Nucleon-pair coupling scheme in Elliott's SU(3) model
aa r X i v : . [ nu c l - t h ] J a n Nucleon-pair coupling scheme in Elliott’s SU(3) model
G. J. Fu ∗ , Calvin W. Johnson † , P. Van Isacker ‡ , and Zhongzhou Ren § School of Physics Science and Engineering, Tongji University, Shanghai 200092, China Department of Physics, San Diego State University,5500 Campanile Drive, San Diego, CA 92182-1233 Grand Acc´el´erateur National d’Ions Lourds, CEA/DRF-CNRS/IN2P3,Boulevard Henri Becquerel, F-14076 Caen, France (Dated: January 28, 2021)Elliott’s SU(3) model is at the basis of the shell-model description of rotational motion in atomicnuclei. We demonstrate that SU(3) symmetry can be realized in a truncated shell-model spaceif constructed in terms of a sufficient number of collective S , D , G , . . . pairs (i.e., with angularmomentum zero, two, four, . . . ) and if the structure of the pairs is optimally determined either bya conjugate-gradient minimization method or from a Hartree-Fock intrinsic state. We illustrate theprocedure for 6 protons and 6 neutrons in the pf ( sdg ) shell and exactly reproduce the level energiesand electric quadrupole properties of the ground-state rotational band with SDG ( SDGI ) pairs.The SD -pair approximation without significant renormalization, on the other hand, cannot describethe full SU(3) collectivity. A mapping from Elliott’s fermionic SU(3) model to systems with s , d , g , . . . bosons provides insight into the existence of a decoupled collective subspace in terms of S , D , G , . . . pairs. Atomic nuclei exhibit a wide variety of behaviors, rang-ing from single-particle motion to superconducting-likepairing to vibrational and rotational modes. To a largeextent the story of nuclear structure is the quest to en-compass the widest range of behaviors within the fewestdegrees of freedom. In the early stage of nuclear physics,the spherical nuclear shell model [1, 2] stressed the single-particle nature of the nucleons in a nucleus, while thegeometric collective model [3, 4] and the Nilsson mean-field model [5] pointed the way to describing rotationalbands by emphasizing permanent quadrupole deforma-tions [6] in “intrinsic” states. The reconciliation betweenthese two pictures has been one of the most importantadvances in our understanding of the structure of nu-clei. It was in large part due to Elliott who showed,on the basis of an underlying SU(3) symmetry, how toobtain deformed “intrinsic” states in a finite harmonic-oscillator single-particle basis occupied by nucleons thatinteract through a quadrupole-quadrupole force [7]. Thismajor step forward provided a microscopic interpretationof rotational motion in the context of the spherical shellmodel and, more recently, led to the symmetry-adaptedno-core shell model [8].Although the spherical shell model does provide a gen-eral framework to reproduce rotational bands [9] and ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] shape coexistence [10] in light- and medium-mass nuclei,it is computationally still extremely challenging to de-scribe deformation in heavier-mass regions [11]. Approx-imations must be sought. A tremendous simplification ofthe shell model occurs by considering only pairs of nucle-ons with angular momentum 0 and 2, and treating themas ( s and d ) bosons. This approximation, known as theinteracting boson model (IBM) [12, 13], is particularlyattractive because of its symmetry treatment in terms ofa U(6) Lie algebra, which allows a spherical U(5), a de-formed SU(3), and an intermediate SO(6) limit. Whilethe IBM has been connected to the shell model for spher-ical nuclei [14, 15], such relation has never been estab-lished for deformed nuclei, in which case the IBM hasrather been derived from mean-field models [16, 17].The nucleon-pair approximation (NPA) [18, 19] is onepossible truncation scheme of the shell-model configura-tion space. The building blocks of the NPA are fermionpairs with certain angular momenta. Calculations arecarried out in a fully fermionic framework, albeit in aseverely reduced model space defined by the most im-portant degrees of freedom in terms of pairs. The NPAtherefore can be considered as intermediate between thefull-configuration shell model and models that adopt thesame degrees of freedom as the nucleon pairs but interms of bosons. While the NPA has been successfulfor nearly spherical nuclei [20–27], previous studies forwell-deformed nuclei are not satisfactory. For example,in the fermion dynamical symmetry model [28, 29] anSU(3) limit with Sp(6) symmetry can be constructed interms of S and D pairs but their symmetry-determinedstructure is far removed from that of realistic pairs [30].Also, the binding energy, moment of inertia, and elec-tric quadrupole ( E
