Effective Nuclear Hamiltonians for ab initio Nuclear Structure Calculations

Transcription

1 Effective Nuclear Hamiltonians for ab initio Nuclear Structure Calculations Angelo Calci Hirschegg 2015 Nuclear Structure and Reactions: Weak, Strange and Exotic

2 Introduction Hamilton Matrix pre-diagonalization(srg) and basis transformations QCD-based interaction realistic NN+3N+4N interactions (χeft) many-body problem nuclear structure confront predictions with experiment

3 µ µ µ ¾ ¾ µ É ¼ ¼ É É É Ç Ç Ç Ä Ç Ä Ä Æ Ä ¾ Æ Æ Chiral NN+3N Interactions standard interaction: Weinberg, van Kolck, Machleidt, Entem, Meissner, Epelbaum, Krebs, Bernard,... NN 3N 4N N 3 LO: Entem & Machleidt, 500MeV cutoff LO N 2 LO: Navr, atil, local, 500MeV cutoff, fit to Triton standard interaction with modified 3N: N 3 LO: Entem & Machleidt, 500MeV cutoff NLO N 2 LO: Navr, atil, local, with modified LECs and cutoffs, fit to 4 He consistent N 2 LO interaction: N 2 LO NN: Epelbaum et al., 450,...,600MeV cutoff 3N: Epelbaum et al., 450,...,600MeV cutoff, nonlocal N 3 LO consistent N 3 LO interaction: 3N recently obtained by the LENPIC Collaboration

4 No-Core Shell Model (NCSM) solving the eigenvalue problem H int Ψ n =E n Ψ n many-body basis: Slater determinants ν composed of harmonic oscillator single-particle states (m-scheme) Ψ n = ν C n ν ν model space: spanned by m-scheme states ν with unperturbed excitation energy of up to N max ħω. protons neutrons e=1 e=0 e=3 e=2 problem enormous increase of model space with particle number A converged calculation limited to small A

5 Importance-Truncated NCSM start with reference state Ψ reƒ as approximation of target state Ψ n from limited reference space M reƒ a priori determination of relevant basis states ϕ ν / M reƒ via firstorder perturbation theory νhintψref κ ν = ε ν ε ref importance truncated space M(κ min ) spanned by basis states with κ ν κ min solving eigenvalue problem in M(κ min ) provides improved approximation for target state extrapolation apply to iterartive of κ min sheme 0 recovers effect of omitted contributions provides same results as the full NCSM keeping all its advantages expands application range to higher A and N max

6 Similarity Renormalization Group in Three-Body Space Roth, Langhammer, AC et al. Phys. Rev. Lett. 107, (2011) Roth, Neff, Feldmeier Prog. Part. Nucl. Phys. 65, 50 (2010) Jurgenson, Navrátil, Furnstahl Phys. Rev. Lett. 103, (2009) Bogner, Furnstahl, Perry Phys. Rev. C (R) (2007)

7 Similarity Renormalization Group (SRG) accelerate convergence by pre-diagonalizing the Hamiltonian with respect to the many-body basis unitary transformation leads to evolution equation d dα e e Hα = ηα, Hα with advantages of SRG: ηα = (2μ) 2 e Tint, Hα = η α flexibility and simplicity 3B-Jacobi HO matrix elements 0 α = 0.00 fm α = 0.08 fm 4 α = 0.32 fm e α Tint E JT E JT H 4 Jπ = ,T 2 = 1 2 ℏΩ = 24 MeV E E E (E, ) (E, ) (E, ) [MeV] (E, ) E 4

8 SRG Evolution in A-Body Space assume initial Hamiltonian and intrinsic kinetic energy are twobody operators written in second quantization H 0 =..., T int =T T cm =... perform single evolution step Δα in Fock-space representation H Δα = H 0 +Δα T int, H 0, H 0 =... +Δα unitary transformation induces many-body contributions in the Hamiltonian

9 Hamiltonian in A-Body Space cluster decomposition: decompose evolved Hamiltonian into irreducible n-body contributions H [n] α H α = H [1] α + H [2] α + H [3] α + + H [n] α + A-body unitarity: transformation is unitary only if all terms up to n=a are kept, then all eigenvalues are independent of α cluster truncation: can construct contributions up to n = 3 from evolution in 2B and 3B space, but have to discard n>3 α-dependence of eigenvalues of cluster-truncated α-variation provideshamiltonian a measures impact of discarded many-body diagnosticcontributions tool to assess the contributions of omitted many-body interactions

