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


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Jerry P. Draayer , Tomas Dytrych, Kristina D. Sviratcheva, Chairul Bahri (LSU) James P. Vary (ISU). Status: Ab-inito Symplectic No-core Shell Model. From quarks/gluons to UNIVERSE. Quantum Chromodynamics. quarks gluons. Universe. equation of state, dark/dense matter, nucleosynthesis.

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

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Jerry p draayer tomas dytrych kristina d sviratcheva chairul bahri lsu james p vary isu

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

James P. Vary (ISU)

Status: Ab-inito

Symplectic No-core

Shell Model


Jerry p draayer tomas dytrych kristina d sviratcheva chairul bahri lsu james p vary isu

From quarks/gluons to UNIVERSE

Quantum Chromodynamics

quarks

gluons

Universe

equation of state,

dark/dense matter,

nucleosynthesis

Many-nucleon systems

shell structure,

-cluster modes,

collective rotations

low-energy

regime: non

perturbative


Jerry p draayer tomas dytrych kristina d sviratcheva chairul bahri lsu james p vary isu

Challenge: QCD-tied Nuclear Models

Nucleosyntesis

Nucleosynthesis Stellar life-cycles

chandra.harvard.edu

Detector ingredients

Many-nucleon systems

12C & 16O:

neutrino experiments

29Si (37Ge):

dark matter search

“femto-Lab”

Quantum Chromodynamics

quarks

gluons

parity violation in weak hadronic physics


Emerging symmetries in complex systems

Emerging symmetries in complex systems

Symplectic Sp(3,R)

realistic interaction(s)

either possess Sp(3,R) symmetry …

or / and

… the nuclear many-body system

filters out Sp(3,R) symmetry-breaking effects


Ab initio no core shell model

Ab initio No-Core Shell Model

No-Core

Shell-Model

(multi- 

-

4h

( ) limit

-

2h

( ) limit

-

0h

( ) limit

Solve

Schrödinger equation

in infinite space

-

h

horizontal

slices

Valence shell

Filled shells

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

  • In principle, exact solutions

  • Reproduction of binding energies and spectral features of light nuclei


Achievements of no core shell model ncsm

Achievements of No Core Shell Model (NCSM)

12C

ħ=15MeV

Interaction: JISP16 (other results available - ahead)

A. M. Shirokov et al., Phys. Letts. B 621, 96(2005)


Computational limits of no core shell model

Computational Limits of No Core Shell Model

Highly deformed modes,

-cluster structures,

B(E2) with NO effective charge

‘Larger’ model Spaces

Heavier Nuclei

can reach…

Symplectic-NCSM


Ncsm and sp 3 r shell model

NCSM and Sp(3,R) Shell Model

Multi-shell

symplectic

slice

No-Core

Shell-Model

(multi- 

-

4h

( ) limit

-

2h

( ) limit

-

0h

( ) limit

SU(3) limit ( )

-

-

h

0h

Symplectic Sp(3,R)

Shell-Model

(multi-ħ

vertical

slices

monopole & quadrupole collective excitations

horizontal

slices

Valence shell

Filled shells

  • Spurious center-of-mass motion free

  • Relation to NCSM basis: microscopic!

  • Reproduction of rotational energy spectra and electromagnetic transitionswithout effective charges

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

  • In principle, exact solutions

  • Reproduction of binding energies and spectral features of light nuclei


Sp 3 r su 3 model

Sp(3,R)SU(3) Model

20Ne

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

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

20-year “IBM” Pause


Symplectic no core shell model

Symplectic No-Core Shell Model

.…

.…

n

2

0

2

0

n

n

0

2

2

n

0

:

:

:

:

.…

.…

P

Q

:

:

U

U†=

P

Q

Q

:

:

Spherical harmonic oscillatorbasis

Symplecticbasis

Revisit below - SRG …


Why symmetries simple illustration

Why Symmetries? – Simple Illustration!

k

l

m

m

Familiar one-dimensional harmonic oscillator problem

Coupled equations of motion (ignore symmetry):

Solve

eigenvalue problem

Uncoupled equations of motion (with symmetry):

Easy as !

physics-animations.com

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

Use symmetry to reduce a two-variable problem to a one-variable (S) problem


Why symplectic symmetries

Why Symplectic Symmetries?

Canonical coordinates…

Hamiltonian:

+ potential energy

Nucleus with A nucleons


Why symplectic symmetries1

Why Symplectic Symmetries?