2) transitions calculated in an SD -pairapproximation are much smaller than those obtained inElliott’s SU(3) limit for the pf and sdg shells [31].In this Letter we successfully apply the NPA of theshell model to well-deformed nuclei. We show that thelow-energy excitations of many-nucleon systems in El-liott’s SU(3) limit can be exactly reproduced with a suit-able choice of pairs in the NPA. We obtain an under-standing of this observation through a mapping to a cor-responding boson model.We consider an example system with even numbersof protons and neutrons in a degenerate pf or sdg shell,interacting through a quadrupole-quadrupole force of theform, V Q = − ( Q π + Q ν ) · ( Q π + Q ν ) , (1)where Q π ( Q ν ) is the quadrupole operator for protons(neutrons), Q = − X αβ h n α l α j α k r Y k n β l β j β i√ r (cid:0) a † α × ˜ a β (cid:1) (2) . (2)Greek letters α , β, . . . denote harmonic-oscillator single-particle orbits labeled by n , l , j , and j z ; a † α and ˜ a β arethe nucleon creation operator and its time-reversed formfor the annihilation operator, respectively; and r is theharmonic-oscillator length. As shown in Ref. [7], the in-teraction V Q is a combination of the Casimir operators ofSU(3) and SO(3), and its eigenstates are therefore clas-sified by (irreducible) representations of these algebraswith eigenenergies given by − π (cid:20)
12 ( λ + λµ + µ + 3 λ + 3 µ ) − L ( L + 1) (cid:21) , (3)in terms of the SU(3) labels ( λ, µ ) and the SO(3) label L , the total orbital angular momentum. Several usefulSU(3) representations for low-lying states can be foundin Ref. [31].In the following we discuss in detail the case of 6 pro-tons and 6 neutrons (6p-6n) in the NPA of the shell modeland subsequently generalize to other numbers of nucle-ons. A nucleon-pair state of 6 protons is written as | ϕ ( I π ) i = (cid:16) ( A ( J ) † × A ( J ) † ) ( I ) × A ( J ) † (cid:17) ( I π ) | i , (4)where I is an intermediate angular momentum and A ( J ) † is the creation operator of a collective pair with angularmomentum J : A ( J ) † = X α ≤ β y J ( αβ ) (cid:16) a † α × a † β (cid:17) ( J ) , (5) where y J ( αβ ) is the pair-structure coefficient. For sys-tems with protons and neutrons, we construct the ba-sis by coupling the proton and neutron pair states toa state with total angular momentum I , i.e., | ψ ( I ) i = (cid:0) | ϕ ( I π ) i × | ϕ ( I ν ) i (cid:1) ( I ) . Level energies and wave functionsare obtained by diagonalization of the Hamiltonian ma-trix in the space spanned by (cid:8) | ψ ( I ) i (cid:9) , that is, from aconfiguration-interaction calculation. If a sufficient num-ber of pair states are considered in Eq. (4), the NPAmodel space can be made exactly equivalent to the fullshell-model space. The interest of the NPA, however, isto restrict to the relevant pairs and describe low-energynuclear structure in a truncated shell-model space.The selection of relevant pairs with the correct struc-ture in Eq. (5) has been a long standing problem inNPA calculations. Recent applications choose pairs bythe generalized seniority scheme (GS). Specifically, oneoptimizes the structure coefficients of the S pair by min-imizing the expectation value of the Hamiltonian in the S -pair condensate and one obtains other pairs by diag-onalizing the Hamiltonian matrix in the space spannedby GS-two (i.e., one-broken-pair) states [25, 32]. Thecollective pairs obtained with the GS approach providea good description of nearly-spherical nuclei but, as rec-ognized in Ref. [33] and as will also be shown below,they are inappropriate in deformed nuclei. Instead we usethe conjugate gradient (CG) method [34, 35], where thestructure coefficients of all pairs considered in the basisare simultaneously optimized by minimizing the ground-state energy in a series of iterative NPA calculations fora given Hamiltonian. The initial pairs in this iterativeprocedure are SU(3) tensors, obtained by diagonalizing V Q in a two-particle basis and retaining the lowest-energypair.Figure 1 shows, for a 6p-6n system in the pf shell, theresults of various NPA calculations concerning excitationenergies and E e π = 1 . e ν = 0 .
