Ab Initio Nuclear Structure Theory with Chiral NN+3N Interactions. Robert Roth

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1 Ab Initio Nuclear Structure Theory with Chiral NN+3N Interactions Robert Roth

2 From QCD to Nuclear Structure Nuclear Structure Low-Energy QCD

3 From QCD to Nuclear Structure Nuclear Structure NN+3N Interaction from Chiral EFT Low-Energy QCD chiral EFT based on the relevant degrees of freedom & symmetries of QCD provides consistent NN, 3N, interaction plus currents

4 From QCD to Nuclear Structure Nuclear Structure Unitary / Similarity Transformation NN+3N Interaction from Chiral EFT adapt Hamiltonian to truncated low-energy model space tame short-range correlations improve convergence behavior transform Hamiltonian & observables consistently Low-Energy QCD

5 From QCD to Nuclear Structure Nuclear Structure Exact & Approx Many-Body Methods Unitary / Similarity Transformation accurate solution of the manybody problem for light & intermediate masses (NCSM, CC,) controlled approximations for heavier nuclei (MBPT,) all rely on truncated model spaces & benefit from unitary transformation NN+3N Interaction from Chiral EFT Low-Energy QCD

6 From QCD to Nuclear Structure Nuclear Structure Exact & Approx Many-Body Methods Importance Truncated NCSM & Coupled-Cluster with NN+3N Unitary / Similarity Transformation Similarity Renormalization Group with NN+3N NN+3N Interaction from Chiral EFT focus on consistent inclusion of chiral 3N interaction Low-Energy QCD

7 Nuclear Interactions from Chiral EFT

8 µ µ µ ¾ ¾ µ É ¼ ¼ É É É Ç Ç Ç Ä Ç Ä Ä Æ Ä ¾ Æ Æ Nuclear Interactions from Chiral EFT low-energy effective field theory for relevant degrees of freedom (π,n) based on symmetries of QCD LO NN 3N 4N NN at N 3 LO long-range pion dynamics explicitly short-range physics absorbed in contact terms, low-energy constants fitted to experiment (NN, πn,) hierarchy of consistent NN, 3N, interactions (plus currents) many ongoing developments NLO N 2 LO Entem & Machleidt 500 MeV cutoff 3N at N 2 LO Navrátil, 3 H fit 500 MeV cutoff 3N interaction at N 3 LO explicit inclusion of Δ-resonance formal issues: power counting, renormalization, cutoff choice, N 3 LO

9 Similarity Renormalization Group Roth, Langhammer, Calci et al Phys Rev Lett 107, (2011) Roth, Neff, Feldmeier Prog Part Nucl Phys 65, 50 (2010) Roth, Reinhardt, Hergert Phys Rev C 77, (2008) Hergert, Roth Phys Rev C 75, (R) (2007)

10 Similarity Renormalization Group continuous transformation driving Hamiltonian to band-diagonal form with respect to a chosen basis simplicity and flexibility are great advantages of unitary transformation of Hamiltonian (and other observables) the SRG approach H α =U α HU α evolution equations for H solve SRG evolution α and U α depending equations using generator two- & η α d H α = η α, H d α dα dα U three-body Jacobi HO α= U representation α η α dynamic generator: commutator with the operator in whose eigenbasis H shall be diagonalized η α =(2μ) 2 T int, H α

11 SRG Evolution in Three-Body Space 3B-Jacobi HO matrix elements chiral NN+3N α = 0000 fm4 0 E N3 LO + N2Λ LO, triton-fit, = fm MeV Jπ = 18 = 1, ℏΩ 2 = 28 MeV NCSM ground state 3 H E [MeV] (E, ) 1+,T E (E, ) Nm x

12 SRG Evolution in Three-Body Space (E, ) 0 E B-Jacobi HO matrix elements E [MeV] α=0320fm 4 J π = 1 2 Λ=133fm 1 +,T= 1 2,ħΩ=28MeV NCSM ground state 3 H significant improvement of convergence behavior 28 suppression of off-diagonal coupling ˆ= pre-diagonalization 0 E (E, )

13 Calculations in A-Body Space evolution induces n-body contributions H [n] α to Hamiltonian H α = H [1] α + H [2] α + H [3] α + H [4] α + truncation of cluster series inevitable formally destroys unitarity and invariance of energy eigenvalues (independence of α) Three SRG-Evolved Hamiltonians NN only: start with NN initial Hamiltonian and keep two-body terms only NN+3N-induced: start with NN initial Hamiltonian and keep twoand three-body terms α-variation provides a diagnostic tool to assess NN+3N-full: start with NN+3N initial Hamiltonian and keep twoand three-body terms the contributions of omitted many-body interactions

