New Horizons in Ab Initio Nuclear Structure Theory. Robert Roth

Transcription

1 New Horizons in Ab Initio Nuclear Structure Theory Robert Roth

2 New Era of Low-Energy Nuclear Physics Experiment new facilities and experiments to produce nuclei far-off stability and study a range of observables Quantum Chromodynamics chiral effective field theory and lattice simulations access low-energy QCD and nuclear interactions Nuclear Many-Body Theory novel theoretical and computational methods allow for ab initio description of many more nuclei

3 Ab Initio Nuclear Structure Nuclear Structure Observables NEW Nuclear Lattice Sim chiral EFT on lattice Exact Ab-Initio Solutions few-body et al IMPROVED NEW Exact Ab-Initio Solutions IMPROVED few-body, no-core shell model, etc Approx Many- Body Methods controlled & improvable schemes Similarity Transformations physics-conserving transform of observables NEW IMPROVED Chiral Interactions consistent & improvable NN, 3N, interactions Energy-Density- Functional Theory IMPROVED guided by chiral EFT NEW Chiral Effective Field Theory systematic low-energy effective theory of QCD Low-Energy Quantum Chromodynamics

4 Ab Initio Nuclear Structure Nuclear Structure Observables Nuclear Lattice Sim chiral EFT on lattice Exact Ab-Initio Solutions few-body et al Exact Ab-Initio Solutions few-body, no-core shell model, etc PREDICTION Approx Many- Body Methods controlled & improvable schemes VALIDATION Similarity Transformations physics-conserving transform of observables Chiral Interactions consistent & improvable NN, 3N, interactions Energy-Density- Functional Theory guided by chiral EFT Chiral Effective Field Theory systematic low-energy effective theory of QCD Low-Energy Quantum Chromodynamics

5 Ab Initio Nuclear Structure Nuclear Structure Observables Nuclear Lattice Sim chiral EFT on lattice Exact Ab-Initio Solutions few-body et al Exact Ab-Initio Solutions few-body, no-core shell model, etc Approx Many- Body Methods controlled & improvable schemes Similarity Transformations physics-conserving transform of observables Chiral Interactions consistent & improvable NN, 3N, interactions Energy-Density- Functional Theory guided by chiral EFT Chiral Effective Field Theory systematic low-energy effective theory of QCD Low-Energy Quantum Chromodynamics

6 Nature of the Nuclear Interaction 16fm NN-interaction is not fundamental analogous to van der Waals interaction between neutral atoms induced via mutual polarization of quark & gluon distributions acts only if the nucleons overlap, ie at short ranges genuine 3N-interaction is important ρ 1/3 =18fm

7 Nuclear Interaction from Lattice QCD NN wave function φ(r) V C (r) [MeV] φ(x,y,z=; 1 S ) x[fm] 1 S 3 S S r [fm] 3 S y[fm] N Ishii et al, PRL 99, 221 (27) first steps towards construction of a nuclear interaction through lattice QCD simulations compute relative two-nucleon wavefunction on the lattice invert Schrödinger equation to obtain local effective twonucleon potential schematic results so far (unphysical quark masses, S-wave interactions only,) r [fm]

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 5 MeV cutoff 3N at N 2 LO Navrátil, A=3 fit 5 MeV cutoff 3N interaction at N 3 LO explicit inclusion of Δ-resonance formal issues: power counting, renormalization, cutoff choice, N 3 LO

9 Ab Initio Nuclear Structure Nuclear Structure Observables Nuclear Lattice Sim chiral EFT on lattice Exact Ab-Initio Solutions few-body et al Exact Ab-Initio Solutions few-body, no-core shell model, etc Approx Many- Body Methods controlled & improvable schemes Similarity Transformations physics-conserving transform of observables Chiral Interactions consistent & improvable NN, 3N, interactions Energy-Density- Functional Theory guided by chiral EFT Chiral Effective Field Theory systematic low-energy effective theory of QCD Low-Energy Quantum Chromodynamics