5) forthe lowest rotational band. These are compared to theexact results of Elliott’s model, where the ground bandbelongs to the SU(3) representation ( λ, µ ) = (24 , SDG -pair approximation of the shell modelin the CG approach (denoted as
SDG CG ) reproduces theexact binding energy, 810 /π MeV according to Eq. (3),to a precision of eight digits, as well as the exact excita-tion energies for the entire ground band. One can under-stand the occurrence of the (24 ,
0) representation fromthe coupling of (12 ,
0) for the six protons and six neu-trons separately and, in fact, all bands contained in theproduct (12 , × (12 , , , , SDG CG -pair truncated space. Exact SD GS SDG GS SD CG SDS’D’ CG SDG CG E x ( M e V ) I (b) B ( E ) ( e b ) (a) FIG. 1: (a) Excitation energy and (b) electric quadrupole re-duced transition probability B ( E I → I −
2) for the groundrotational band of 6 protons and 6 neutrons in the pf shell inElliott’s SU(3) model. The subscript “GS” stands for gener-alized seniority and “CG” for conjugate gradient (see text). We also find that the results of the
SDG -pair approxima-tion are close to the exact results if the pairs are SU(3)tensors. For example, with such pairs the calculation re-produces 98% of the exact binding energy, 99% of theexact moment of inertia, and 97% of the exact B ( E SDG -pair ap-proximation deteriorate if the pairs are obtained withthe GS approach (denoted as
SDG GS ), which repro-duces only 76% of the exact binding energy. Further-more, SDG GS fails to describe the quadrupole collectiv-ity: The moment of inertia predicted by SDG GS is only ∼
43% of the exact one, the predicted B ( E
2) values aretoo small, and the yrast states with angular momentum I ≥
10 do not follow the behavior of a quantum rotor.One concludes that the structure of the collective pairs,as determined by the GS approach, is not suitable for thedescription of well-deformed nuclei.It is also of interest to investigate the standard SD -pair approximation of the shell model and results ofthe SD GS -, SD CG -, and SDS ′ D ′ CG -pair approximationsare shown in Fig. 1. Here S ′ and D ′ are collectivepairs with angular momentum 0 and 2 but orthogo-nal to the S and D pairs, respectively. While the CGapproach provides the numerically optimal solution in Exact SD GS SDGI GS SD CG SDG CG SDGI CG E x ( M e V ) I (b) B ( E ) ( e b ) (a) FIG. 2: Same as Fig. 1 for the sdg shell. SD CG - and SDS ′ D ′ CG -pair approximations, the resultsnonetheless are underwhelming. In the SD GS , SD CG ,and SDS ′ D ′ CG spaces only 76%, 83%, and 84% of theexact binding energy are reproduced, respectively, andthe predicted moments of inertia and B ( E
2) strengthsare evidently smaller than the exact SU(3) results. Weconclude that the collective SD pairs cannot fully explainthe quadrupole collectivity of the SU(3) states. Interest-ingly, the excitation energies of the yrast states predictedby the SD -pair approximations follow an I ( I +1) rule andthe B ( E
2) strength exhibits a nearly-parabolic shape [seeFig. 1(b)], two typical features of rotational motion. Thisraises the hope that an effective
