. Energy functional E = E SU sdg (3) /N as a function of β and β . (a) plot for ( α sd = 1 , α dg =1) with γ = 0 ◦ and ( α sd = − , α dg = − 1) with γ = 60 ◦ . Note that the energy functional is samefor both of these choices as can be seen from Eq.(30). (b) Same as (a) but for ( α sd = 1 , α dg = − γ = 0 ◦ and ( α sd = − , α dg = +1) with γ = 60 ◦ . A. Geometry of multiple SU (3) algebras in sd IBM and sdg IBM In order to have some insight into the multiple SU (3) algebras in IBM, let us examinethe geometric shapes generated by them using coherent states. Starting with sd IBM, thecoherent state is | N ; β ; γ i = h N ! (cid:0) β (cid:1) N i − / ( s † + β " cos γ d † + r 12 sin γ (cid:16) d † + d †− (cid:17) N , (26)where β ≥ ◦ ≤ γ ≤ ◦ . Now, let us consider the SU (3) Hamiltonian H α sd SU sd (3) = − Q ( α sd ) · Q ( α sd ) (27)and − Q ( α sd ) · Q ( α sd ) = − C ( SU α sd (3)) + L · L . It is important to note that H SU sd (3) generates the same spectrum for the two choices of α sd . In the N → ∞ limit, the coherent20tate expectation value of H SU sd (3) is given by E SU αsdsd (3) ( N ; β , γ ) = h N ; β , γ | − Q ( α sd ) · Q ( α sd ) | N ; β , γ i = − N (1 + β ) (cid:20) β + β √ α sd β cos 3 γ (cid:21) . (28)Minimizing the SU (3) energy functional E SU sd (3) ( N ; β , γ ) gives the equilibrium solutions( β , γ ) to be β = √ γ = 0 ◦ for α sd = +1 and γ = 60 ◦ for α sd = − 1. Also, forboth situations the equilibrium energy is − N and this is same as the large N eigenvalueof − C ( SU (3)) in the h.w. (2 N, 0) irrep [also for the lowest weight (0 , N ) irrep]. Notethat the eigenvalue of C ( SU (3)) in a SU (3) irrep ( λµ ) is simply λ + µ + λµ + 3( λ + µ ).Also, the formula in Eq. (28) is good in the limit N → ∞ and in this limit L · L will notcontribute as only terms of the order of N will survive. Thus, α sd = ± sd IBM are well known [25, 36].First non-trivial situation happens with sdg IBM and for this we will consider the threeparameter coherent state used in [44, 52] in terms of ( β , β , γ ) parameters for a N bosonsystem, | N ; β ; β , γ i = h N ! (1 + β + β ) N i − / n s † + β h cos γ d † + q sin γ (cid:16) d † + d †− (cid:17)i + β h (5 cos γ + 1) g † + q sin 2 γ (cid:16) g † + g †− (cid:17) + q sin γ (cid:16) g † + g †− (cid:17)io N | i . (29)Note that β ≥ −∞ ≤ β ≤ + ∞ and 0 ◦ ≤ γ ≤ ◦ respectively. Using the results given[26, 52], the SU (3) energy functional is given by E SU α sdg (3) ( N ; β , β , γ ) = h N ; β ; β , γ | − Q ( α ) · Q ( α ) | N ; β ; β , γ i = − N β + β ) " α sd β + 384 √ α sd α dg β β + 352 √ α sd β cos 3 γ + 64 √ α sd β β cos 3 γ + 3456245 α dg β β + 1056 √ α dg β β cos 3 γ + 484147 β + 192 √ α dg β β cos 3 γ + 880441 (cid:0) − cos γ (cid:1) β β + 4001323 (cid:0) − γ (cid:1) β (cid:21) . (30)Note that α = ( α sd , α dg ). Minimizing E SU sdg (3) ( N ; β , β , γ ) with respect to β , β and γ willgive the equilibrium (ground state) shape parameters ( β , β , γ ) and the correspondingequilibrium energy E SU sdg (3) . Results are given in Table VI. As seen from the Table VI,the four values of ( α sd , α dg ) generate four combinations of ( β , β , γ ). These can be easily21 ABLE VI. Equilibrium shapes for the four SU (3) algebras in sdg IBM. For ( α sd , α dg ) = ( − , +1)and ( − , − β and β values for both γ = 0 ◦ and 60 ◦ and they are equivalent. α sd α dg β β γ E SU sdg (3) +1 +1 p / p / ◦ − N +1 -1 p / − p / ◦ − N -1 +1 p / − p / ◦ − N − p / − p / ◦ − N -1 -1 p / p / ◦ − N − p / p / ◦ − N understood from the symmetries under β → − β , β → − β and γ = 0 ◦ → ◦ . We have forexample E ( β , β , γ ; α sd = 1 , α dg = 1) = E ( β , − β , γ ; α sd = 1 , α dg = − E ( β , β , γ ; α sd =1 , α dg = 1) = E ( β , β , γ + 60 ◦ ; α sd = − , α dg = − 1) = E ( − β , − β , γ ; α sd = − , α dg = 1) = E ( − β , β , γ ; α sd = − , α dg = − γ = 60 ◦ can bechanged to γ = 0 ◦ with β → − β as given in Table VI. More importantly, for all the foursolutions, the E SU sdg (3) = − N . This energy value is same as the large N eigenvalue of − C ( SU (3)) in (4 N, 0) irrep. This then implies that the internal structure of the (4 N, β and β for γ = 0 ◦ and 60 ◦ for the four choices of ( α sd , α gd ). B. Large N results for quadrupole moments