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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. 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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  1. Jerry P. Draayer, Tomas Dytrych, Kristina D. Sviratcheva, Chairul Bahri (LSU) James P. Vary (ISU) Status: Ab-inito Symplectic No-core Shell Model

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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)

  7. 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

  8. 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

  9. 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

  10. 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 …

  11. 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

  12. Why Symplectic Symmetries? … … Canonical coordinates… Hamiltonian: + potential energy Nucleus with A nucleons

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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 …

  19. 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)

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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)

  25. “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)

  26. 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 (%)

  27. 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

  28. 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)

  29. 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

  30. 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

  31. 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

  32. 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

  33. 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

  34. 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!

  35. 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'

  36. 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

  37. 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

  38. 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

  39. 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

  40. 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

  41. 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

  42. 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

  43. 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

  44. 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)

  45. 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

  46. Alpha-cluster Model Extra 1 -CM 12C 1st O+ State

  47. Alpha-cluster Model Extra 2 -CM 12C 2nd O+ State

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