( l, m

( b, g)

m

b

g

l

nucleon system

HO Hamiltonian

(p2+x2)/2

Bohr-Mottelson Model

SU(3)

Model

vorticity

(from irrotational

to rigid rotor flows)

mass quadrupole

moment

angular momentum

SO(3)

multi-shell monopole

and quadrupole

collective vibrations

many-particle

kinetic energy

collective

microscopic

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


Jerry p draayer tomas dytrych kristina d sviratcheva chairul bahri lsu james p vary isu

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

( l, m

( b, g)

m

b

g

l

Elliott Model (single shell)

collective

microscopic

(11)

…….

Angular Momentum

Quadrupole Moment

Number Operator

Multi-shell Coupling

SO(3)

L1,M

Q2,M

N

AL,M

BL,M

(11)

………

SU(3)

xi, pi

……

(00)

essentially HO Hamiltonian

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

(20)

(02)

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

  • Higher-lying excitations: monopole & quadrupole modes

  • Kinetic energy:

  • Microscopic formulation of the Bohr-Mottelson model:

3-body interaction!

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


Jerry p draayer tomas dytrych kristina d sviratcheva chairul bahri lsu james p vary isu

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

(11)

L1,M

Q2,M

N

AL,M

BL,M

A’s raise 2ħ

1+5 = 6

(11)

N & L & Q

valence space

xi, pi

(00)

1+3+5 = 9

(20)

1+5 = 6

B’s lower 2ħ

(02)

9+6+6 = 21

SO(3)

SU(3)

U(3)

Sp(3,R)

collective

microscopic

  • 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


Symplectic sp 3 r basis

Symplectic {Sp(3,R)} Basis

12C

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

.

.

.

6ħ

N=6 sdgi

N=5 pfh

4ħ

N=4 sdg

N=3 pf

2ħ

N=2 sd

N=1 p

Valence shell

Filled shell

N=0 s

Examples for protons and proton-neutron excitations

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


Are the lowest energy np nh states missing

Are the lowest-energy np-nh States Missing?

12C

Examples for proton excitations

e.g., 2ħ2p-2h, 4ħ4p-4h, …

.

.

.

N=6 sdgi

N=5 pfh

4ħ

N=4 sdg

N=3 pf

2ħ

N=2 sd

N=1 p

Valence shell

Filled shell

N=0 s


Multi particle multi hole sp 3 r slices

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

12C

.

.

.

6ħ

N=6 sdgi

Model space of all possible Sp(3,R) vertical slices = NCSM space

N=5 pfh

4ħ

N=4 sdg

N=3 pf

2ħ2p-2h vertical slice

N=2 sd

N=1 p

Valence shell

Filled shell

N=0 s

Revisit below --CM

… or, CCX theory …


Achievements of no core shell model ncsm1

Achievements of No Core Shell Model (NCSM)

12C

ħ=15MeV

Interaction: JISP16 (other results available - ahead)

A. M. Shirokov et al., Phys. Letts. B 621, 96(2005)


Symplectic symmetry in many body wave functions

Symplectic Symmetry in Many-body Wave Functions

12C

Only 3 vertical slices: ~80%

NCSM wave function projected

onto Symplectic basis

6ħ

4ħ

2ħ

No-Core Shell-Model wave function probability distribution

(0+gs)

ħ=15MeV

0ħ


Symplectic symmetry in many body wave functions1

Symplectic Symmetry in Many-body Wave Functions

12C

16O

12C

Only a few symplectic slices:

6ħ

4ħ

0+gs, 2+1, 4+1

2ħ

0+gs

0ħ

NCSM wave function projected

onto Symplectic basis

6ħ

4ħ

2ħ

No-Core Shell-Model wave function probability distribution

(0+gs)

  • ~85%-90% overlap

  • ~100% B(E2) values

1 vertical slice

(most deformed, S=0) ~65%

ħ=15MeV

0ħ


Jerry p draayer tomas dytrych kristina d sviratcheva chairul bahri lsu james p vary isu

Probability Distribution: Ground State – 85-90%

100%

of 0ħ

Only 3 0p-0h symplectic irreps: ~80%

(04) “slice”

Sp(3,R)

NCSM

(00) “slice”

0gs Probability distribution (%)

Validity of Elliott’s SU(3)

N ()

2ħ 2p-2h Sp(3,R) irreps:

~4% (12C)

~10% (16O)

24.5(04) : most deformed

24.5(12)2: spin one states


Major reductions in model space

Major Reductions in Model Space

Dimension of Model Space

1012

3 0p-0h

all 0p-0h

12C

16O

1010

NCSM

dominant

NCSM

0p-0h + 2p-2h

108

NCSM

106

Sp(3,R)

Sp(3,R)

104

102

1

Compared to NCSM

0ħ

4ħ

8ħ

12ħ

0ħ

4ħ

8ħ

12ħ

Model Space

Model Space

Dimension of model space

0.009% for 12C

0.0004% for 16O

T. Dytrych. KDS,

C. Bahri, J.P. Draayer, J.P. Vary,

Phys. Rev. Lett. 98 (2007) 162503


Matching dynamics to geometry

Matching “Dynamics” to “Geometry”

12C

g.st.

( l, m

( b, g)

m

b

g

l

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

ħ

=15 MeV

Dominant modes:

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

2p-2h: (2 4)


Classical peek at a quantum nuclear system

“Classical” Peek At a Quantum Nuclear System

12C

16O

Ground state

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

0+2

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

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

ħ

( =15 MeV)


Jerry p draayer tomas dytrych kristina d sviratcheva chairul bahri lsu james p vary isu

Spin Distribution in NCSM Eigenstates

2+

0+

12C

4+

90

90

80

80

70

70

60

60

50

50

40

40

Bare

Bare

Bare

11

11

11

12

12

12

13

13

13

14

14

14

15

15

15

16

16

16

17

17

17

18

18

18

30

30

20

20

10

10

0

0

Spin=0

Spin=1

Spin=2

Probability amplitude (%)

Probability amplitude (%)


Jerry p draayer tomas dytrych kristina d sviratcheva chairul bahri lsu james p vary isu

Independence of Oscillator Strength

0+

2+

4+

0+

Spin components of converged states

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

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

Spin=0

Spin=1

Symplectic structure is not altered by Lee-Suzuki transformation

Spin=0

Spatial wavefunctions: independent of whether bare or effective interaction is used


Jerry p draayer tomas dytrych kristina d sviratcheva chairul bahri lsu james p vary isu

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

collective

microscopic

(11)

L1,M

Q2,M

N

AL,M

BL,M

A’s raise 2ħ

1+5 = 6

(11)

N & L & Q

valence space

xi, pi

(00)

1+3+5 = 9

(20)

1+5 = 6

B’s lower 2ħ

(02)

9+6+6 = 21

Sp(3,R)

Spin

X

S

HSp(2)=a{XX}+bL.S+cS2

Symplectic Sp(3,R) symmetry preserving Hamiltonian

(the most general)

Compare with

JISP16 realistic interaction (and others)


Spectral distribution theory correlation coefficients

Spectral Distribution Theory: Correlation Coefficients

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


Correlation coefficients

Correlation Coefficients

1

Perfect!

0.9

0.8

0.7

‘good’

0.6

0.5

0.4

0.3

0.2

‘poor’

0.1

0

nearly perfect

very large

Correlation coefficient

large

medium

small

trivial


T 0 symplectic symmetry in jisp16

T=0 Symplectic Symmetry in JISP16

‘very large’

N=5 pfh

N=4 sdg

N=3 pf

N=2 sd

N=1 p

N=0 s

Effective JISP16, 6 shells (ħ=15 MeV)

Bare JISP16

Effective JISP16, 4 shells (ħ=15 MeV)

JISP16 NN interaction

Symmetry breaking

Symplectic interaction

Correlation coefficient


T 1 symplectic symmetry in jisp16

T=1 Symplectic Symmetry in JISP16

JISP16 NN interaction

Effective JISP16, 6 shells (ħ=15 MeV)

Bare JISP16

Effective JISP16, 4 shells (ħ=15 MeV)

Symmetry breaking

Symplectic interaction

‘very large’

Correlation coefficient

N=5 pfh

N=4 sdg

N=3 pf

N=2 sd

N=1 p

N=0 s


Symplectic symmetry in other interactions

Symplectic Symmetry in Other Interactions

Pairing interaction

Bare JISP16

Effective JISP16, 4 shells (ħ=15 MeV)

Pairing interaction

Symmetry breaking

Symplectic interaction

‘very large’

Correlation coefficient

N=5 pfh

‘poor’

N=4 sdg

N=3 pf

N=2 sd

N=1 p

N=0 s


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% overlap

      • 100% 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!


