Evidence for Symplectic Symmetry in Ab Initio No-Core Shell Model
  Results for Light Nuclei
Evidence for Symplectic Symmetry in Ab Initio
No-Core Shell Model Results for Light Nuclei
Tomáš Dytrych, Kristina D. Sviratcheva, Chairul Bahri, and Jerry P. Draayer
Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA
James P. Vary
Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA
Lawrence Livermore National Laboratory, L-414,
7000 East Avenue, Livermore, California, 94551, USA and
Stanford Linear Accelerator Center, MS81, 2575 Sand Hill Road, Menlo Park, California, 94025, USA
Clear evidence for symplectic symmetry in low-lying states of 12C and 16O is reported. Eigenstates
of 12C and 16O, determined within the framework of the no-core shell model using the JISP16
NN realistic interaction, typically project at the 85-90% level onto a few of the most deformed
symplectic basis states that span only a small fraction of the full model space. The results are nearly
independent of whether the bare or renormalized effective interactions are used in the analysis. The
outcome confirms Elliott’s SU(3) model which underpins the symplectic scheme, and above all, points
to the relevance of a symplectic no-core shell model that can reproduce experimental B(E2) values
without effective charges as well as deformed spatial modes associated with clustering phenomena
in nuclei.
Recently developed realistic interactions, such as J-
matrix inverse scattering potentials [1] and modern two-
and three-nucleon potentials derived from meson ex-
change theory [2] or by using chiral effective field theory
[3], succeed in modeling the essence of the strong inter-
action for the purpose of input into microscopic shell-
model calculations that target reproducing characteris-
tic features of light nuclei. The ab initio No-Core Shell
Model (NCSM) [4] which employs such modern realistic
interactions, yields a good description of the low-lying
states in few-nucleon systems [5] as well as in more com-
plex nuclei like 12C [4, 6]. In addition to advancing our
understanding of the propagation of the nucleon-nucleon
force in nuclear matter and clustering phenomena [7, 8],
modeling the structure of 12C, 16O and similar nuclei
is also important for gaining a better understanding of
other physical processes such as parity-violating electron
scattering from light nuclei [9] and results gained through
neutrino studies [10] as well as for making better predic-
tions for capture reaction rates that figure prominently,
for example, in the burning of He in massive stars [11].
In this letter we report on investigations that show that
realistic eigenstates for low-lying states determined in
NCSM calculations for light nuclei with the JISP16 realis-
tic interaction [1], predominantly project onto few of the
most deformed Sp(3,R)-symmetric basis states that are
free of spurious center-of-mass motion. This reflects the
presence of an underlying symplectic sp(3,R) ⊃ su(3) ⊃
so(3) algebraic structure [22], which is not a priori im-
posed on the interaction and furthermore is found to re-
main unaltered after a Lee-Suzuki similarity transforma-
tion used to accommodate the truncation of the infinite
Hilbert space by renormalization of the bare interaction.
This in turn provides insight into the physics of a nucleon
system and its geometry. Specifically, nuclear collective
states with well-developed quadrupole and monopole vi-
brational modes and rotational modes are described nat-
urally by irreducible representations (irreps) of Sp(3,R).
The present study points to the possibility of achieving
convergence of higher-lying collective modes and reach-
ing heavier nuclei by expanding the NCSM basis space
beyond its current limits through Sp(3,R) basis states
that span a dramatically smaller subspace of the full
space. In this way, the symplectic no-core shell-model
(Sp-NCSM) with realistic interactions and with a mixed
Sp(3,R) irrep extension will allow one to account for even
higher ~Ω configurations required to realize experimen-
tally measured B(E2) values without an effective charge,
and to accommodate highly deformed spatial configura-
tions [12] that are required to reproduce α-cluster modes,
which may be responsible for shaping, e.g., the second 0+
state in 12C and 16O [8].
