Emergent Sp(3,R) dynamical symmetry in the nuclear many-body system from an ab initio description
EEmergent
Sp(3 , R ) dynamical symmetry in the nuclear many-body system from an abinitio description Anna E. McCoy,
1, 2
Mark A. Caprio, Tom´aˇs Dytrych,
3, 4 and Patrick J. Fasano TRIUMF, Vancouver, British Columbia V6T 2A3, Canada Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USA Nuclear Physics Institute, Academy of Sciences of the Czech Republic, 250 68 ˇReˇz, Czech Republic Department of Physics and Astronomy, Louisiana State University, Louisiana 70803, USA (Dated: August 14, 2020)
Ab initio nuclear theory provides not only a microscopic framework for quantitative descriptionof the nuclear many-body system, but also a foundation for deeper understanding of emergentcollective correlations. A symplectic Sp(3 , R ) ⊃ U(3) dynamical symmetry is identified in ab initio predictions, from a no-core configuration interaction approach, and found to provide a qualitativeunderstanding of the spectrum of Be. Low-lying states form an Elliott SU(3) spectrum, while anSp(3 , R ) excitation gives rise to an excited rotational band with strong quadrupole connections tothe ground state band. The nucleus is a complex many-body system, whichnonetheless exhibits simple patterns indicative of emer-gent collective degrees of freedom [1–4].
Ab initio nu-clear theory now provides accurate quantitative predic-tions for observables in light nuclei [5–14]. Signaturesof collective phenomena including clustering [14–19] androtation [12, 20–23] emerge from ab initio calculations.This leaves us with the question of understanding theunderlying physical nature of the collective correlationsgiving rise to these patterns.In a system exhibiting dynamical symmetry [24–28],simple patterns arise naturally, since the spectrum ofeigenstates is organized according to irreducible repre-sentations ( irreps ) of the dynamical symmetry group. Inheavier nuclei, dynamical symmetries have played a cen-tral role in characterizing nuclear correlations and col-lective phenomena [29, 30]. In intermediate-mass nuclei,described by the shell model, Elliott’s SU(3) dynamicalsymmetry [31–33] provides a mechanism for the emer-gence of rotation. In the lightest nuclei, accessible by ab initio theory, we may now seek to identify the role ofSp(3 , R ) ⊃ U(3) dynamical symmetry [34–36] in definingthe structure of the excitation spectrum.The symplectic group Sp(3 , R ), associated with the co-ordinates and momenta in three dimensions, has longbeen proposed as an organizing scheme for the nuclearmany-body problem [34–37]. Through its U(3) subgroup,the symmetry group of the harmonic oscillator, it is inti-mately connected to the nuclear shell model [34, 35, 38–44]. In its contraction limit, Sp(3 , R ) yields a microscopicformulation of nuclear collective dynamics, in terms ofcoupled rotational and giant monopole and quadrupolevibrational degrees of freedom [45, 46].Wave functions obtained in ab initio calculations havealready been identified as having specific dominant U(3)and Sp(3 , R ) symmetry components [9, 47–54]. In thisletter, calculations carried out in a symplectic no-coreconfiguration interaction (SpNCCI) framework demon-strate that the symmetry of individual states moreover fits into an overall Sp(3 , R ) ⊃ U(3) dynamical symmetrypattern of the spectrum as a whole.In particular, for Be, beyond the well-known K = 1 / ab initio calculations as a sym-plectic collective excitation. The remainder of the low-lying spectrum follows an Elliott SU(3) dynamical sym-metry pattern, where the rotational structure is, how-ever, dressed by multishell symplectic excitations. Pre-liminary results were presented in Refs. [55, 56].Sp(3 , R ) ⊃ U(3) dynamical symmetry.
