Jerry P. Draayer , Tomas Dytrych, Kristina D. Sviratcheva, Chairul Bahri (LSU) James P. Vary (ISU)

Status: Ab-inito Symplectic No-core Shell

Model

Seminar: Ab-Initio Nuclear Structure - Where do we

stand?

Bad Honnef, GermanyJuly 28-30, 2008

Jerry P. Draayer, Tomas Dytrych, Kristina D. Sviratcheva, Chairul Bahri (LSU)

James P. Vary (ISU)


Model


stand?


From quarks/gluons to UNIVERSEFrom quarks/gluons to UNIVERSE

Universe

Quantum Chromodynamics quarks

gluonsquarksgluons

shell structure,-cluster modes,

collective rotations

shell structure,-cluster modes,

collective rotations

Many-nucleon

systems

low-energy regime: nonperturbative

low-energy regime: nonperturbative

equation of state,dark/dense matter,

nucleosynthesis

equation of state,dark/dense matter,

nucleosynthesis


Model


stand?


Many-nucleon

systems

Challenge: QCD-tied Nuclear ModelsChallenge: QCD-tied Nuclear Models

Nucleosynthesis Stellar life-cycles

Starssohowww.nascom.nasa.gov

chandra.harvard.edu

Nucleosyntesis

Detector ingredients

12C & 16O: neutrino experiments

29Si (37Ge):dark matter search “femto-

Lab”

parity violation in weak hadronic physics

Quantum Chromodynamics quarks

gluonsquarksgluons


Model


stand?


Emerging symmetries in complex systemsEmerging symmetries in complex systems

realistic interaction(s) either possess Sp(3,R) symmetry …

… the nuclear many-body system filters out Sp(3,R) symmetry-breaking effects

or / and

Symplectic Sp(3,R)

QuickTime™ and aAnimation decompressor

are needed to see this picture.


Model


stand?


horizontalslices

No-CoreShell-Model (multi- h-

Realistic interaction (local/nonlocal;NN,NNN,…)

In principle, exact solutions Reproduction of binding energies

and spectral features of light nuclei




Filled shells

Ab initio No-Core Shell Model Ab initio No-Core Shell Model

Valence shell Solve Schrödinger equation

in infinite space

€

HΨ(1,2,...,A) = EΨ(1,2,...,A)

( ) limit 4h-

( ) limit 2h-

( ) limit 0h-

…


Model


stand?


Achievements of No Core Shell Model (NCSM)


ħ =15MeV

12C

Interaction: JISP16 (other results available - ahead) A. M. Shirokov et al., Phys. Letts. B 621, 96(2005)


Model


stand?


QuickTime™ and aTIFF (LZW) decompressor


‘Larger’ model Spaces‘Larger’ model Spaces

Heavier NucleiHeavier Nuclei

Symplectic-NCSMSymplectic-NCSM

can reach…

Computational Limits of No Core Shell Model Computational Limits of No Core Shell Model

2H4He6,7Li8,9Be9,10B12C14N16O18F20Ne24Mg40,48Ca56Ni80Zr

Highly deformed modes,-cluster structures,

B(E2) with NO effective charge

Highly deformed modes,-cluster structures,

B(E2) with NO effective charge


Model


stand?


No-CoreShell-Model (multi- h-







Filled shells

Spurious center-of-mass motion free Relation to NCSM basis: microscopic! Reproduction of rotational energy

spectra and electromagnetic transitions without effective charges

Spurious center-of-mass motion free Relation to NCSM basis: microscopic! Reproduction of rotational energy

spectra and electromagnetic transitions without effective charges

NCSM and Sp(3,R) Shell ModelNCSM and Sp(3,R) Shell Model

SU(3) limit ( )0h-

verticalslices

monopole & quadrupole collective excitations

Symplectic Sp(3,R)Shell-Model (multi-ħ

Multi-shellsymplectic

slice

Multi-shellsymplectic

slice

Valence shell

horizontalslices

( ) limit 4h-

( ) limit 2h-

( ) limit 0h-

…


Model


stand?