We focus on the 0+gs ground state and the lowest
2+(≡2+
) and 4+(≡4+
) states in the oblate 12C nucleus
as well as the 0+gs in the ‘closed-shell’
16O nucleus. The
NCSM eigenstates for these states are reasonably well
converged in the Nmax = 6 (or 6~Ω) model space with an
effective interaction based on the JISP16 realistic interac-
tion [1], which typically leads to rapid convergence in the
NCSM evaluations, describes NN data to high accuracy
and is consistent with, but not constrained by, meson ex-
change theory, QCD or locality. In addition, calculated
binding energies as well as other observables for 12C such
as B(E2;2+
→0+gs), B(M1;1
→0+gs), ground-state proton
rms radii and the 2+
quadrupole moment all lie reason-
ably close to the measured values. While symplectic al-
gebraic approaches have achieved a very good reproduc-
tion of low-lying energies and B(E2) values in light nuclei
[13, 14] and specifically in 12C using phenomenological
interactions [15] or truncated symplectic basis with sim-
http://arxiv.org/abs/0704.1108v1
plistic (semi-) microscopic interactions [16, 17], here, for
the first time, we establish, the dominance of the sym-
plectic Sp(3,R) symmetry in light nuclei, and hence their
propensity towards development of collective motion, as
unveiled through ab initio calculations of the NCSM type
starting with realistic two-nucleon interactions.
The symplectic shell model [18, 19] is based on the
noncompact symplectic sp(3,R) algebra. The classical
realization of this symmetry underpins the dynamics of
rotating bodies and has been used, for example, to de-
scribe the rotation of deformed stars and galaxies [20].
In its quantal realization it is known to underpin the
successful Bohr-Mottelson collective model and has also
been shown to be a multiple oscillator shell generaliza-
tion of Elliott’s SU(3) model. Consequently, symplectic
basis states bring forward important information about
nuclear shapes and deformation in terms of (λ, µ), which
serve to label the SU(3) irreps within a given Sp(3,R)
irrep, for example, (0, 0), (λ, 0) and (0, µ) describe spher-
ical, prolate and oblate shapes, respectively.
The significance of the symplectic symmetry for a mi-
croscopic description of a quantum many-body system
emerges from the physical relevance of its 21 generators
constructed as bilinear products of the momentum (pα)
and coordinate (qβ) operators, e.g. pαpβ , pαqβ , and qαqβ
with α, β = x, y, and z for the 3 spatial directions. Hence,
the many-particle kinetic energy, the mass quadrupole
moment operator, and the angular momentum are all
elements of the sp(3,R) ⊃ su(3) ⊃ so(3) algebraic struc-
ture. It also includes monopole and quadrupole collective
vibrations reaching beyond a single shell to higher-lying
and core configurations, as well as vorticity degrees of
freedom for a description of the continuum from irrota-
tional to rigid rotor flows. Alternatively, the elements of
the sp(3,R) algebra can be represented as bilinear prod-
ucts in harmonic oscillator (HO) raising and lowering op-
erators, which means the basis states of a Sp(3,R) irrep
can be expanded in a 3-D HO (m-scheme) basis which
is the same basis used in the NCSM, thereby facilitating
calculations and symmetry identification.
The basis states within a Sp(3,R) irrep are built by
applying symplectic raising operators to a np-nh (n-
particle-n-hole, n = 0, 2, 4, ...) lowest-weight Sp(3,R)
state (symplectic bandhead), which is defined by the
usual requirement that the symplectic lowering operator
annihilates it. The raising operator induces a 2~Ω 1p-1h
monopole or quadrupole excitation (one particle raised
by two shells) together with a smaller 2~Ω 2p-2h correc-
tion for eliminating the spurious center-of-mass motion.
If one were to include all possible lowest-weight np-nh
starting state configurations (n ≤ Nmax), and allowed all
multiples thereof, one would span the full NCSM space.