10 Why Similarity Transformations? momentum-space matrix elements 3 S 1 strong coupling of low- and highmomentum modes 3 S 1 3 D 1 5 makes solution 4of the many-body ϕl(r)[arb units] problem 3 very difficult 2 1 Argonne V18 J π =1 +,T= deuteron wave-function strong short-range correlations in many-body states L= L= r[fm]

11 Why Similarity Transformations? momentum-space matrix elements 3 S 1 3 S 1 3 D 1 ϕl(r)[arb units] chiral N 3 LO Entem & Machleidt, 5 MeV J π =1 +,T= deuteron wave-function L= L= r[fm]

12 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 matrix 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

13 SRG Evolution in Two-Body Space momentum-space matrix elements 3 S 1 3 S 1 3 D 1 ϕl(r)[arb units] α=fm chiral NN 4 Entem & Machleidt Λ= fmn 1 3 LO, 5 MeV J π =1 +,T= deuteron wave-function L= L= r[fm]

14 SRG Evolution in Two-Body Space momentum-space matrix elements 3 S 1 3 S 1 3 D 1 ϕl(r)[arb units] α=32fm 4 Λ=133fm 1 J π =1 +,T= deuteron wave-function suppression of off-diagonal coupling 4 ˆ= pre-diagonalization elimination of L= L=2 short-range correlations r[fm]

15 SRG Evolution in Three-Body Space 3B-Jacobi HO matrix elements chiral NN+3N α = fm4 E N3 LO + N2Λ LO, triton-fit, = fm 1 5 MeV Jπ = 18 = 1, ℏΩ 2 = 28 MeV NCSM ground state 3 H E [MeV] (E, ) 1+,T E (E, ) Nm x

16 SRG Evolution in Three-Body Space (E, ) E B-Jacobi HO matrix elements E [MeV] α=32fm 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 E (E, )

17 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 induced three-body terms α-variation provides a diagnostic tool to assess NN+3N-full: start with NN+3N initial Hamiltonian and keep twoand all three-body terms the contributions of omitted many-body interactions

18 Ab Initio Nuclear Structure Nuclear Structure Observables Nuclear Lattice Sim chiral EFT on lattice Exact Ab-Initio Solutions few-body et al Exact Ab-Initio Solutions few-body, no-core shell model, etc Approx Many- Body Methods controlled & improvable schemes Similarity Transformations physics-conserving transform of observables Chiral Interactions consistent & improvable NN, 3N, interactions Energy-Density- Functional Theory guided by chiral EFT Chiral Effective Field Theory systematic low-energy effective theory of QCD Low-Energy Quantum Chromodynamics

19 No-Core Shell Model Basics many-body basis: Slater determinants ν composed of harmonic oscillator single-particle states (m-scheme) Ψ = C ν ν ν model space: spanned by basis states ν with unperturbed excitation energies of up to ħω numerical solution of matrix eigenvalue problem for the intrinsic Hamiltonian H within truncated model space H Ψ =E Ψ ν H μ C μ =E C ν model spaces of up to 1 9 basis states are used routinely

20 Importance Truncated NCSM converged NCSM calculations essentially restricted to lower/mid p-shell full 1ħΩ calculation for 16 O getting very difficult (basis dimension >1 1 ) Importance Truncation reduce model space to the relevant basis states using an a priori importance measure derived from MBPT E [MeV] E [MeV] O NN-only α=4fm 4 ħω=2mev IT-NCSM + full NCSM

21 Importance Truncation: General Idea given an initial approximation (m) Ψ ref for the target states measure the importance of individual basis state ν via firstorder multiconfigurational perturbation theory ν H Ψ (m) κ (m) ref = ν Δε ν construct importance truncated space spanned by basis states with κ (m) ν κ min and solve eigenvalue problem for κ min the sequential scheme: construct importance full NCSMtruncated model space for next using previous eigenstates as reference space and thus the exact solution (m) Ψ is recoveredref a posteriori threshold extrapolation and perturbative correction used to recover contributions from discarded basis states