To recognizethe role of Sp(3 , R ) ⊃ U(3) dynamical symmetry in ab initio calculated spectra, we must first be familiarwith some basic properties of Sp(3 , R ) irreps. Elliott’sU(3) = U(1) × SU(3) group considered here is the prod-uct of an SU(3) generated by the orbital angular mo-mentum operator and a quadrupole tensor Q , and theU(1) group of the harmonic oscillator Hamiltonian. ThenSp(3 , R ) augments these generators with symplectic rais-ing and lowering operators, which physically representcreation and annihilation operators for giant monopoleand quadrupole vibrations.An Sp(3 , R ) irrep is comprised of an infinite tower ofU(3) irreps. Starting from a single U(3) irrep with somelowest number of oscillator quanta, or lowest grade irrep (LGI), the remaining U(3) irreps are obtained by repeat-edly acting with the symplectic raising operator, whichadds two oscillator quanta at a time. Each U(3) irrep ischaracterized by a fixed number of oscillator quanta andby SU(3) quantum numbers ( λ, µ ), which are related tothe nuclear deformation [58]. A U(3) irrep may there-fore be labeled by quantum numbers ω ≡ N ω, ex ( λ ω , µ ω ),where N ex denotes the number of oscillator excitationsrelative to the lowest Pauli-allowed oscillator configura-tion. The entire Sp(3 , R ) irrep is then uniquely labeled bythe U(3) quantum numbers σ ≡ N σ, ex ( λ σ , µ σ ) of its LGI.The 2 (cid:126) ω and 4 (cid:126) ω U(3) irreps arising through the actionof the symplectic raising operator on the σ = 0(3 ,
0) LGIin Be is illustrated in Fig. 1(b). a r X i v : . [ nu c l - t h ] A ug ( b ) L = (cid:1) S = ( ) / / / L = (cid:1) S = ( ) / / / ( c ) / / / / ( ) / / / / / / / ( ) / / / / / / / / ( ) / / / / / ( ) / / ( ) / / ( ) / ( a ) σ S = ( ) / ( R ) ⊃ U ( ) dynamical symmetry ( d ) / / / / / J FIG. 1. Low-lying spectrum in an Sp(3 , R ) ⊃ U(3) dynamical symmetry description of Be. (a) Energies. Parameters inthe dynamical symmetry Hamiltonian (2) are chosen for approximate consistency with the experimental [57] and ab initio calculated spectra of Be. States are grouped by U(3) irreps (labeled by ωS ). The excited ωS = 2(5 , / σS = 0(3 , / , R ) irrep is shown. To facilitate comparison with Fig. 2, Sp(3 , R )irreps are tagged by symbols defined there for σS . (b) Organization of Sp(3 , R ) irrep σ = 0(3 ,
0) into U(3) irreps (dots),connected by the symplectic raising operator (lines). Shown through N ex = 4. (c) Branching of the U(3) irrep 0(3 ,
0) to orbitalangular momenta L , followed by coupling with spin ( S = 1 /
2) to give total angular momenta J . (d) Quadrupole transitionstrengths (isoscalar), within the σS = 0(3 , / , R ) irrep, with B ( E
2) strength indicated by line thickness (and shading).
The full subgroup chain, taking into account angularmomenta, is (cid:2)
Sp(3 , R ) σ (cid:122) (cid:125)(cid:124) (cid:123) N σ, ex ( λ σ ,µ σ ) ⊃ U(3) ω (cid:122) (cid:125)(cid:124) (cid:123) N ω, ex ( λ ω ,µ ω ) ⊃ SO(3) L (cid:3) × SU S (2) S ⊃ SU J (2) J . (1)Each U(3) irrep contains states of orbital angular mo-menta L according to the SU(3) ⊃ SO(3) branchingrule [31]. Fermionic antisymmetry defines the possibletotal spins S [32, 59, 60] for each U(3) irrep realized in thenuclear many-body system. Then L and S combine togive total angular momenta J , as illustrated in Fig. 1(c)for the ω = 0(3 ,
0) irrep in Be, where L = 1 , S = 1 / J = 1 / , / , / , / Be is illustrated in Fig. 1(a).In the 0 (cid:126) ω (or valence) space, the U(3) irreps which arisehave ω = 0(3 , ,