Hamiltonian and effec-tive charges can be derived in the restricted SD CG space,which takes into account the coupling with the excludedspace. This conclusion is in line with a more phenomeno-logical approach [17], in which an L · L term is added tothe Hamiltonian, such that properties of low-lying statesof well-deformed nuclei are reproduced in sd -IBM.Figure 2 shows the corresponding results of for the 6p-6n system in the sdg shell. In this case the SDGI CG -pairapproximation of the shell model reproduces exactly theSU(3) results and all states belonging to the coupled rep-resentation (18 , × (18 , , , , SDGI CG -pair truncated space.Again, if the pairs are SU(3) tensors, the SDGI -pair ap-proximation is close to the exact result and reproduces99% of the exact binding energy, 97% of the exact mo-ment of inertia, and 99% of the exact B ( E
2) values. The
SDG CG -pair approximation yields 96% of the bindingenergy and 57% of the moment of inertia. The pre-dicted B ( E
2) strength in the
SDG CG -pair approxima-tion is close to the exact result for low angular momentabut deteriorates as angular momentum I increases. Thenecessity of renormalization is even larger in the SD CG -pair approximation.Let us now try to understand the above results. Specif-ically, why is it that the SU(3) results in the pf shell areexactly reproduced with SDG but not with SD pairs?Similarly, why is it that SU(3) in the sdg shell cannot berepresented with SD or SDG but requires
SDGI pairs?To explain these findings, we invoke a mapping to a sys-tem with corresponding s , d , g , and i bosons (denotedas sd -, sdg -, or sdgi -IBM) and the bosonic realization ofSU(3). The mapping is further specified by the fact thatthe quadrupole-quadrupole interaction V Q is an SU(4) in-variant and, consequently, one aims to realize the symme-tries associated with Wigner’s supermultiplet model [36]in terms of bosons. An SU(4)-invariant boson model,known as IBM-4 [37], requires to assign to each bosona spin-isospin of ( s, t ) = (0 ,
1) or (1 , st (6).The SU(3) limit can be realized in terms of bosons byfirst decoupling the orbital angular momentum from thespin-isospin of the bosons. For an n b -boson state thisleads to the classificationU(6Λ) ⊃ U(Λ) ⊗ U st (6) ↓ ↓ ↓ [ n b ] (cid:2) ¯ h (cid:3) ≡ [ h , ..., h ] (cid:2) ¯ h (cid:3) ≡ [ h , ..., h ] , (6)with Λ = 6, 15, and 28 for sd -, sdg -, and sdgi -IBM,respectively. The six labels [¯ h ] are a partition of n b suchthat h ≥ h ≥ · · · ≥ h ; they specify the representationsof U(Λ) and U st (6), which by virtue of the overall U(6Λ)symmetry of the bosons must be identical. For all abovevalues of Λ (i.e., Λ = 6, 15, and 28), Elliott’s SU(3)appears as a subalgebra of U(Λ),U(Λ) ⊃ U(3) ⊃ SU(3) ⊃ SO(3) ↓ ↓ ↓ ↓ (cid:2) ¯ h (cid:3) [ h ′′ , h ′′ , h ′′ ] ( λ, µ ) K L , (7)while Wigner’s SU(4) occurs as a subalgebra of U st (6),U st (6) ⊃ SU st (4) ⊃ SU s (2) ⊗ SU t (2) ↓ ↓ ↓ ↓ (cid:2) ¯ h (cid:3) ( λ ′ , µ ′ , ν ′ ) S T . (8)The quantum numbers ( λ, µ ), K , and L in Eq. (7) and( λ ′ , µ ′ , ν ′ ), S , and T in Eq. (8) have an interpretationidentical to that in Elliott’s fermionic SU(3) model [7, 38]. The SU(3) labels ( λ, µ ) in the different versions of theIBM can be worked out with the following procedure [39].For a given number of bosons n b , one enumerates allpossible Young diagrams [¯ h ] of U(Λ) or U st (6). Foreach [¯ h ] one obtains the SU st (4) labels ( λ ′ , µ ′ , ν ′ ) fromthe branching rule U(6) ⊃ SU(4), and retains only theones that contain the favored supermultiplet. Finally,the SU(3) labels ( λ, µ ) for the above [¯ h ] are found fromthe U(Λ) ⊃ SU(3) branching rule.Let us apply this procedure to the 6p-6n system inthe pf shell. The lowest eigenstates of the quadrupole-quadrupole interaction belong to the favored SU(4)supermultiplet ( λ ′ , µ ′ , ν ′ ) = (0 , ,