22 Ab Initio Nuclear Structure Nuclear Structure Observables Nuclear Lattice Sim chiral EFT on lattice Exact Ab-Initio Solutions few-body et al Exact Ab-Initio Solutions few-body, no-core shell model, etc Approx Many- Body Methods controlled & improvable schemes Similarity Transformations physics-conserving transform of observables Chiral Interactions consistent & improvable NN, 3N, interactions Energy-Density- Functional Theory guided by chiral EFT Chiral Effective Field Theory systematic low-energy effective theory of QCD Low-Energy Quantum Chromodynamics

23 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=2mev E [MeV] Exp α=4fm 4 α=5fm 4 α=625fm 4 α=8fm 4 α=16fm 4 Λ=224fm 1 Λ=211fm 1 Λ=2fm 1 Λ=188fm 1 Λ=158fm 1

24 6 Li: Ground-State Energies NN only NN+3N-induced NN+3N-full -22 ħω=2mev E [MeV] Exp α=4fm 4 α=5fm 4 α=625fm 4 α=8fm 4 α=16fm 4 Λ=224fm 1 Λ=211fm 1 Λ=2fm 1 Λ=188fm 1 Λ=158fm 1

25 12 C: Ground-State Energies NN only NN+3N-induced NN+3N-full -6 ħω=2mev -7 E [MeV] -8-9 WORLD RECORD Exp WORLD RECORD α=4fm 4 α=5fm 4 α=625fm 4 α=8fm 4 α=16fm 4 Λ=224fm 1 Λ=211fm 1 Λ=2fm 1 Λ=188fm 1 Λ=158fm 1

26 16 O: Ground-State Energies -8 NN only NN+3N-induced NN+3N-full ħω=2mev -1 E [MeV] Exp caused by long-range 2π terms (c ) clear signature of induced 4N originating from initial 3N α=4fm 4 α=5fm 4 α=625fm 4 α=8fm 4 α=16fm 4 Λ=224fm 1 Λ=211fm 1 Λ=2fm 1 Λ=188fm 1 Λ=158fm 1

27 Spectroscopy of 12 C 18 NN only NN+3N-induced NN+3N-full E [MeV] C ħω=16mev Exp Exp Exp α=4fm 4 α=8fm 4 Λ=224fm 1 Λ=188fm 1

28 Spectroscopy of 12 C 18 NN only NN+3N-induced NN+3N-full E [MeV] C ħω=16mev Exp Exp spectra largely insensitive to induced 4N Exp α=4fm 4 α=8fm 4 Λ=224fm 1 Λ=188fm 1

29 The Bottom Line beyond the lightest nuclei, SRG-induced 4N contributions affect the absolute energies (but not the excitation energies) with the inclusion of the leading 3N interaction we already obtain a good description of spectra (and ground states) breakthrough in computation, transformation and management of 3N matrix-elements next-generation SRG: can we find new SRG-generators that do not induce as much 4N but still give good convergence? next-generation chiral 3N: how will N3LO or Δ-full chiral 3N interactions affect the picture? applications: which experiment-related applications are in reach with the present framework?

30 Outlook: Sensitivity on Initial 3N 18 standard 3N modified 3N interaction with shifted c i 4 MeV cutoff E [MeV] C Exp 2 NN+3N-full α=8fm 4 ħω=16mev Exp 2 spectra of A 1 nuclei are a very sensitive benchmark for chiral 3N Exp interactions 2 c E refit to 4 He ground-state energy

31 Outlook: Carbon Isotopic Chain E [MeV] C C C Exp Exp Exp 1 E [MeV] ? NN+3N-full Λ 3N =4MeV α=8fm 4 ħω=16mev C 1 18 C 1 2 C Exp Exp Exp 4

