ab initio valence space hamiltonians for exotic nuclei
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Ab Initio Valence-Space Hamiltonians for Exotic Nuclei Jason D son D. H . Holt olt R. Stroberg S. Bogner H. Hergert A. Schwenk J. Menendez Frontiers and Impact of Nuclear Science Aim of ab initio


  1. Ab Initio Valence-Space Hamiltonians for Exotic Nuclei Jason D son D. H . Holt olt R. Stroberg S. Bogner H. Hergert A. Schwenk J. Menendez

  2. Frontiers and Impact of Nuclear Science Aim of ab initio nuclear theory: Develop unified first-principles picture of structure and reactions - Nuclear forces (QCD/strong interaction at low energies) - Electroweak physics - Nuclear many-body problem 82 � 126 � 50 � protons 82 � 28 � 20 � 50 � 8 � 28 � 2 � 20 � 8 � 2 � neutrons

  3. Medium- and Heavy-Mass Exotic Nuclei What are the properties of proton/neutron-rich matter? What are the limits of existence of matter? How do magic numbers form and evolve? Worldwide joint experimental/theoretical effort! 82 � Advanc nces in m s in many-body m ny-body methods thods Treatm tment of nt of n nuc ucle lear f r for orces s π π π 126 � 3N f for orces e s esse ssentia ntial l for e or exotic otic n nuc ucle lei i 50 � protons 82 � 28 � 20 � 50 � 8 � 28 � 2 � 20 � 8 � 2 � neutrons

  4. The Nuclear Many-Body Problem Nucleus strongly interacting many-body system – A -body problem impossible H ψ n = E n ψ n Quasi-exact solutions in light nuclei ( GFMC , (IT)NCSM , …) Large space : controlled approximations to full Schrödinger Equation Large-space approach Limited range: Closed shell ±1 Even-even Limited properties: Ground states only Some excited state Coupled Cluster In-Medium SRG Green’s Function

  5. The Nuclear Many-Body Problem Nucleus strongly interacting many-body system – A -body problem impossible H ψ n = E n ψ n Quasi-exact solutions in light nuclei ( GFMC , (IT)NCSM , …) Large space : controlled approximations to full Schrödinger Equation Valence space : diagonalize exactly with reduced number of degrees of freedom Large-space approach Valence-space approach Limited range: All nuclei near closed-shell cores Closed shell ±1 Even-even All properties: Ground states Limited properties: Excited states Ground states only Some excited state EW transitions Coupled Cluster Coupled Cluster In-Medium SRG In-Medium SRG Green’s Function Perturbation Theory

  6. In-Medium Similarity Renormalization Group Continuous unitary trans (basis change) decouples “off-diagonal” physics H ( s ) = U ( s ) HU † ( s ) ≡ H d ( s ) + H od ( s ) → H d ( ∞ ) Interaction in new basis is simple Tsukiyama, Bogner , Schwenk, PRL (2011)

  7. In-Medium SRG Continuous unitary trans (basis change) decouples “off-diagonal” physics H ( s ) = U ( s ) HU † ( s ) ≡ H d ( s ) + H od ( s ) → H d ( ∞ ) Interaction in new basis is simple Can always write , for some generator U = e η η ✓ ◆ θ 0 For incline plane: η = − θ 0 Tsukiyama, Bogner , Schwenk, PRL (2011)

  8. ���� ���� ���� ���� ���� ���� Life Is Difficult: Particle/Hole Excitations Consider basis states as excitations from uncorrelated reference state 1p-1h excitation 2p-2h excitation Ref. Slater Determinant Unoccupied (Particles) ε F Occupied (Holes) 16 O N a † Y a a i a † i i = a † � � Φ ab = a † | Φ 0 i = i | 0 i | Φ a ↵ a a i | Φ 0 i b a j | Φ 0 i ij i =1 ����� ����� ����� ����� Hamiltonian schematically in terms of ph excitations ����� ����� ����� ����� ����� � � ����� � � ����� ����� � � � � Ground-state coupled to excitations is difficult ����� ����� ����� ����� h i | H | j i H od = h p | H | h i + h pp | H | hh i + · · · + h . c . � � � � � � � � � � � �

  9. In-Medium SRG for Nuclei For nuclear Hamiltonian, take η = H od with U = e η ∆ + h . c . Perform multiple rotations until U N = e η N · · · e η 2 e η 1 η N = 0 h n p n h | ˜ h i | H | j i H | Φ 0 i = 0 h Φ 0 | ˜ Fully correlated ground state : one matrix element H | Φ 0 i d H ( s ) Also flow equation approach = [ η ( s ) , H ( s )] d s