Sp(3,R)SU(3) ModelSp(3,R)SU(3) Model

J.P. Draayer, K.J. Weeks, G. Rosensteel (1984)

€

+ ε j

j

∑ n j + G0P

G. Rosensteel and D.J. Rowe (1980)

€

12hω

20Ne

H =H0 +b2Q⋅Q+b3(Q×Q)⋅Q+b4 (Q⋅Q)2 20-year “IBM” Pause


Model


stand?


Symplectic No-Core Shell ModelSymplectic No-Core Shell Model

U U†=

€

h0

€

h2

€

hn

::

::

€

h0

€

h2

€

hn.… .…

€

h0

€

h2

€

hn

::

::

€

h0

€

h2

€

hn.… .…

PP

PQ

Q

Q

Spherical harmonic oscillator basis

Symplectic basis

Revisit below - SRG …


Model


stand?


Why Symmetries? – Simple Illustration!Why Symmetries? – Simple Illustration!

physics-animations.com

Normal modes ... associated with the symmetry of the pendulum motion

Coupled equations of motion (ignore symmetry):

€

˙ ̇ θ 1 = −k

mθ1 −θ2( ) −

g

lθ1

˙ ̇ θ 2 = −k

mθ2 −θ1( ) −

g

lθ2

€

θS = θ1 + θ2 ⇒ ˙ ̇ θ S = −g

lθS

€

θA = θ1 −θ2 ⇒ ˙ ̇ θ A = −2k

m−

g

l

⎛

⎝ ⎜

⎞

⎠ ⎟θA

€

ωS =g

l

€

ωA = 2k

m+

g

l

Use symmetry to reduce a two-variable problem to a one-variable (θS) problemUse symmetry to reduce a two-variable problem to a one-variable (θS) problem

Familiar one-dimensional harmonic oscillator problemFamiliar one-dimensional harmonic oscillator problem



θ1 θ2

k

m m

l

Easy as !Easy as !

Uncoupled equations of motion (with symmetry):

Solve eigenvalue problem


Model


stand?


Why Symplectic Symmetries? Why Symplectic Symmetries?

€

rx

€

rp

Canonical coordinates…QuickTime™ and a

TIFF (LZW) decompressorare needed to see this picture.

θ1 θ2



θ1 θ2



θ1 θ2



θ1 θ2



θ1 θ2

…

…

€

θ1, p1

€

θ2, p2

€

θ3, p3

€

θ4 , p4

€

θA−1, pA−1

€

θA , pA

€

H = H θ1,θ2,...,θA , p1, p2,..., pA( )Hamiltonian:

€

p12

2ml2+

p22

2ml2+ ...+

pA2

2ml2

Nucleus with A nucleons

+ potential energy

€

psi psj

s

∑ , xsi psj

s

∑ ± xsj psi, xsixsj

s

∑

€

psi psj

s

∑ , xsi psj

s


s

∑


Model


stand?


collectivecollectivemicroscopicmicroscopic

Why Symplectic Symmetries? Why Symplectic Symmetries?

many-particle kinetic energy many-particle kinetic energy

angular momentumSO(3)

angular momentumSO(3)

mass quadrupolemoment

mass quadrupolemoment

HO Hamiltonian (p2+x2)/2

HO Hamiltonian (p2+x2)/2

€

psi psj

s

∑ , xsi psj

s


s

∑

€

psi psj

s

∑ , xsi psj

s


s

∑G. Rosensteel and D.J. Rowe, Phys. Rev. Lett. 38 (1977) 10

€

rx

€

rp

nucleon system

Bohr-Mottelson Model

SU(3)Model

vorticity (from irrotational

to rigid rotor flows)

vorticity (from irrotational

to rigid rotor flows)multi-shell monopole

and quadrupole collective vibrations

multi-shell monopole and quadrupole

collective vibrations


Model


stand?