14 Sounds easy, but ➊ computation of initial 2B/3B-Jacobi HO matrix elements of chiral NN+3N interactions we use Petr Navratil s ManyEff code for computing 3B-Jacobi matrix elements and corresponding CFPs ➋ SRG evolution in 2B/3B space and cluster decomposition efficient implementation using adaptive ODE solver & BLAS; largest block takes a few hours on single node ➌ transformation of 2B/3B Jacobi HO matrix elements into JT-coupled representation formulated transformation directly into JT-coupled scheme; highly efficient implementation; can handle E 3m x =16 in JT-coupled scheme ➍ data management and on-the-fly decoupling in many-body codes invented optimized storage scheme for fast on-the-fly decoupling; can keep all matrix elements up to E 3m x =16 in memory

15 Importance Truncated NCSM Roth, Langhammer, Calci et al Phys Rev Lett 107, (2011) Navrátil, Roth, Quaglioni Phys Rev C 82, (2010) Roth Phys Rev C 79, (2009) Roth, Navrátil Phys Rev Lett 99, (2007)

16 Importance Truncated NCSM NCSM is one of the most powerful and universal exact ab-initio methods construct matrix representation of Hamiltonian using a basis of HO Slater determinants truncated wrt HO excitation energy ħω solve large-scale eigenvalue problem for a few extremal eigenvalues all relevant observables can be computed from the eigenstates range of applicabilitylimited by factorial growth of basis with & A adaptive importance truncation extends the range of NCSM by reducing the model space to physically relevant states we have developed a parallelized IT-NCSM/NCSM code capable of handling 3N matrix elements up to E 3m x =16

17 4 He: Ground-State Energies NN only NN+3N-induced NN+3N-full -23 strong α-dependence: induced 3N interactions no α-dependence: no induced 4N interactions no α-dependence: no induced 4N interactions -24 ħω=20mev E [MeV] Exp α=004fm 4 α=005fm 4 α=00625fm 4 α=008fm 4 α=016fm 4 Λ=224fm 1 Λ=211fm 1 Λ=200fm 1 Λ=188fm 1 Λ=158fm 1

18 6 Li: Ground-State Energies NN only NN+3N-induced NN+3N-full -22 ħω=20mev E [MeV] Exp α=004fm 4 α=005fm 4 α=00625fm 4 α=008fm 4 α=016fm 4 Λ=224fm 1 Λ=211fm 1 Λ=200fm 1 Λ=188fm 1 Λ=158fm 1

19 12 C: Ground-State Energies NN only NN+3N-induced NN+3N-full -60 ħω=20mev -70 E [MeV] Exp α=004fm 4 α=005fm 4 α=00625fm 4 α=008fm 4 α=016fm 4 Λ=224fm 1 Λ=211fm 1 Λ=200fm 1 Λ=188fm 1 Λ=158fm 1

20 16 O: Ground-State Energies -80 NN only NN+3N-induced NN+3N-full ħω=20mev -100 E [MeV] Exp caused by long-range 2π terms (c ) clear signature of induced 4N originating from initial 3N α=004fm 4 α=005fm 4 α=00625fm 4 α=008fm 4 α=016fm 4 Λ=224fm 1 Λ=211fm 1 Λ=200fm 1 Λ=188fm 1 Λ=158fm 1

21 6 Li: Excitation Energies NN only NN+3N-induced NN+3N-full E [MeV] ħω=20mev α=004fm 4 α=005fm 4 α=00625fm 4 α=008fm 4 α=016fm 4 Λ=224fm 1 Λ=211fm 1 Λ=200fm 1 Λ=188fm 1 Λ=158fm 1

22 Spectroscopy of 12 C 20 NN only NN+3N-induced NN+3N-full E [MeV] ħω=20mev Exp Exp 0 0 spectra largely insensitive to induced 4N Exp 0 0 α=008fm 4 α=00625fm 4

23 Outlook: Sensitivity on Initial 3N 18 standard 3N modified 3N interaction with shifted c i 400 MeV cutoff E [MeV] Exp PRELIMINARY NN+3N-full α=008fm 4 ħω=20mev Exp spectra of A 10 nuclei are a very sensitive benchmark for chiral 3N interactions Exp