and B ( E ’s For further understanding of the four solutions for SU sdg (3), we have examined quadrupolemoments and B ( E 2) values in the ground K = 0 band generated by the four solutions inTable VI. Note that the intrinsic state structure for the K = 0 ground band is | N ; K = 0 i = ( N !) − / (cid:16) x s † + x d † + x g † (cid:17) N | i (31)where x = p / x = β / √ x = β / √ γ = 0 ◦ . It is easy to construct theangular momentum projected states | N ; K = 0 , L, M i and calculate quadrupole moments Q ( L ) and B ( E L → L − 2) for the ground band. The formulation for these is given in22etail in [53] and valid to order 1 /N where N is the boson number. Then we have, Q ( L ) = h LL | Q | LL i = h LL | LL i√ L + 1 (cid:10) L || Q || L (cid:11) ,B ( E L → L − 2) = 516 π |h L − || Q || L i| (2 L + 1) ; h N ; K = 0 , L f || Q || N ; K = 0 , L i i = h N p (2 L i + 1) i h L i | L f , i × (cid:20) B + N (cid:18) B − B − B a (cid:19) − L i ( L f + 1) aN (cid:26) B + F a δ L f ,L i − F a δ L f ,L i +2 (cid:27)(cid:21) ; L f = L i or L f = L i + 2 B mn = X ℓ ′ ,ℓ [ ℓ ′ ( ℓ ′ + 1)] m [ ℓ ( ℓ + 1)] n h ℓ ′ ℓ | i t ℓ ′ ,ℓ x ℓ ′ x ℓ ,F = B − B − B + 12 B , F = B − B + 6 B − B ,a = X ℓ ℓ ( ℓ + 1) ( x ℓ ) , ℓ = 0 , , . (32)In Eq. (32), the t ℓ ′ ,ℓ are the coefficients in the E T E = X ℓ ′ ,ℓ t ℓ ′ ℓ (cid:16) b † ℓ ′ ˜ b ℓ (cid:17) q = Q q ( α sd = +1 , α dg = +1) . (33)See Eq. (25) for Q q ( α sd = +1 , α dg = +1). Using the T E , the solutions in Table VI andEq. (32), results are obtained for Q (2 +1 ), Q (4 +1 ), B ( E 2; 2 +1 → +1 ) and B ( E 2; 4 +1 → +1 ) fora 10 boson system and the results are given in Table VII. It is seen that the SU (+ , +) (3)and SU (+ , − ) (3) are closer generating prolate shape and SU ( − , − ) (3) generating oblate shape.The SU ( − , +) (3) though generates prolate shape, the quadrupole moments are very small.Thus, sdg IBM substantiates the general structures observed in sdg shell model examplespresented in Section III. V. CONCLUSIONS Multiple SU (3) algebras appear in both shell model and interacting boson model spacesand they open a new paradigm in the applications of SU (3) symmetry in nuclei. In the firstdetailed attempt made in this paper, using three ( sdg ) space examples in SM, we showedthat the four SU (3) algebras in this space exhibit quite different properties with regard toquadrupole collectivity as brought out by the quadrupole moments Q ( J ) and B ( E ABLE VII. Quadrupole moments and B ( E 2) values for low-lying states in the ground band fora 10 boson system generated by the four SU (3) algebras in sdg IBM. Note that T E Eq. (33) isunit-less and therefore Q ( L ) and B ( E α sd α dg Q (2 +1 ) Q (4 +1 ) B ( E 2; 2 +1 → +1 ) B ( E 2; 4 +1 → +1 )+1 +1 − . − . 43 45 . 68 64 . − . − . 89 9 . 15 12 . − . − . 98 1 . 05 1 . . 38 6 . 55 7 . 33 10 . ground K = 0 band in even-even systems (see Tables III-V). The SM and DSM calculationsare restricted to the examples with the leading SU (3) irrep of the type ( λ SU (3) algebras found using SM and DSMis further substantiated by coherent state analysis and asymptotic formulas for quadrupolemoments and B ( E sdg IBM. Also, the results from Q ( J ) and B ( E SU (3) algebras are consistent with the correlation coefficients betweenthe four different Q.Q operators in the sdg space of SM. Results in Tables III-V and VIImay be useful in finding empirical examples for multiple SU (3) algebras in sdg and largerSM spaces and in sdg IBM.Going beyond the present investigations, in future the structure of the low-lying γ (also β ) band generated by the multiple SU (3) algebras will be investigated using SM and DSM.Here, we need to deal with the SU (3) integrity basis operators that are 3 and 4-body, as theleading SU (3) irrep in general will be of the type ( λµ ) with µ = 0 [14]. For example, as seenfrom Table II, for 8 nucleons with T = 0 the leading SU (3) irrep is (24 , H Q ’s in Eq. (16) to quantum phase transitions (QPT) may give newinsights. For example, using H = P α c α Q ( α ) · Q ( α ) and varying the parameters c α , itis possible to study QPT; for a similar study using multiple pairing algebras in SM and IBMsee [37]. 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