1) and 0(0 , S as shown in Fig. 1(a). Eachserves as the LGI of an Sp(3 , R ) irrep (with N σ, ex = 0).The U(3) irrep with ωS = 2(5 , / σS = 0(3 , / , R ) irreps originate at higher N σ, ex .The energy spectrum in Fig. 1 is determined by a sim-ple dynamical symmetry Hamiltonian constructed fromthe Casimir operators for the subgroup chain (1): H = αC Sp(3 , R ) + εH + βC SU(3) + a L L + a S S + ξ L · S . (2) Here, C Sp(3 , R ) is the Casimir operator of Sp(3 , R ) [44],while the SU(3) Casimir operator C SU(3) = (1 / Q · Q +3 L ) incorporates the classic Elliott quadrupole Hamilto-nian [32]. The J angular momentum Casimir operatoris absorbed into L · S = ( J − L − S ).A K = 1 / Be (and mirror nuclide Li) [57, 61],with an exaggerated Coriolis energy staggering lead-ing to an inverted angular momentum sequence ( J =3 / , / , / , / i.e. , β < , Be ground-state band. The staggering is qualita-tively reproduced via the L · S dependence in (2).Dynamical symmetry provides concrete predictionsalso for transition strengths [27, 28]. The isoscalar partof the quadrupole operator is a linear combination ofSp(3 , R ) generators. Thus, Sp(3 , R ) ⊃ U(3) dynamicalsymmetry implies strong E , R ) ir-rep. Predictions for isoscalar E , R ) generator matrix elements [62, 63],with no free parameters, as illustrated in Fig. 1(d) for σS = 0(3 , / Ab initio SpNCCI results for Be . The present Sp-NCCI framework for ab initio calculations makes use of
Rotational Be Daejeon16 N max = (cid:1)(cid:0) =
15 MeV ( a ) E x ( M e V ) / / / / / J = ( , ) = ( , ) ( )( ) σ S ( ) / ( ) / ( ) / ( ) / ( ) / N σ ,ex ≥ ( b ) / / / / / / J FIG. 2.
Ab initio calculated negative parity energy spectrum of Be: (a) Rotational bands (red squares). Strengths (linethickness and shading) are indicated for all J -decreasing E J f < J i or J f = J i and E f < E i ). Energies are plotted against angular momentum scaled as J ( J + 1), as appropriate for rotationalanalysis. Fits to the rotational energy formula with Coriolis staggering are shown (dashed/dotted lines). (b) Most significantSp(3 , R ) contribution σS (indicated by symbol shape and color, see legend) for each state. States with the same largest U(3)contribution ωS are connected by dashed lines. Close-lying states may represent degenerate subspaces involving differentinternal spin couplings (square brackets, with a numeral 2 indicating degenerate doublets indistinguishable in the plot) or mayundergo significant two-state mixing (angled brackets). Experimental energies [57] are shown for context (horizontal lines).Calculation is for the Daejeon16 interaction, with N max = 6 and oscillator basis parameter (cid:126) ω = 15 MeV [64]. N ex = N ex = U ( ) / - ( a ) P r ob a b ilit y N ex = N ex = Sp ( ) / - ( b ) N ex = N ex = / - ( c ) P r ob a b ilit y N ex = N ex = / - ( d ) N ex = N ex = / - ( e ) P r ob a b ilit y ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ω S N ex = N ex = / - ( f ) ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / ( , ) / σ S FIG. 3. Decompositions of calculated Be wave functionsby U(3) (left) and Sp(3 , R ) (right) contributions, for the ωS = 0(3 , / / − (bottom),the ωS = 2(5 , / / − (middle),and the strongly connected 11 / − (top). Contributions arearranged by N ex ( λ, µ ) S and shown through N ex = 2. a symmetry-adapted basis for the fermionic many-bodyspace, one which reduces the subgroup chain (1) and isfree of center-of-mass excitations. Matrix elements ofthe Hamiltonian and other operators are obtained recur- sively in terms of matrix elements between the LGIs,building on ideas of Reske, Suzuki, and Hecht [65–67]. These seed matrix elements are calculated usingthe U(3)-coupled symmetry-adapted no-core shell model(SA-NCSM) [9, 52]. Details may be found in Ref. [56].Here we carry out SpNCCI calculations for Be withthe Daejeon16 internucleon interaction [68], in a basisincorporating all Sp(3 , R ) irreps with LGIs with up to6 quanta ( N σ, ex ≤ N ω, ex ≤ N max = 6 no-core shell model (NCSM) space [71], andthe spectroscopic results, shown in Fig. 2(a), are identicalto those of a traditional N max = 6 calculation.Although