10 4 He: Ground-State Energies NN only NN+3N ind 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 IT-NCSM E [MeV] ħω=20mev N mx N mx N mx α=0.04fm 4 α=0.05fm 4 α=0.0625fm 4 α=0.08fm 4 α=0.16fm 4 λ=2.24fm 1 λ=2.11fm 1 λ=2.00fm 1 λ=1.88fm 1 λ=1.58fm 1

11 16 O: Ground-State Energies -80 NN only NN+3N ind NN+3N full -100 IT-NCSM E [MeV] ħω=20mev N mx initial 3N interactions induce sizable 4N contributions N mx N mx α=0.04fm 4 α=0.05fm 4 α=0.0625fm 4 α=0.08fm 4 α=0.16fm 4 λ=2.24fm 1 λ=2.11fm 1 λ=2.00fm 1 λ=1.88fm 1 λ=1.58fm 1

12 16 O: Origin of Induced 4N switch off individual contributions of the 3N interaction E [MeV] N mx NN+3N full ħω=20mev two-pion exchange term of the chiral 3N (in particular the c 3 -term) N mx N mx is responsible for induced 4N α=0.04fm 4 α=0.08fm 4 α=0.16fm 4 Λ=2.24fm 1 Λ=1.88fm 1 Λ=1.58fm 1 N mx

13 16 O: Lowering the Initial 3N Cutoff standard 500 MeV 450 MeV 400 MeV 350 MeV c D = 0.2 c E = c D = 0.2 c E = c D = 0.2 c E =0.098 c D = 0.2 c E = E [MeV] N mx α dependence: by induced 4N from initial 3N N mx N mx lowering the initial 3N cutoff N suppresses induced mx 4N terms α=0.04fm 4 α=0.05fm 4 α=0.0625fm 4 α=0.08fm 4 NN+3N full Λ=2.24fm 1 Λ=2.11fm 1 Λ=2.00fm 1 Λ=1.88fm 1 ħω=20mev

14 Towards Heavy Nuclei with NN+3N Interactions Binder, Langhammer, AC, Roth Phys. Lett. B 736, (2014) Binder, Piecuch, AC, Langhammer, Navrátil, Roth Phys. Rev. C 88, (2013) Hagen, Papenbrock, Dean, Hjorth-Jensen Phys. Rev. C 82, (2010) Taube, Bartlett J. Chem. Phys. 128, (2008)

15 Coupled-Cluster Approach exponential Ansatz to solve Eigenvalue problem Ψ =e T ref =e T 1 +T 2 +T 3 + +T A ref T n : npnh excitation (cluster) operator 1 2 μ T n = 1...μ n n! ν 1...ν n μ 1 μ n νn ν1 t ν 1...νn μ 1...μ n similarity-transformed Eigenvalue problem H N ref =ΔE ref with H N =e T H N e T CC equations: coupled system of non-linear equations ref H N ref =ΔE μ 1..μ n ν 1..ν n HN ref =0, ν1...ν n, μ 1...μ n

16 Cluster Truncation exponential Ansatz to solve Eigenvalue problem Ψ =e T ref =e T 1 +T 2 +T 3 + +T A ref CCSD: singles and doubles excitations, T=T 1 +T 2 exponential Ansatz: excitation beyond 2p2h included by products of excitation operators CCSDT: include triples excitations, T=T 1 +T 2 +T 3 explicit inclusion to expensive use approximations to estimate triples effects Λ-CCSD(T) and CR-CC(2,3) Ab Initio Coupled Cluster: applicable far beyond the sd shell typically restricted to ground states of closed-shell nuclei

17 Oxygen Istopes H. Hergert, S. Binder, AC, J. Langhammer, R. Roth Phys. Rev. Lett. 110, (2013) E[ MeV]. E[ MeV] (a) NN+3N-induced ind (b) NN+3N-full MR-IM-SRG(2) IT-NCSM CCSD Λ-CCSD(T) A investigate effect of 3N interactions quantify uncertainties of medium mass approaches access nuclei up to driplines with ab initio approaches Λ 3N =400MeV optimal ħω e mx =14 E 3mx =14

18 SRG: Basis Representation accelerate convergence by pre-diagonalizing the Hamiltonian unitary transformation driven by with respect to the many-body basis d d H α = η α, H α with ηα =(2μ) 2 T int, (2μ) 2 T int H α H α 2H α T int H α + H E JT H α EJT = α H α T int dα dα E (SRG) (2μ) 2 max E JT TintE JT E JT H α E JT E JT H α EJT E,E, 2 E JT H α E JT E JT Tint E JT E JT H α EJT + E JT H α E JT E JT H α E JT E JT T int EJT SRG model space truncated E E (SRG) max