The lowest-lying eigenstates of 12C and 16O were calcu-
lated using the NCSM as implemented through the Many
Fermion Dynamics (MFD) code [21] with an effective
interaction derived from the realistic JISP16 NN poten-
0 2 4 6 8 10 12
3 0p-0h
all 0p-0h
dominant 
0p-0h+2p-2h
Sp(3,R)
0 2 4 6 8 10 12M
Sp(3,R)
FIG. 1: NCSM space dimension as a function of the maxi-
mum ~Ω excitations, Nmax, compared to that of the Sp(3,R)
subspace: (a) J = 0, 2, and 4 for 12C, and (b) J = 0 for 16O.
tial [1] for different ~Ω oscillator strengths. For both
nuclei we constructed all of the 0p-0h and 2~Ω 2p-2h
(2 particles raised by one shell each) symplectic band-
heads and generated their Sp(3,R) irreps up to Nmax = 6
(6~Ω model space). Analysis of overlaps of the symplec-
tic states with the NCSM eigenstates for 2~Ω, 4~Ω, and
6~Ω model spaces (Nmax = 2, 4, 6) reveals the dominance
of the 0p-0h Sp(3,R) irreps. For the 0+gs and the lowest
2+ and 4+ states in 12C there are nonnegligible overlaps
for only 3 of the 13 0p-0h Sp(3,R) irreps, namely, the
leading (most deformed) representation specified by the
shape deformation of its symplectic bandhead, (0 4), and
carrying spin S = 0 together with two (1 2) S = 1 ir-
reps with different bandhead constructions for protons
and neutrons. For the ground state of 16O there is only
one possible 0p-0h Sp(3,R) irrep, (0 0) S = 0. In addi-
tion, among the 2~Ω 2p-2h Sp(3,R) irreps only a small
fraction contributes significantly to the overlaps and it in-
cludes the most deformed configurations that correspond
to oblate shapes in 12C and prolate ones in 16O.
The typical dimension of a symplectic irrep in the
Nmax = 6 space is on the order of 10
2 as compared to
107 for the full NCSM m-scheme basis space. As Nmax is
increased the dimension of the J = 0, 2, and 4 symplec-
tic space built on the 0p-0h Sp(3,R) irreps for 12C grows
very slowly compared to the NCSM space dimension (Fig.
1a). The dominance of only three irreps additionally re-
duces the dimensionality of the symplectic model space,
which remains a small fraction of the NCSM basis space
even when the most dominant 2~Ω 2p-2h Sp(3,R) irreps
are included. The space reduction is even more dramatic
in the case of 16O (Fig. 1b). This means that a space
spanned by a set of symplectic basis states is computa-
tionally manageable even when high-~Ω configurations
are included.
The overlaps of the most dominant symplectic states
with investigated NCSM eigenstates for the 12C and the
16O in the 0, 2, 4 and 6~Ω subspaces are given in Table I
and II. In order to speed up the calculations, we retained
only the largest amplitudes of the NCSM states, those
sufficient to account for at least 98% of the norm which
is quoted also in the table. The results show that approx-
imately 85% of the NCSM eigenstates for 12C (16O) fall
within a subspace spanned by the few most significant
0p-0h and 2~Ω 2p-2h Sp(3,R) irreps, with the 2~Ω 2p-
2h Sp(3,R) irreps accounting for 5% (10%) and with the
leading irrep, (0 4) for 12C and (0 0) for 16O, carrying
close to 70% (75%) of the NCSM wavefunction.
Furthermore, the S = 0 part of all three NCSM eigen-
states for 12C is almost entirely projected (95%) onto only
six S = 0 symplectic irreps included in Table I, with as
much as 90% of the spin-zero NCSM states accounted for
solely by the leading (0 4) irrep. The S = 1 part is also
remarkably well described by merely two Sp(3,R) irreps.
Similar results are observed for the ground state of 16O.