Jerry p draayer tomas dytrych kristina d sviratcheva chairul bahri lsu james p vary isu

Similarity Renormalization Group and VNN

SRG

dHs

ds

Hs

d

ds

O

[O, Hs]

Renormalized interaction Hs

Bare interaction (VNN, VNNN, …) Hs=0

Unitary

transformation

where+= -(antihermitian)

s is called the 'flow parameter'

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

'flow equation'


Similarity renormalization group and v nn

Similarity Renormalization Group and VNN

SRG

Renormalized interaction Hs

Bare interaction (VNN, VNNN, …) Hs=0

Unitary

transformation

 flow equation

flow parameter 

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

SU(3) basis states:

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

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


J 0 t 1 up through the pf shell

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

SRG

6

MeV

4

2

iHifor the i SU(3) state:

( )=(0 0) L=0 S=0

0

-

2

-

4

Flow parameter s=1/2

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Exact solution for the lowest-lying state

0.4

120

0.2

100

0.0

Number of non-

diagonal matrix

elements

80

-

0.2

60

40

-

0.4

20

-

0.6

5

10

15

20

25

30


J 0 t 1 8 shells

J=0, T=1 (8 shells)

SRG

5

0

-

5

0.0

0.1

0.2

0.3

0.4

0.5

 = 0.71 MeV

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

MeV

iHifor the i SU(3) state:

( )=(0 0) L=0 S=0

 Exact solution for the lowest-lying state

Flow parameter s=1/2


J 1 t 0 up through the pf shell

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

SRG

MeV

iHifor the i SU(3) state:

( )=(0 0) L=0 S=1

500

400

Flow parameter s=1/2

10

300

Exact solution for the lowest-lying state

8

0.6

200

6

0.4

100

4

0.2

5

10

15

20

25

30

2

0.0

Number of non-

diagonal matrix

elements

0

-

0.2

-

2

-

0.4

0.00

0.01

0.02

0.03

0.04

0.05


J 1 t 0 8 shells

J=1, T=0 (8 shells)

SRG

5

0

-

5

0.00

0.05

0.10

0.15

0.20

 = 2.24 MeV

MeV

iHifor the i SU(3) state:

( )=(0 0) L=0 S=1

 Exact solution for the lowest-lying state

Flow parameter s=1/2


Similarity renormalization group summary

Similarity Renormalization Group Summary

SRG

  • 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


Jerry p draayer tomas dytrych kristina d sviratcheva chairul bahri lsu james p vary isu

Comparison with a simple -cluster model

-CM

Possible SU(3) symmetry of cluster wave functions

Constituent clusters “frozen” to SU(3)-symmetric ground states

Relative motion of clusters (carries Q≥4 oscillator quanta)

100% overlap with the leading symplectic bandheads!

Hecht, Phys. Rev C 16 (1977) 2401

Suzuki, Nucl. Phys. A 448 (86) 395


Jerry p draayer tomas dytrych kristina d sviratcheva chairul bahri lsu james p vary isu

Comparison with -cluster wavefunctions

-CM

16O

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

g.st.

Projection onto cluster wave functions

……………31%

..………… ..65%

6 ħ

4 ħ

2 ħ

0 ħ

Probability distribution, %

………… .100%

Project at…

ħ, MeV


Jerry p draayer tomas dytrych kristina d sviratcheva chairul bahri lsu james p vary isu

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

-CM

16O

Probability, %

2p-2h

4p-4h

Sp(3,R) slices built on most deformed

np-nh

bandheads

ħ

ħ

ħ

ħ

ħ

ħ

ħ

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

-cluster modes constitute:

30% of [(A(20)  A(20))(00)];

67% of [(A(20))(42)];

100% of (84) (the (84) Sp(3,R) bandhead)


Jerry p draayer tomas dytrych kristina d sviratcheva chairul bahri lsu james p vary isu

Alpha-cluster Model Summary

-CM

  • 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 symplecticbandheads need to be included


Jerry p draayer tomas dytrych kristina d sviratcheva chairul bahri lsu james p vary isu

Alpha-cluster Model Extra 1

-CM

12C

1st O+ State


Jerry p draayer tomas dytrych kristina d sviratcheva chairul bahri lsu james p vary isu

Alpha-cluster Model Extra 2

-CM

12C

2nd O+ State


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