Symplectic {Sp(3,R) SU(3) SO(3)} Model


Angular Momentum

Quadrupole Moment

Number Operator

Multi-shell Coupling

Angular Momentum

Quadrupole Moment

Number Operator

Multi-shell Coupling

xi, pi

collectivecollective

€

1

12Q⊗Q[ ]

L= 0=

3

20πA2 R0

b

⎛

⎝ ⎜

⎞

⎠ ⎟4

β 2

−1

108

7

2Q⊗Q⊗Q[ ]

L= 0=

1

20π 5πA3 R0

b

⎛

⎝ ⎜

⎞

⎠ ⎟6

β 3 cos3γ

Higher-lying excitations: monopole & quadrupole modes

Kinetic energy:

Microscopic formulation of the Bohr-Mottelson model:

Higher-lying excitations: monopole & quadrupole modes

Kinetic energy:

Microscopic formulation of the Bohr-Mottelson model:

€

H00(00)

2−

6

4A00

(20) + B00(02)

( )

essentially HO Hamiltonian

€

rx ×

r p

€

x ix j

…….

………

……

L1,M

Q2,M

N

AL,M

BL,M

(20)

(02)

(00)

(11)

(11)

SU(3)

microscopicmicroscopic

SO(3)

[Hamiltonian H0 = (p2+x2)/2]

[Monopole, L = 0 & Quadrupole, L = 2]…

Elliott Model (single shell)

G. Rosensteel and D.J. Rowe, Phys. Rev. Lett. 38 (1977) 10

3-body interaction!


Model


stand?




3 Ang. Mom. Ops: L+1, L0, L–1

5 Quadrupole Ops: Q+2, Q+1, Q0, Q–1, Q–2

1 Number Operator: N

6 2ħ Raising Ops: A0,0 & A2,+2, A2,+1, A2,0, A2,–1, A2,–2

6 2ħ Lowering Ops: B0,0 & B2,+2, B2,+1, B2,0, B2,–1, B2,–2

21 Total number of independent quadratic scalars operators in x and p

3 Ang. Mom. Ops: L+1, L0, L–1

5 Quadrupole Ops: Q+2, Q+1, Q0, Q–1, Q–2

1 Number Operator: N

6 2ħ Raising Ops: A0,0 & A2,+2, A2,+1, A2,0, A2,–1, A2,–2

6 2ħ Lowering Ops: B0,0 & B2,+2, B2,+1, B2,0, B2,–1, B2,–2

21 Total number of independent quadratic scalars operators in x and p


xi, pi

L1,M

Q2,M

N

AL,M

BL,M

(20)

(02)

(00)

(11)

(11)


SO(3)SU(3)

U(3)

Sp(3,R)

A’s raise 2ħ 1+5 = 6

B’s lower 2ħ 1+5 = 6

N & L & Qvalence space

1+3+5 = 9

9+6+6 = 21


Model


stand?


N=0 sN=1 p

N=2 sd

N=3 pf

N=4 sdg

N=5 pfh

N=6 sdgi

Valence shell

Filled shell

Symplectic {Sp(3,R)} BasisSymplectic {Sp(3,R)} Basis

2ħ2ħ

4ħ4ħ

In addition to the 2ħ 1p-1h excitations: small (~1/A) 2ħ 2p-2h correction for removing spurious center-of-mass motion

Examples for protons and proton-neutron excitations

6ħ6ħ

12C

Vertical slices: 2-shell L=0 and L=2 excitations...

...


Model


stand?


N=0 sN=1 p

N=2 sd

N=3 pf

N=4 sdg

N=5 pfh

N=6 sdgi

Valence shell

Filled shell

Are the lowest-energy np-nh States Missing?Are the lowest-energy np-nh States Missing?

2ħ2ħ

4ħ4ħ

12C

e.g., 2ħ 2p-2h, 4ħ 4p-4h, …...

...