Another striking property of the low-lying eigenstates
is revealed when the spin projections of the converged
NCSM states are examined. Specifically, as shown in
Fig. 2, their Sp(3,R) symmetry and hence the geometry
of the nucleon system being described is nearly indepen-
dent of the ~Ω oscillator strength. The symplectic sym-
metry is present with equal strength in the spin parts of
the NCSM wavefunctions for 12C as well as 16O regard-
less of whether the bare or the effective interactions are
used. This suggests that the Lee-Suzuki transformation,
which effectively compensates for the finite space trun-
cation by renormalization of the bare interaction, does
not affect the Sp(3,R) symmetry structure of the spa-
tial wavefunctions. Hence, the symplectic structure de-
TABLE I: Probability distribution of NCSM eigenstates for
12C across the dominant 0p-0h and 2~Ω 2p-2h Sp(3,R) irreps,
~Ω=15 MeV.
0~Ω 2~Ω 4~Ω 6~Ω Total
J = 0
Sp(3,R) (0 4)S = 0 46.26 12.58 4.76 1.24 64.84
(1 2)S = 1 4.80 2.02 0.92 0.38 8.12
(1 2)S = 1 4.72 1.99 0.91 0.37 7.99
2~Ω 2p-2h 3.46 1.02 0.35 4.83
Total 55.78 20.05 7.61 2.34 85.78
NCSM 56.18 22.40 12.81 7.00 98.38
J = 2
Sp(3,R) (0 4)S = 0 46.80 12.41 4.55 1.19 64.95
(1 2)S = 1 4.84 1.77 0.78 0.30 7.69
(1 2)S = 1 4.69 1.72 0.76 0.30 7.47
2~Ω 2p-2h 3.28 1.04 0.38 4.70
Total 56.33 19.18 7.13 2.17 84.81
NCSM 56.18 21.79 12.73 7.28 98.43
J = 4
Sp(3,R) (0 4)S = 0 51.45 12.11 4.18 1.04 68.78
(1 2)S = 1 3.04 0.95 0.40 0.15 4.54
(1 2)S = 1 3.01 0.94 0.39 0.15 4.49
2~Ω 2p-2h 3.23 1.16 0.39 4.78
Total 57.50 17.23 6.13 1.73 82.59
NCSM 57.64 20.34 12.59 7.66 98.23
tected in the present analysis for 6~Ωmodel space is what
would emerge in NSCM evaluations with a sufficiently
large model space to justify use of the bare interaction.
11 12 13 14 15 16 17 18 Bare
(a) J=0
h�̄Ω ——(MeV)
11 12 13 14 15 16 17 18 Bare
(b) J=2
h�̄Ω ——(MeV)
11 12 13 14 15 16 17 18 Bare
(c) J=4
h�̄Ω ——(MeV)
12 13 14 15 16 Bare
(d) J=0
h�̄Ω ——(MeV)
FIG. 2: Projection of the S = 0 (blue, left) [and S = 1 (red,
right)] Sp(3,R) irreps onto the corresponding significant spin
components of the NSCM wavefunctions for (a) 0+gs, (b) 2
and (c) 4+
in 12C and (d) 0+gs in
16O, for effective interaction
for different ~Ω oscillator strengths and bare interaction.
In addition, as one varies the oscillator strength ~Ω,
the projection of the NCSM wavefunctions onto the sym-
plectic subspace changes only slightly (see, e.g., Fig. 3 for
the 0+gs state of
12C and 16O). The symplectic structure
is preserved, only the Sp(3,R) irrep contributions change
because the S = 0 (S = 1) part of the NCSM eigen-
states decrease (increase) towards higher ~Ω frequen-
cies. Clearly, the largest contribution comes from the
leading Sp(3,R) irrep (black diamonds), growing to 80%
of the NCSM wavefunctions for the lowest ~Ω. These
results can be interpreted as a strong confirmation of El-
liott’s SU(3) model since the projection of the NCSM
states onto the 0~Ω space [Fig. 3, blue (lowest) bars] is
a projection of the NCSM results onto the SU(3) shell
model. The outcome is consistent with what has been
shown to be a dominance of the leading SU(3) symmetry
for SU(3)-based shell-model studies with realistic inter-
actions in 0~Ω model spaces. It seems the simplest of
TABLE II: Probability distribution of the NCSM eigenstate
for the J = 0 ground state in 16O across the 0p-0h and dom-
inant 2~Ω 2p-2h Sp(3,R) irreps, ~Ω=15 MeV.