0) and the leading(fermionic) SU(3) representation is ( λ, µ ) = (24 , n b = 6 bosons, the U st (6) or U(Λ) representations con-taining this favored supermultiplet (0 , ,
0) are [¯ h ] = [6],[4 , ], and [1 ], which have the SU(3) labels ( λ, µ )as listed in Table I for the sd -, sdg -, and sdgi -IBM. Theleading SU(3) representation (24 ,
0) is not contained in sd -IBM but is present in the [6] representation of U(15),and therefore it is contained in sdg -IBM. Similarly, 6p-6nin the sdg shell give rise to the leading SU(3) represen-tation (36 , sd - nor sdg -IBMbut present in sdgi -IBM.The generalization to the 2p-2n ( n = 4) and 4p-4n( n = 8) systems in the pf and sdg shells is summarizedin Table II. The second column lists the leading fermionicSU(3) representations and the third, fourth, and fifthcolumns indicate whether this representation is containedin sd -, sdg -, and sdgi -IBM, respectively. A dash (—) in-dicates that it is not, in which case an NPA calculationadopting the corresponding SD , SDG , or
SDGI pairsdoes not reproduce the full collectivity of the ground-state band in the fermionic SU(3) model. For n = 4and n = 8 nucleons in the sdg shell no exact mappingcan be realized to sdgi -IBM and bosons with even higherangular momentum are needed. It should be noted, how-ever, that this generally occurs for low nucleon number(e.g., for n = 12 nucleons in the sdg shell the problemdoes not occur), for which NPA calculations with highangular momentum pairs are still feasible.While the best NPA solutions so far have been foundby a numerically intensive optimization, it turns outthey can also be obtained from a deformed “intrinsic”state. Again consider the 6p-6n system in the pf shell.An unconstrained Hartree-Fock (HF) calculation in thissingle-particle shell-model space [40] with a quadrupole-quadrupole interaction provides us with a HF state withan axially symmetric quadrupole deformed shape, a con-sequence of the spontaneous symmetry breaking [41] ofrotational symmetry. One can project out a K = 0 band TABLE I: Leading SU(3) representations for 6 bosons in sd -, sdg -, and sdgi -IBM occurring in the U(Λ) and U st (6) repre-sentations [¯ h ] containing the favored supermultiplet (0 , , n b [¯ h ] ( λ, µ )( sd ) [6] (12 , , (8 , , (4 , , (6 , , (0 , , . . . [4 ,
2] (8 , , (6 , , (7 , , (4 , , (5 , , . . . [2 ] (6 , , (0 , , (3 , , (2 , , (0 , ] (0 , sdg ) [6] (24 , , (20 , , (18 , , (16 , , (18 , , . . . [4 ,
2] (20 , , (18 , , (19 , , (16 , , (17 , , . . . [2 ] (18 , , (15 , , (12 , , (13 , , (14 , , . . . [1 ] (12 , , (8 , , (9 , , (3 , , (7 , , . . . ( sdgi ) [6] (36 , , (32 , , (30 , , (28 , , (30 , , . . . [4 ,
2] (32 , , (30 , , (31 , , (28 , , (29 , , . . . [2 ] (30 , , (27 , , (24 , , (25 , , (26 , , . . . [1 ] (24 , , (20 , , (21 , , (18 , , (19 , , . . . TABLE II: Leading fermionic SU(3) representations ( λ, µ )for n nucleons in the pf and sdg shells and the U(6), U(15),and U(28) representations of the n b = n/ λ, µ ) in sd -, sdg -, and sdgi -IBM.(shell) n ( λ, µ ) sd -IBM sdg -IBM sdgi -IBM( pf ) (12 ,
0) — — [2]( pf ) (16 ,
4) — — [4] , [2 ]( pf ) (24 ,
0) — [6] [6] , [4 , , [2 ] , [1 ]( sdg ) (16 ,
0) — — —( sdg ) (24 ,
4) — — —( sdg ) (36 ,
0) — — [6] with good angular momentum from this HF state [42],which exactly corresponds to the SU(3) representation(24 ,