Examples for proton excitations

⟩⟩⟩


Model


stand?


N=0 sN=1 p

N=2 sd

N=3 pf

N=4 sdg

N=5 pfh

N=6 sdgi

Valence shell

Filled shell

Multi-particle-multi-hole Sp(3,R) SlicesMulti-particle-multi-hole Sp(3,R) Slices

2ħ 2p-2h vertical slice 2ħ 2p-2h vertical slice

4ħ4ħ

6ħ6ħ

12C......

Model space of all possible

Sp(3,R) vertical slices

= NCSM space

Model space of all possible

Sp(3,R) vertical slices

= NCSM space

Revisit below - -CM… or, CCX theory …

npnh ;n =0,2,4,...; where B npnh =0

Am npnh ;m=0,1,2,...; spans full space

npnh ;n =0,2,4,...; where B npnh =0

Am npnh ;m=0,1,2,...; spans full space


Model


stand?




ħ =15MeV

12C

Interaction: JISP16 (other results available - ahead) A. M. Shirokov et al., Phys. Letts. B 621, 96(2005)


Model


stand?


Symplectic Symmetry in Many-body Wave Functions


Only 3 vertical slices: ~80%

Only 3 vertical slices: ~80%

NCSM wave function projected onto Symplectic basis

No-Core Shell-Model wave function probability distribution(0+

gs)

0ħ

2ħ

4ħ

6ħ

12C

ħ =15MeV


Model


stand?


12C16O

0+gs, 2+

1, 4+1

0+gs

Only a few symplectic slices:



~85%-90% overlap

~100% B(E2) values

~85%-90% overlap

~100% B(E2) values

NCSM wave function projectedonto Symplectic basis

No-Core Shell-Model wave function probability distribution(0+

gs)

0ħ

2ħ

4ħ

6ħ

12C

0ħ

2ħ

4ħ6ħ

ħ =15MeV

1 vertical slice (most deformed, S=0)

~65%

1 vertical slice (most deformed, S=0)

~65%


Model


stand?


Probability Distribution: Ground State – 85-90%

Probability Distribution: Ground State – 85-90%

0102030405060708090100

1112131415161718h≈ç≠ ––(MeV)

(a)

12C 6h≠4h≠2h≠0h≠ 1213141516h≈ç≠ ––(MeV)

(b)

16O0 gs P

roba

bilit

y di

stri

butio

n (%

)

h (MeV)–

Only 3 0p-0h symplectic irreps:

~80%

Only 3 0p-0h symplectic irreps:

~80%

N ()

Sp(3,R)Sp(3,R) NCSMNCSM

Validity of

Elliott’s SU(3)

Validity of

Elliott’s SU(3)

(04) “slice”

(00) “slice”

2ħ 2p-2h Sp(3,R) irreps: ~4% (12C) ~10% (16O)

2ħ 2p-2h Sp(3,R) irreps: ~4% (12C) ~10% (16O)

24.5(04) : most deformed

24.5(12)2: spin one states

100%of 0ħ100%of 0ħ

h (MeV)–


Model


stand?


Major Reductions in Model SpaceMajor Reductions in Model Space

Compared to NCSMCompared to NCSM

0ħ 4ħ 8ħ 12ħ

Dimension of Model Space

Model Space

3 0p-0hall 0p-0hdominant 0p-0h + 2p-2hNCSM

Sp(3,R)

NCSM

1

102

104

106

108

1010

1012

0ħ 4ħ 8ħ 12ħModel Space

Sp(3,R)

NCSM

12C 16O

Dimension of model space0.009% for 12C0.0004% for 16O

Dimension of model space0.009% for 12C0.0004% for 16O

T. Dytrych. KDS, C. Bahri, J.P. Draayer, J.P. Vary, Phys. Rev. Lett. 98 (2007) 162503


Model


stand?