19 SRG Model Space for Heavy Nuclei E/A [MeV] ground states CCSD ħ Ω=36MeV ħω=24mev α=0.08fm 4 Λ 3N =400MeV HF basis E 3max =14 e max =12 16 O 40 Ca 48 Ca 56 Ni 68 Ni 78 Ni 90 Zr 100 Sn 114 Sn 120 Sn 132 Sn SRG model-space ramp E (SRG) max B A 13 2 J introduce extended SRG space as standard for heavy mass nuclei SRG space B much larger then model spaces in previous works large angular momenta important for heavy mass nuclei

20 Heavy Nuclei Binder, Langhammer, AC, Roth Phys. Lett. B 736 (2014) E/A [MeV] NN+3N ind NN+3N full Exp. 16 O 36 Ca 48 Ca 54 Ca 56 Ni 62 Ni 68 Ni 88 Sr 106 Sn 100 Sn 116 Sn 120 Sn 24 O 40 Ca 52 Ca 48 Ni 60 Ni 66 Ni 78 Ni 90 Zr 108 Sn 114 Sn 118 Sn 132 Sn many-body method and truncation well under control initial flow-parameter NN interaction dependence induces caused sizable by 4N with increasing induced 4N mass contributions number cancellation between 4N contributions induced by initial NN (attractive) and 3N (repulsive) strongly reduced flow-parameter dependence CR-CC(2,3) ħ Ω=36MeV ħω=24mev α= fm 4 E 3max =18 e max =12 mass trend reproduced throughout nuclear chart

21 SRG in Four-Body Space

22 Induced Four-Body Contributions induced 4N constitute major limitation for applications of chiral interactions ➊ suppress induced 4N contributions by reducing the cutoff Λ 3N circumvention: restriction to 3N interactions with lower cutoffs might not work for all interactions or system (heavy masses) ➋ find alternative SRG generator to exclude induced 4N from the outset promising ideas for a better compromise between induced forces and convergence acceleration Dicaire, Omand, Navratil Phys. Rev. C 90, (2014) ➌ include 4N contributions SRG evolution in four-body space extension of all HO developments and IT-NCSM to treat 4N interactions

23 IT-NCSM with Four-Body Contributions without 4N O IT-NCSM ħω=24 MeV.E [MeV] Exp N mx α=0.04fm 4 α=0.08fm 4 α=0.16fm 4

24 IT-NCSM with Four-Body Contributions without 4N with 4N.E [MeV] O IT-NCSM ħω=24 MeV Exp N mx all partial waves with J= N mx α=0.04fm 4 α=0.08fm 4 α=0.16fm 4 E 4max =6, E (SRG) 4max =21 ħ Ω=40MeV

25 IT-NCSM with Four-Body Contributions without 4N 16 O IT-NCSM ħω=24 MeV with 4N flow-parameter dependence strongly suppresed dominant 4N included.e [MeV] Exp. techniques directly all partial waves with J applicable to initial N N mx force N mx α=0.04fm 4 α=0.08fm 4 α=0.16fm 4 E 4max =6, E (SRG) 4max =21 ħ Ω=40MeV

26 Alternative Chiral Hamiltonians & Uncertainty Quantification

27 Uncertainties of Chiral Interactions ideal start with NN+3N force at consistent chiral orders use sequence of cutoffs and different chiral orders estimate uncertainties practice uncertainties from many-body approach included observables calculated for single chiral Hamiltonian (inconsistent chiral order) quality of chiral forces assessed by agreement with experiment nuclear structure physics approaches new era ongoing progress in construction of consistent NN+3N Hamiltonians N 2 LO [Epelbaum et al., 450,...,600MeV cutoff] N 3 LO [Epelbaum et al., 450MeV cutoff] first applications in nuclear spectroscopy first step towards reliable uncertainty quantification

28 6 Li: Alternative Interactions standard N 3 LO+N 2 LO 500 MeV Epelbaum N 2 LO+N 2 LO 450/500 MeV 600/500 MeV 550/600 MeV 450/700 MeV 600/700 MeV E [MeV] ħω=16 MeV N mx =10 α=0.08fm 4. 0 NN+3N NN+3N NN+3N NN+3N NN+3N NN+3N 1 Exp.