0~Ω 2~Ω 4~Ω 6~Ω Total
Sp(3,R) (0 0)S = 0 50.53 15.87 6.32 2.30 75.02
2~Ω 2p-2h 5.99 2.52 1.32 9.83
Total 50.53 21.86 8.84 3.62 84.85
NCSM 50.53 22.58 14.91 10.81 98.83
11 12 13 14 15 16 17 18
h�̄Ω ——(MeV)
       
       
       
       
12 13 14 15 16
h�̄Ω ——(MeV)
FIG. 3: Ground 0+ state probability distribution over
0~Ω (blue, lowest) to 6~Ω (green, highest) subspaces for
the most dominant 0p-0h + 2~Ω 2p-2h Sp(3,R) irrep case
(left) and NCSM (right) together with the leading irrep con-
tribution (black diamonds), (0 4) for 12C (a) and (0 0) for 16O
(b), as a function of the ~Ω oscillator strength, Nmax = 6.
Elliott’s collective states can be regarded as a good first-
order approximation in the presence of realistic interac-
tions, whether the latter is restricted to a 0~Ω model
space or richer multi-~Ω NCSM model spaces.
The 0+gs and 2
states in 12C, constructed in terms
of the three Sp(3,R) irreps with probability amplitudes
defined by the overlaps with the NCSM wavefunctions
for Nmax = 6 case, were also used to determine B(E2 :
→ 0+gs) transition rates. The latter, increasing from
101% to 107% of the corresponding NCSM numbers with
increasing ~Ω, clearly reproduce the NCSM results.
In summary, we have shown that ab initio NCSM cal-
culations with the JISP16 nucleon-nucleon interaction
display a very clear symplectic structure, which is un-
altered whether the bare or effective interactions for var-
ious ~Ω strengths are used. Specifically, NCSM wave-
functions for the lowest 0+gs, 2
and 4+
states in 12C and
the ground state in 16O project at the 85-90% level onto
a few 0p-0h and 2~Ω 2p-2h spurious center-of-mass free
symplectic irreps. Furthermore, while the dimensionality
of the latter is only ≈ 10−3% that of the NCSM space,
they closely reproduce the NCSM B(E2) estimates. The
wavefunctions for 12C are strongly dominated by the
three leading 0p-0h symplectic irreps, with a clear dom-
inance of the most deformed (0 4)S = 0 collective con-
figuration. The ground state of 16O is dominated by the
single 0p-0h irrep (0 0)S = 0. The results confirm for
the first time the validity of the Sp(3,R) approach when
realistic interactions are invoked in a NCSM space. This
demonstrates the importance of the Sp(3,R) symmetry
in light nuclei while reaffirming the value of the simpler
SU(3) model upon which it is based. The results further
suggest that a Sp-NCSM extension of the NCSM may
be a practical scheme for achieving convergence to mea-
sured B(E2) values without the need for introducing an
effective charge. In short, the NCSM with a modern re-
alistic interaction supports the development of collective
motion in nuclei which is realized through the Sp-NCSM
and as is apparent in its 0~Ω Elliott model limit.
Discussions with many colleagues, but especially Bruce
R. Barrett, are gratefully acknowledged. This work was
supported by the US National Science Foundation, Grant
Nos 0140300 & 0500291, and the Southeastern Universi-
ties Research Association, as well as, in part, by the US
Department of Energy Grant Nos. DE-AC02-76SF00515
and DE-FG02-87ER40371 and at the University of Cal-
ifornia, Lawrence Livermore National Laboratory under
contract No. W-7405-Eng-48. Tomáš Dytrych acknowl-
edges supplemental support from the Graduate School of
Louisiana State University.
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