Matching “Dynamics” to “Geometry”Matching “Dynamics” to “Geometry”

Dominant modes:

0p-0h: (0 4) [oblate]

2p-2h: (2 4)

Dominant modes:

0p-0h: (0 4) [oblate]

2p-2h: (2 4)

12C

0p-0h Sp(3,R) irreps2p-2h Sp(3,R) irreps

(prolate shapes)

(oblate shapes)

ω

ω

0

2

45

1

3

6

0 1 2 3 4 5 6 7 8

0p-0h Sp(3,R) irreps2p-2h Sp(3,R) irreps

(prolate shapes)

(oblate shapes)

ω

ω

0

2

45

1

3

6

0 1 2 3 4 5 6 7 8

g.st.

Area = Probability of dominant Sp(3,R) slicesArea = Probability of dominant Sp(3,R) slices

=15 MeV

ħ


Model


stand?


12C 16OGround state

(0p-0h + 2p-2h symplectic slices)Ground state

(0p-0h + 2p-2h symplectic slices)0+

2 (0p-0h + 2p-2h symplectic slices)

0+2

(0p-0h + 2p-2h symplectic slices)

“Classical” Peek At a Quantum Nuclear System

“Classical” Peek At a Quantum Nuclear System

Larger the probability for a given shape, longer the time it is dispalyed

QuickTime™ and aApple Pixlet Video decompressorare needed to see this picture.

QuickTime™ and aApple Pixlet Video decompressorare needed to see this picture.

( =15 MeV)ħ


Model


stand?


Spin Distribution in NCSM EigenstatesSpin Distribution in NCSM Eigenstates

0102030405060708090

11 12 13 14 15 16 17 18 Bare 11 12 13 14 15 16 17 18 Bare

11 12 13 14 15 16 17 18 Bare0

102030405060708090

12C

Prob

abil

ity

ampl

itud

e (%

)

Prob

abil

ity

ampl

itud

e (%

)

0+ 2+

4+

Spin=0

Spin=1

Spin=2

Spin=0

Spin=1

Spin=2

h (MeV)– h (MeV)–

h (MeV)–


Model


stand?


Independence of Oscillator StrengthIndependence of Oscillator Strength

0102030405060708090100

1112131415161718Bare

(c) J=4

h≈ç≠ ––(MeV)12C 0102030405060708090100

1213141516Bare

(d) J=0

h≈ç≠ ––(MeV)16O

0102030405060708090100

1112131415161718Bare

(a) J=0

12C h≈ç≠ ––(MeV)0102030405060708090100

1112131415161718Bare

(b) J=2

h≈ç≠ ––(MeV)12C

Overlaps (%) of NCSM wavefunctions with dominant Sp(3,R) states

Spin=0

Spin=1

Spin=0

Spin=1

Spin=0Spin=0

0+ 2+

4+ 0+

Symplectic structure is

not altered by Lee-Suzuki

transformation

Symplectic structure is

not altered by Lee-Suzuki

transformation

Spin components of converged statesSpin components of converged states

Spatial wavefunctions: independent of

whether bare or effective

interaction is used

Only 6 Sp(3,R) irreps (3 0p-0h and 3 2p-2h )




Model


stand?


Sp(3,R) + Complementary (spin-isospin) symmetry

Sp(3,R) + Complementary (spin-isospin) symmetry

X S

HSp(2)=a{XX}+bL.S+cS2HSp(2)=a{XX}+bL.S+cS2

Sp(3,R) Spin

Symplectic Sp(3,R) symmetry preserving Hamiltonian

(the most general) JISP16 realistic interaction (and others)

Compare with


xi, pi

L1,M

Q2,M

N

AL,M

BL,M

(20)

(02)

(00)

(11)

(11)


A’s raise 2ħ 1+5 = 6

B’s lower 2ħ 1+5 = 6

N & L & Qvalence space

1+3+5 = 9

9+6+6 = 21


Model


stand?