symmetry-adapted bases combined withphysically motivated truncation schemes can yield im-proved convergence of calculations [49, 52], our interesthere lies in understanding how the dynamical symmetrystructure of Be underlies the features of the ab initio cal-culated spectrum. Since the basis reduces the subgroupchain (1), SpNCCI calculations provide immediate ac-cess to the Sp(3 , R ) and U(3) symmetry decompositionsof the calculated wave functions, as illustrated in Fig. 3.Further decompositions are provided in the SupplementalMaterial [72].Notably, rotational features emerge in the spectrum.A K = 1 / J = 1 / / E ab initio calculated spectrum [Fig. 2(a), lower dashedline], as in earlier NCSM calculations [20–22]. Calculatedexcitation energies within the band are already largelyinsensitive to N max even though absolute energies arenot well-converged (see Ref. [73]).Moreover, two higher angular momentum states (9 / − and 11 / − ) have strong E / − and 11 / − states also have en-hanced transitions to a particular high-lying 5 / − stateand 7 / − state, well off the yrast line. Tracing E J reveals that these 5 / − and 7 / − states belong to an excited K = 1 / / − and 11 / − states belong to the excited ro-tational band, albeit with energies below those expectedfor this band [Fig. 2(a), upper dotted line].Returning to the Sp(3 , R ) ⊃ U(3) decompositions ofFig. 3 for insight, the wave functions of the ground stateband members are dominated by a single U(3) irrep,namely, ωS = 0(3 , /
2, as expected (above) from a dy-namical symmetry picture. About 60–70% of the prob-ability (or norm) of these states comes from this U(3)irrep, as illustrated for the ground state [Fig. 3(e)], withthe exception of the 5 / − band member, which lies in aclose doublet and undergoes two-state mixing.This 0 (cid:126) ω Elliott U(3) description of the ground stateband is dressed by 2 (cid:126) ω and higher excitations. We seethat excitations within the same Sp(3 , R ) irrep accountfor much of the remaining probability. For the groundstate [Fig. 3(f)], the σS = 0(3 , / , R ) irrep ac-counts for over 80% of the probability, which comes from, e.g. , the ωS = 2(5 , / , / , R ) irrep [recall Fig. 1(b)].For the excited band, the largest U(3) contributioncomes from ωS = 2(5 , / e.g. , ∼
40% for the 7 / − band member [Fig. 3(c)]. This again suggests an Elliottrotational description, but now in the 2 (cid:126) ω space ratherthan in the traditional 0 (cid:126) ω shell model valence space.The U(3) symmetry is more diluted than for the groundstate band, and dressing with higher excitations is againsignificant.Moreover, we see that the excited band members liealmost entirely within the same σS = 0(3 , / , R )irrep as the ground state band, e.g. , ∼
70% for the 7 / − band member [Fig. 3(d)]. While there are 8 different U(3) irreps with quantum numbers ωS = 2(5 , / Be,the 2(5 , / σS = 0(3 , / , R ) ⊃ U(3) dynamical symmetry (Fig. 1).Indeed the Sp(3 , R ) symmetry is significantly better pre-served than the U(3) symmetry.Turning to the 9 / − and 11 / − states with strongtransitions to both bands, these have predominantly ωS = 2(5 , / ∼ σS = 0(3 , / , R ) irrep [Fig. 3(b)].Thus, Sp(3 , R ) ⊃ U(3) dynamical symmetry providesa context for understanding both the emergent rota-tional features and the incomplete description providedfor these features by a simple adiabatic rotational picture.Qualitatively, a 0 (cid:126) ω ground state band [ ωS = 0(3 , / (cid:126) ω excited band [ ωS = 2(5 , /