29 6 Li: Dependence on Chiral Order standard N 3 LO+N 2 LO 500 MeV Epelbaum N 2 LO+N 2 LO E [MeV] small cutoff dependence reasonable predictions. 1 0 ħω=16 MeV N mx =10 α=0.08fm 4 NN+3N NN+3N Exp. 1

30 6 Li: Dependence on Chiral Order standard N 3 LO+N 2 LO 500 MeV Epelbaum N 2 LO+N 2 LO Epelbaum N 3 LO+N 3 LO 450/500 MeV 5 2 small cutoff dependence E [MeV] ħω=16 MeV N mx =10 α=0.08fm reasonable predictions further cutoffs and orders required. 0 NN+3N NN+3N NN+3N Exp. 1

31 12 C: Alternative Interactions standard N 3 LO+N 2 LO 500 MeV Epelbaum N 2 LO+N 2 LO 450/500 MeV 600/500 MeV 550/600 MeV E [MeV] ħω=16 MeV N mx =8 α=0.08fm NN +3N NN +3N NN +3N NN +3N Exp. 0 0

32 12 C: Alternative Interactions standard N 3 LO+N 2 LO 500 MeV Epelbaum N 2 LO+N 2 LO 450/500 MeV 600/500 MeV 550/600 MeV E [MeV] ħω=16 MeV N mx =8 α=0.08fm 4 3N lowers first 1 + -excitation energy NN +3N NN +3N NN +3N NN +3N Exp. 0 0

33 12 C: Alternative Interactions standard N 3 LO+N 2 LO 500 MeV Epelbaum N 2 LO+N 2 LO 450/500 MeV 600/500 MeV 550/600 MeV E [MeV] ħω=16 MeV N mx =8 α=0.08fm NN +3N NN +3N NN +3N NN +3N Exp. 0 0

34 12 C: Alternative Interactions standard N 3 LO+N 2 LO 500 MeV Epelbaum N 2 LO+N 2 LO 450/500 MeV 600/500 MeV 550/600 MeV E [MeV] ħω=16 MeV N mx =8 α=0.08fm 4 small cutoff dependence for NN+3N NN +3N NN +3N NN +3N NN +3N Exp. 0 0

35 12 C: Dependence on Chiral Order Epelbaum N 2 LO+N 2 LO 450/500 MeV Epelbaum N 3 LO+N 3 LO 450/500 MeV E [MeV] ħω=16 MeV α=0.08fm first results with NN+3N force at N 3 LO reasonable predictions cutoff dependence need to be studied NN +3N NN +3N Exp.

36 Conclusions consistent three- and four-body SRG evolution successful application of chiral 3N forces promising ideas for alternative generators heavy nuclei accessable with ab initio approaches mass systematics can be reproduced p-shell spectra provide powerful testbed for chiral potentials first nuclear structure application of 3N force at N 3 LO constraints for interactions

37 Outlook exciting progress in construction of chiral forces N 2 LO sat NN+3N: fit LECs to many-body observables Ekström, Machleidt et al. self-contained local NN up to N 2 framework to employ LO for quantum Monte Carlo present and future chiral NN+3N+4N Gezerlis, Tews, Epelbaum et al. Phys. Rev. C 90, (2014) interactions in a variety of many-body minimally non-local NNmethods up to N 3 LO include Δ in TPE component Piarulli, Girlanda, Marcucci et al. arxiv: LENPIC Collaboration improved NN up to N 4 LO Epelbaum et al. arxiv: ; arxiv: N up to N 3 LO allow to vary cutoff and chiral order to quantify uncertainty

38 Outlook: Improved NN LO NLO N 2 LO N 3 LO N 4 LO 4 He NCSM bare NN R=1.2fm N max = N max =16PRELIMINARY sequence of cutoffs and chiral orders can be studied 3N forces in progress

39 Epilogue thanks to my collaborators S. Binder, J. Langhammer, K. Hebeler, R. Roth, S. Schulz Institut für Kernphysik, TU Darmstadt P. Navrátil, J. Holt TRIUMF Vancouver, Canada J. Vary, P. Maris Iowa State University, USA G. Hupin, S. Quaglioni LLNL Livermore, USA H. Hergert Michigan State University, USA H. Feldmeier GSI Helmholtzzentrum LENPIC Low-Energy Nuclear Physics International Collaboration JUROPA LOEWE-CSC HOPPER COMPUTING TIME