Spectral Distribution Theory: Correlation Coefficients

Spectral Distribution Theory: Correlation Coefficients

€

cosθ =r v ⋅ ′

r v

r v ′

r v

QuickTime™ and a decompressor


00.2

0.4 0.60.8

0

0.2

0.4

0.6

0.8

0

0.1

0.2

0.3

0.4

0.5

00.2

0.4 0.60.8

HSp(4)

HQ⊥

1GXPF-CD Bonn

- +3CD Bonn terms

1f7/2

Excellent method for comparing •any two microscopic interactions

•the way the two interactions govern many-nucleon systems

•Comparison of global properties•Revealing underlying symmetries/

symmetry breaking patterns in realistic interactions

•Large correlation coefficients yield similar energy spectra

•Comparison of global properties•Revealing underlying symmetries/

symmetry breaking patterns in realistic interactions

•Large correlation coefficients yield similar energy spectra


Model


stand?


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Correlation CoefficientsCorrelation Coefficients

Perfect!C

orre

lati

on c

oeff

icie

nt

small

medium

large

very large

nearly perfect

trivial

‘good’

‘poor’


Model


stand?


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

s,ps,p

sds,p

sdsd

pfs,p

pfsd

pfpf

sdgs,p

sdgsd

sdgpf

sdgsdg

pfhs,p

pfhsd

pfhpf

pfhsdg

pfhpfh

T=0 Symplectic Symmetry in JISP16T=0 Symplectic Symmetry in JISP16

N=0 sN=1 p

N=2 sd

N=3 pf

N=4 sdg

N=5 pfh

Cor

rela

tion

coe

ffic

ient

Effective JISP16, 6 shells (ħ =15 MeV)Bare JISP16Effective JISP16, 4 shells (ħ =15 MeV)

JISP16 NN interaction

Symplectic interaction

Symmetry breaking

‘very large’


Model


stand?


T=1 Symplectic Symmetry in JISP16T=1 Symplectic Symmetry in JISP16

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

s,ps,p

sds,p

sdsd

pfs,p

pfsd

pfpf

sdgs,p

sdgsd

sdgpf

sdgsdg

pfhs,p

pfhsd

pfhpf

pfhsdg

pfhpfh

N=0 sN=1 p

N=2 sd

N=3 pf

N=4 sdg

N=5 pfh

Cor

rela

tion

coe

ffic

ient

Effective JISP16, 6 shells (ħ =15 MeV)Bare JISP16Effective JISP16, 4 shells (ħ =15 MeV)

‘very large’

JISP16 NN interaction


Symmetry breaking


Model


stand?


Symplectic Symmetry in Other InteractionsSymplectic Symmetry in Other InteractionsC

orre

lati

on c

oeff

icie

nt

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

s,ps,p

sds,p

sdsd

pfs,p

pfsd

pfpf

sdgs,p

sdgsd

sdgpf

sdgsdg

pfhs,p

pfhsd

pfhpf

pfhsdg

pfhpfh

Pairing interactionBare JISP16Effective JISP16, 4 shells (ħ =15 MeV)

N=0 sN=1 p

N=2 sd

N=3 pf

N=4 sdg

N=5 pfh

‘poor’

‘very large’

Pairing interaction


Symmetry breaking


Model


stand?


Summary Summary

Ab-initio No Core Shell Model: successfully reproduces (low-lying) features of the deuteron, alpha particle, 12C and even 16O

Comparison of converged NCSM eigenstates with Sp(3,R)-symmetric states shows:• Reproduction of NCSM results by a few Sp(3,R) states –

85%-90% overlap100% B(E2: 21

+ 01

+)• Dramatic reduction in model space (several orders of magnitude)

Symplectic-NCSM: effective model space reduction scheme

Sp(3,R) symmetry found dominant in ab initio realistic solutions

Symplectic-NCSM … simply matching “geometry” to “dynamics”

Ab-initio No Core Shell Model: successfully reproduces (low-lying) features of the deuteron, alpha particle, 12C and even 16O

Comparison of converged NCSM eigenstates with Sp(3,R)-symmetric states shows:• Reproduction of NCSM results by a few Sp(3,R) states –

85%-90% overlap100% B(E2: 21

+ 01

+)• Dramatic reduction in model space (several orders of magnitude)

Symplectic-NCSM: effective model space reduction scheme

Sp(3,R) symmetry found dominant in ab initio realistic solutions

Symplectic-NCSM … simply matching “geometry” to “dynamics”Where do we stand? … We’ve learned a lot; we’ve go lots to learn!