24 Outlook: Carbon Isotopic Chain E [MeV] E [MeV] C Exp 16 C C Exp PRELIMINARY 2 3? C C Exp NN+3N-full Λ 3N =400MeV α=008fm 4 ħω=16mev 20 C Exp Exp Exp 0 4

25 Normal-Ordered 3N Interaction & Coupled-Cluster Method Roth, Binder, Vobig et al arxiv: (2011)

26 Normal-Ordered 3N Interaction avoid technical challenge of including explicit 3N interactions in many-body calculation idea: write 3N interaction in normal-ordered form with respect to an A-body reference Slater-determinant (0ħΩ state) V 3N = V 3N =W 0B + + W 1B { }+ W 2B { } W 3B { } question: if we neglect the normal-ordered 3B term, how well does this approximation work?

27 Benchmark of Normal-Ordered 3N 4 He 16 O E [MeV] NN+3N-ind NN+3N-ind compare IT-NCSM results with full 3N to normal-ord 3N truncated at the 2B level NN+3N-full NN+3N-full typical deviations up to 2% for 4 He and 1% for 16 O E [MeV] full / NO2B / α=004fm 4 / α=005fm 4 / α=00625fm 4 / α=008fm ħω=20mev

28 16 O: Coupled-Cluster with 3N NO2B -80 NN only NN+3N-induced NN+3N-full -100 E [MeV] CCSD ħω=20mev E 3m =14 Exp α=004fm 4 α=005fm 4 α=00625fm 4 α=008fm 4 Λ=224fm 1 Λ=211fm 1 Λ=200fm 1 Λ=188fm 1

29 16 O: Coupled-Cluster with 3N NO2B -80 NN only NN+3N-induced NN+3N-full -100 E [MeV] CCSD Exp ħω=20mev E 3m 3N =14 interaction with 400 MeV cutoff, c E refitted to 4 He ground state α=004fm 4 α=005fm 4 α=00625fm 4 α=008fm 4 Λ=224fm 1 Λ=211fm 1 Λ=200fm 1 Λ=188fm 1

30 24 O: Coupled-Cluster with 3N NO2B NN only NN+3N-induced NN+3N-full E [MeV] CCSD Exp -220 ħω=20mev E 3m = α=004fm 4 α=005fm 4 α=00625fm 4 α=008fm 4 Λ=224fm 1 Λ=211fm 1 Λ=200fm 1 Λ=188fm 1

31 40 Ca: Coupled-Cluster with 3N NO2B -250 NN only NN+3N-induced NN+3N-full -300 E [MeV] CCSD ħω=20mev E 3m =14 Exp -650 α=004fm 4 α=005fm 4 α=00625fm 4 α=008fm 4 Λ=224fm 1 Λ=211fm 1 Λ=200fm 1 Λ=188fm 1

32 48 Ca: Coupled-Cluster with 3N NO2B NN only NN+3N-induced NN+3N-full -300 E [MeV] CCSD ħω=20mev E 3m =14 Exp α=004fm 4 α=005fm 4 α=00625fm 4 α=008fm 4 Λ=224fm 1 Λ=211fm 1 Λ=200fm 1 Λ=188fm 1

33 Outlook: Chiral 3N for Heavy Nuclei Ni 56 Ni 68 Ni 78 Ni E/A [MeV] E/A [MeV] Sn PRELIMINARY 114 Sn Sn CCSD(HF) NN+3N-full NO2B Λ 3N =400MeV α=008fm 4 ħω=36mev

34 Conclusions

35 Conclusions new era of ab-initio nuclear structure and reaction theory connected to QCD via chiral EFT chiral EFT as universal starting point some issues remain consistent inclusion of 3N interactions in similarity transformations & many-body calculations breakthrough in computation & handling of 3N matrix elements innovations in many-body theory: extended reach of exact methods & improved control over approximations versatile toolbox for different observables & mass ranges many exciting applications ahead

36 Epilogue thanks to my group & my collaborators S Binder, A Calci, B Erler, E Gebrerufael, A Günther, H Krutsch, J Langhammer, S Reinhardt, C Stumpf, R Trippel, K Vobig, R Wirth Institut für Kernphysik, TU Darmstadt P Navrátil TRIUMF Vancouver, Canada S Quaglioni LLNL Livermore, USA P Piecuch Michigan State University, USA C Forssén Chalmers University, Sweden H Feldmeier,T Neff GSI Helmholtzzentrum H Hergert Ohio State University, USA P Papakonstantinou IPN Orsay, F COMPUTING TIME JUROPA LOEWE-CSC