32 Outlook: Carbon Isotopic Chain B(E2,2 + + ) [e 2 fm 4 ] C C Nm x C C B(E2,2 + + ) [e 2 fm 4 ] B(E2, gs ) 18 C B(E2,2+2 +gs) Nm x C NN+3N-full NEW EXPERIMENT NSCL, Petri et al NEW EXPERIMENT ANL, McCutchan et al NEW EXPERIMENT NSCL, Voss et al NEW EXPERIMENT NSCL, Petri et al Λ3N=4MeV α=8fm 4 ħω=16mev Nm x

33 Ab Initio Nuclear Structure Nuclear Structure Observables Nuclear Lattice Sim chiral EFT on lattice Exact Ab-Initio Solutions few-body et al Exact Ab-Initio Solutions few-body, no-core shell model, etc Approx Many- Body Methods controlled & improvable schemes Similarity Transformations physics-conserving transform of observables Chiral Interactions consistent & improvable NN, 3N, interactions Energy-Density- Functional Theory guided by chiral EFT Chiral Effective Field Theory systematic low-energy effective theory of QCD Low-Energy Quantum Chromodynamics

34 Heavy Nuclei with 3N Interactions ab initio calculations for heavier nuclei require alternative many-body tools and approximate treatment of 3N interactions coupled-cluster method for ground states of closed-shell nuclei exponential ansatz for many-body states using singles and doubles excitations (CCSD) normal-ordering approximation of the 3N interaction truncated at the two-body level summation over reference state converts part of 3N interaction to zero-, one- and two-body terms both approximations are controlled and systematically improvable

35 16 O: Coupled-Cluster with 3N NO2B -8 NN only NN+3N-induced NN+3N-full -1 E [MeV] CCSD(HO) Exp Λ 3N =4MeV -16 ħω=2mev E 3m =14 Λ 3N =5MeV e m x e m x e m x α=4fm 4 α=5fm 4 α=625fm 4 α=8fm 4 Λ=224fm 1 Λ=211fm 1 Λ=2fm 1 Λ=188fm 1

36 24 O: Coupled-Cluster with 3N NO2B NN only NN+3N-induced NN+3N-full E [MeV] CCSD(HO) Exp -22 ħω=2mev E 3m = e m x e m x e m x α=4fm 4 α=5fm 4 α=625fm 4 α=8fm 4 Λ=224fm 1 Λ=211fm 1 Λ=2fm 1 Λ=188fm 1

37 4 Ca: Coupled-Cluster with 3N NO2B -25 NN only NN+3N-induced NN+3N-full -3 E [MeV] CCSD(HO) ħω=2mev E 3m =14 Exp e m x e m x e m x α=4fm 4 α=5fm 4 α=625fm 4 α=8fm 4 Λ=224fm 1 Λ=211fm 1 Λ=2fm 1 Λ=188fm 1

38 48 Ca: Coupled-Cluster with 3N NO2B NN only NN+3N-induced NN+3N-full -3 E [MeV] CCSD(HO) ħω=2mev E 3m =14 Exp e m x e m x e m x α=4fm 4 α=5fm 4 α=625fm 4 α=8fm 4 Λ=224fm 1 Λ=211fm 1 Λ=2fm 1 Λ=188fm 1

39 Outlook: Chiral 3N for Heavy Nuclei E/A [MeV] PRELIMINARY CCSD(HF) HeVERY e m x =12 NN-only E 3m x =12 α=4fm 4 NN+3N-ind Λ 3N =4MeV NN+3N-full 4 16 O 24 O 48 Ca 56 Ni 78 Ni 1 Sn 132 Sn 4 Ca 48 Ni 68 Ni 9 Zr 114 Sn first ab initio calculations with chiral NN+3N Hamiltonians for heavy nuclei realistic mass systematics without phenomenological adjustments α-dependence might hold surprises

40 Conclusions

41 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

42 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 J Vary, P Maris Iowa State University, USA S Quaglioni LLNL Livermore, USA P Piecuch Michigan State University, USA H Hergert, K Hebeler Ohio State University, USA P Papakonstantinou IPN Orsay, F C Forssén Chalmers University, Sweden H Feldmeier,T Neff GSI Helmholtzzentrum JUROPA LOEWE-CSC HOPPER COMPUTING TIME