Model


stand?


Similarity Renormalization Group and VNNSimilarity Renormalization Group and VNN

Unitary

transformation

Bare interaction (VNN, VNNN, …)

Hs=0

Renormalized interaction

Hs

where + = - (antihermitian)s is called the 'flow parameter'

typically choose = [O, Hs] where O is a physically relevant operator, e.g. K.E.dH s

ds= O,Hs[ ],Hs⎡⎣ ⎤⎦

dH s

ds= O,Hs[ ],Hs⎡⎣ ⎤⎦

H s =e–sHs=0esH s =e–sHs=0es

'flow equation'

Hs

Oθ

[O, Hs]

dHs

ds

dθds

SRG


Model


stand?


Similarity Renormalization Group and VNNSimilarity Renormalization Group and VNN

Trel+NN interaction (bare CD-Bonn), ħ =15 MeV

€

dHs

ds= Csu(3)

(2) ,Hs[ ],Hs[ ]

€

dHs

ds= Csu(3)

(2) ,Hs[ ],Hs[ ]

Unitary

transformation

flow equationflow parameter

Bare interaction (VNN, VNNN, …)

Hs=0

Renormalized interaction

Hs

Second-order invariant operator of SU(3) (diagonal in SU(3) basis)

SU(3) basis states:

€

1

Νa(n1 0)st

† × a(n2 0)st†

[ ]κL;JM

(λμ )S,TT0

0

€

1

Νa(n1 0)st

† × a(n2 0)st†

[ ]κL;JM

(λμ )S,TT0

0

Flow equation solved for interaction matrix elements in SU(3) basis

SRG


Model


stand?


J=0, T=1 (up through the pf-shell)J=0, T=1 (up through the pf-shell)

0.00 0.05 0.10 0.15 0.20 0.25 0.30

-4

-2

0

2

4

6MeV

Flow parameter s=1/2

iHi⟩ for the i⟩ SU(3) state:( )=(0 0) L=0 S=0


5 10 15 20 25 30

20

40

60

80

100

120

Number of non-diagonal matrix

elements


elements

€

=1

s

SRG

-- 0.60.6

-- 0.40.4

-- 0.20.2

0.00.0

0.20.2

0.40.4

Exact solution for the lowest-lying state



Model


stand?


J=0, T=1 (8 shells)J=0, T=1 (8 shells)

0.0 0.1 0.2 0.3 0.4 0.5

- 5

0

5


MeV


= 0.71 MeV = 0.71 MeV

iHi⟩ for the i⟩ SU(3) state: ( )=(0 0) L=0 S=0

iHi⟩ for the i⟩ SU(3) state: ( )=(0 0) L=0 S=0

Structure dictated by the kinetic energy T & quadrupole operator Q which both belong to Sp(3,R) …

SRG


Model


stand?


J=1, T=0 (up through the pf-shell)J=1, T=0 (up through the pf-shell)

0.00 0.01 0.02 0.03 0.04 0.05

- 2

0

2

4

6

8

10MeV




5 10 15 20 25 30

100

200

300

400

500


elements


elements

€

=1

s

SRG

-- 0.40.4

-- 0.20.2

0.00.0

0.20.2

0.40.4

0.60.6




Model


stand?


J=1, T=0 (8 shells)J=1, T=0 (8 shells)


MeV


= 2.24 MeV = 2.24 MeV



0.00 0.05 0.10 0.15 0.20

- 5

0

5

SRG


Model


stand?