0) [7]. We use a and ¯ a to denote the HF single-particle orbit and its time-reversal partner, respectively,and we write the creation operator of a nucleon as c † a . ASlater determinant for an even number 2 N of protons orneutrons can be written as a pair condensate: N Y a =1 c † a c † ¯ a | i = N X a v a c † a c † ¯ a ! N | i . (9)The pair in the deformed HF state is a superposition ofcollective pairs of good angular momentum in the shellmodel [43]: X a v a c † a c † ¯ a = X JM A ( J ) M † . (10)For the appropriate v a one obtains SDG pairs, whichare the same as the
SDG pairs obtained by the CG-NPAcalculations. Similarly, the
SDGI pairs responsible for(36,0) for 6p-6n in the sdg shell can be also projectedout from a deformed HF pair. The CG approach pro-vides numerically optimal solutions in the NPA but is Fe E x ( M e V ) + + + + + + SDGSMExpt.
FIG. 3: The ground rotational band of Fe. The experi-mental energies are taken from Ref. [44] and the shell-modelresults are obtained with the GXPF1 interaction. I π Expt. SM
SDG + + + + + B ( E I → I −
2) values (in W.u.) for the groundrotational band of Fe. The experimental values are takenfrom Ref. [44]and the shell-model results are obtained with the GXPF1interaction. computationally heavy due to hundreds, even thousandsof iterations. The HF approach derives pairs using anunconstrained HF calculation and the decomposition ofpairs according to Eq. (10) has a very low computationalcost.Finally, we show that the NPA with CG-pairs providesa good description of low-lying states of rotational nucleialso if a realistic shell-model interaction is taken. We ex-emplify this with the nucleus Fe, considered as a 6p-6nsystem in the pf shell with the GXPF1 effective interac-tion [45]. Figure 3 and Table III compare, for the groundrotational band of Fe, the experimental data [44], thefull configuration shell model (SM), and the
SDG CG -pairapproximation. Both the level energies and the B ( E SDG CG are in good agreement withthe data and with the shell model.In summary, we construct in the NPA a collective sub-space of the full shell-model space such that the for-mer exactly reproduces, without any renormalization, theproperties of the low-energy states of the latter. This con-struction is valid for an SU(3) quadrupole-quadrupoleHamiltonian and is achieved by determining the struc-ture of the pairs with the conjugate-gradient minimiza-tion technique or on the basis of a deformed HF calcula-tion. Exact correspondence is achieved only if a sufficientnumber of different pairs is considered. For example, a6p-6n system in the pf ( sdg ) shell is reproduced exactlywith SDG ( SDGI ) pairs; with just SD pairs, an im-portant renormalization of all operators is required. Wehave analytic understanding of this result: The collec-tive subspace of the NPA exactly captures the collectiv-ity of the full space if and only if the mapping to a modelconstructed with bosons corresponding to the pairs givesrise to a leading bosonic SU(3) representation that is alsoleading in fermionic SU(3).For many years a central problem in nuclear structurehas been the construction of a collective subspace that de-couples from the full shell-model space. With this workthe conditions necessary for this decoupling to be ex-act are now understood for an SU(3) Hamiltonian. Thisunderstanding will pave the way for the construction ofviable collective subspaces for more realistic shell-modelinteractions. It will also clarify the derivation of bosonHamiltonians appropriate for quadrupole deformed nu-clei. 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