  10. IM-SRG for Valence-Space Hamiltonians Tsukiyama, Bogner , Schwenk, PRC (2012) Separate p states into valence states ( v ) and those above valence space ( q ) 50 0g 9/2 ����� ����� ����� ����� ����� ����� ����� ����� 0f 5/2 q H eff ����� ����� 1p 1/2 1p 3/2 p 28 ����� ����� 0f 7/2 20 ����� ����� 0d 3/2 v 1s 1/2 0d 5/2 ����� ����� 8 16 O � � � � � � h 0p 3/2 H ( s = 0) → H ( ∞ ) 0p 1/2 � � � � � � � � �� � �� � �� � �� � � � � � ������ Redefine H od to decouple valence space from excitations outside v �� �� �� H od = h p | H | h i + h pp | H | hh i + h v | H | q i + h pq | H | vv i + h pp | H | hv i + h . c . Core Energy Single-particle Two-body valence particle energies interaction matrix elements

  11. Ground-State Energies in Oxygen Isotopes Large/valence-space methods with same SRG-evolved NN+3N-full forces -130 -130 Bogner et al, PRL (2014) AME 2012 obtained in large many-body spaces NN+3N-full -140 -140 Energy (MeV) Energy (MeV) -150 -150 -160 -160 MR-IM-SRG IT-NCSM NN+3N-ind -170 -170 SCGF NN+3N-full Lattice EFT AME 2012 CC -180 -180 16 18 20 22 24 26 28 16 18 20 22 24 26 28 Mass Number A Mass Number A Hebeler, JDH, Menéndez, Schwenk, ARNPS (2015) Agreement between all methods with same input forces Clear improvement with NN+3N-full Still significant discrepancy between valence/large-space results

  12. � � � � � � � �� � ��� � ��� � �� � � � � � �� �� ��� ��� � ���� �� ������ How Do We Handle 3N Forces? �� � � Normal-ordered 3N : contribution from core with valence particles � � � � � �� � ��� � ��� � �� � � � � � �� �� ��� ��� � ���� �� ������ �� N.O. 1-body N.O. 0-body N.O. 2-body � � � � � � � �� � ��� � ��� � �� � � � � � �� �� ��� ��� � ���� �� ������ �� 16 O core O core O core + + + Neglect 3N forces between valence particles – significant as N v ∼ N c

  13. � � � � � � � �� � ��� � ��� � �� � � � � � �� �� ��� ��� � ���� �� ������ Targeted Normal Ordering �� � � Normal-ordered 3N : contribution from core with valence particles � � � � � �� � ��� � ��� � �� � � � � � �� �� ��� ��� � ���� �� ������ �� N.O. 1-body N.O. 0-body N.O. 2-body � � � � � � � �� � ��� � ��� � �� � � � � � �� �� ��� ��� � ���� �� ������ �� 16 O core O core O core + + + Neglect 3N forces between valence particles – significant as N v ∼ N c Capture these effects with new Targeted N.O. 16 O core ore 16 O

  14. � � � � � � � �� � ��� � ��� � �� � � � � � �� �� ��� ��� � ���� �� ������ Targeted Normal Ordering �� � � Normal-ordered 3N : contribution from core with valence particles � � � � � �� � ��� � ��� � �� � � � � � �� �� ��� ��� � ���� �� ������ �� N.O. 1-body N.O. 0-body N.O. 2-body � � � � � � � �� � ��� � ��� � �� � � � � � �� �� ��� ��� � ���� �� ������ �� 16 O core O core O core + + + Neglect 3N forces between valence particles – significant as N v ∼ N c Capture these effects with new Targeted N.O. Initial N.O. wrt nearest closed shell Still decouple standard sd valence space 16 O core ore 16 O → 22 O

  15. Ground-State Energies in Oxygen Isotopes Large/valence-space methods with same SRG-evolved NN+3N-full forces -130 -130 Bogner et al, PRL (2014) AME 2012 obtained in large many-body spaces NN+3N-full -140 -140 Energy (MeV) Energy (MeV) -150 -150 -160 -160 MR-IM-SRG IT-NCSM NN+3N-ind -170 -170 SCGF NN+3N-full Lattice EFT AME 2012 CC -180 -180 16 18 20 22 24 26 28 16 18 20 22 24 26 28 Mass Number A Mass Number A Hebeler, JDH, Menéndez, Schwenk, ARNPS (2015) Agreement between all methods with same input forces Clear improvement with NN+3N-full Still significant discrepancy between valence/large-space results

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