Similarity Renormalization Group Summary Similarity Renormalization Group Summary

We suggest use of SU(3) basis together with the second-order SU(3) invariant as the evolution operator for the SRG approach

SRG in SU(3) basis appears to be a very effective scheme for renormalization of the NN interaction used in the Sp-NCSM

Preliminary results point to the possibility of achieving an exact unitary transformation of realistic interactions, using SRG in the many-body Sp(3,R) basis

We suggest use of SU(3) basis together with the second-order SU(3) invariant as the evolution operator for the SRG approach

SRG in SU(3) basis appears to be a very effective scheme for renormalization of the NN interaction used in the Sp-NCSM

Preliminary results point to the possibility of achieving an exact unitary transformation of realistic interactions, using SRG in the many-body Sp(3,R) basis

SRG


Model


stand?


-CMComparison with a simple -cluster modelComparison with a simple -cluster model

Constituent clusters “frozen” to SU(3)-

symmetric ground states

Constituent clusters “frozen” to SU(3)-

symmetric ground states

Relative motion of clusters

(carries Q≥4 oscillator quanta)

Relative motion of clusters

(carries Q≥4 oscillator quanta)

Possible SU(3)

symmetry of cluster

wave functions

Possible SU(3)

symmetry of cluster

wave functions

100% overlap with the leading symplectic bandheads!

100% overlap with the leading symplectic bandheads!

Hecht, Phys. Rev C 16 (1977) 2401Suzuki, Nucl. Phys. A 448 (86) 395


Model


stand?


-CM

0

10

20

30

40

50

60

70

80

1 2 3 4 5 6 7 8 9 10 11 12 13 14

6hΩ4hΩ2hΩ0hΩ

12 13 14 15 16

hΩ, MeV

J=0

Comparison with -cluster wavefunctionsComparison with -cluster wavefunctions

(00) Sp(3,R) slice in NCSM 0+ state (00) Sp(3,R) slice in NCSM 0+ state g.st.

Projection onto cluster wave functions

Projection onto cluster wave functions

Probability distribution, %

………… .100%

..………… ..65%……………31%

Project at…

16O

ħ, MeV

6 ħ4 ħ2 ħ0 ħ


Model


stand?


-CMProjection of Sp(3,R) slices on cluster states Projection of Sp(3,R) slices on cluster states

2p-2h4p-4h

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12

(00)(42)(84)

Sp(3,R) slices built on most

deformed np-nh

bandheads

Probability, %

-cluster modes constitute:30% of [(A(20) A(20))(00)⟩];67% of [(A(20))(42)⟩]; 100% of (84)⟩ (the (84) Sp(3,R) bandhead)

16O

(00) Sp(3,R) slice insufficient – (42) & (84) slices must be included also

(00) Sp(3,R) slice insufficient – (42) & (84) slices must be included also

ħ ħ ħ ħ ħ ħ ħ


Model


stand?


-CMAlpha-cluster Model SummaryAlpha-cluster Model Summary

Deformed symplectic states possess appreciable overlaps with cluster wave functions

100% overlap for the most deformed symplectic bandheads

0p-0h Sp(3,R) slices are not sufficient to reproduce -cluster modes – Sp(3,R) slices build over highly deformed symplectic bandheads need to be included

Deformed symplectic states possess appreciable overlaps with cluster wave functions

100% overlap for the most deformed symplectic bandheads

0p-0h Sp(3,R) slices are not sufficient to reproduce -cluster modes – Sp(3,R) slices build over highly deformed symplectic bandheads need to be included


Model


stand?




-CMAlpha-cluster Model Extra 1Alpha-cluster Model Extra 1

12C 1st O+ State


Model


stand?


-CMAlpha-cluster Model Extra 2Alpha-cluster Model Extra 2



12C 2nd O+ State

Jerry P. Draayer , Tomas Dytrych, Kristina D. Sviratcheva, Chairul Bahri (LSU) James P. Vary (ISU)

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