2] lie within thesame symplectic irrep [ σS = 0(3 , / / − and 11 / − stateswould simply be part of the excited band. Enhancedtransitions among these states are a consequence of dy-namical symmetry [Fig. 1(d)], reflecting the role of theisoscalar E N = ± , R ) irrep.Yet, mixing of U(3) irreps within the Sp(3 , R ) irrep,which becomes significant for the off-yrast excited bandmembers ( J ≤ / E J excited band members [Fig. 2(a)], compared tothe dynamical symmetry predictions [Fig. 1(d)], as wellas the discontinuity in energies between the low- J andhigh- J members of this band.For the remaining low-lying states in Fig. 2(a), theoverall pattern of the spectrum is again qualitatively de-scribed by Sp(3 , R ) ⊃ U(3) dynamical symmetry. InFig. 2(b), the symbols identify the largest Sp(3 , R ) com-ponent, while dashed lines (where practical) connectstates sharing the same largest U(3) component.For many of the states near the yrast line the largestU(3) and Sp(3 , R ) components contribute the preponder-ance of the probability. However, as we move to higherenergy and away from the yrast line, contributions fromother U(3) and Sp(3 , R ) components become increasinglyimportant (see Supplemental Material [72]). Further-more, recall that, when two states are nearly degeneratein energy, they may undergo two-state mixing [brack-ets in Fig. 2(b)]. This serves both to mix the dynamicalsymmetry content and lift the energy degeneracy throughlevel repulsion [3, 74].The dynamical symmetry picture accounts for the fullset of states in the calculated low-lying (0 (cid:126) ω ) spectrumand the overall pattern of their energies. In comparingFig. 1(a) with Fig. 2(b), it is helpful to focus on the“constellations” formed when the calculated states areclassified by their predominant Sp(3 , R ) ⊃ U(3) contri-butions. For instance, the calculated states [Fig. 2(b)]with largest component ωS = 0(1 , / ωS = 0(1 , / ωS = 0(0 , / ωS = 0(0 , / e.g. , theJISP16 [75] and Entem-Machleidt N LO chiral pertur-bation theory [76] interactions in the Supplemental Ma-terial [72].
Conclusion.
We have seen that Sp(3 , R ) ⊃ U(3) dy-namical symmetry, as laid out in Fig. 1, provides an or-ganizing scheme for understanding the entire low-lying ab initio calculated spectrum of Be, as shown in Fig. 2.Symmetry is reflected not merely in the decompositionsof the individual wave functions (Fig. 3), but in the over-all arrangement of energies, which is remarkably consis-tent with a simple dynamical symmetry Hamiltonian (2),and in the E (cid:126) ω spectrum well-described in an ElliottU(3) picture, but dressed by 2 (cid:126) ω and higher contribu-tions from within the Sp(3 , R ) irrep; and excited statesreflecting 2 (cid:126) ω excitations within the Sp(3 , R ) irrep, re-lated to the lower-lying states by strong quadrupole tran-sitions. Although the purity of U(3) symmetry falls offaway from the yrast line, as reflected in the inability of asimple rotational description to simultaneously describeboth the low- J members of the excited band and thestrongly connected 9 / − and 11 / − states, the persis-tence of Sp(3 , R ) symmetry explains the presence of thesestrong transitions.The connection between the ground state and excitedbands by the symplectic raising operators, which physi-cally represent creation operators for the giant monopoleand quadrupole resonances, is suggestive of the emer-gence of collective vibrational degrees of freedom. In thelight, weakly bound Be system, such an interpretationcan at most be approximate. In the present bound stateformalism, it moreover remains uncertain how the struc-ture of the calculated excited band relates to the struc-ture of physical resonances [19].Nonetheless, the presence of rotational bands withstrong E , R ) symmetry as an organizing scheme for nu-clear structure in light nuclei provides a link to more purely collective interpretations of the dynamics throughthe rotation-vibration degrees of freedom which naturallyarise in the classical (large quantum number) limit of thesymplectic description [36, 45, 77–79]. ACKNOWLEDGEMENTS
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