Ab initio Coupled Cluster Calculations of Nuclei: Status and Outlook

Gaute Hagen

Annual UNEDF collaboration meeting

East Lansing, MI, June 19-24 2011 Research partly funded by the US Department of Energy

and

UNEDF: J. Holt (UT), T. Papenbrock (UT/ORNL)

CS support: H. A. Nam (ORNL)

External collaborators: D. J. Dean (ORNL), M. Hjorth-Jensen (Oslo), G. Baardsen (Oslo), G. Jansen (Oslo), O. Jensen (Bergen), T. Lesinski (Seattle)

1. Status and main accomplishments since last meeting

2. Status regarding high-performance computing

3. Roadmap for remainder of year 5 and outlook

Personnel

UNEDF collaborators: G. Hagen, J. Holt, T. Papenbrock, T. Lesinski (UW)

CS support: Hai Ah Nam (ORNL)

Development of coupled-cluster method with relevance for UNEDF (by students supervised by M. Hjorth-Jensen and G. Hagen)

•Øyvind Jensen (student at University of Bergen – Graduated July 2011; one-nucleon overlap functions and spectroscopic factors in J-coupled scheme)

•Gustav Jansen (student at University of Oslo; closed shell ±2 nucleons)

•Gustav Baardsen (student at University of Oslo; coupled-cluster theory for infinite nuclear matter)

Publication SummaryPublished / accepted since July 20101. Quenching of spectroscopic factors for proton removal in oxygen isotopes, Ø. Jensen, G. Hagen, M. Hjorth-

Jensen, B. Alex Brown, and A. Gade, Phys. Rev. Lett., in press (2011), arXiv:1104.15522. Ab initio coupled-cluster approach to nuclear structure with modern nucleon-nucleon interactions, G. Hagen,

T. Papenbrock, D. J. Dean, and M. Hjorth-Jensen, Phys. Rev. C 82, 034330 (2010).3. Fermionic-Bosonic Couplings in a Weakly-Deformed Odd-Mass Nucleus, 93Nb, J. N. Orce, J. D. Holt, A.

Linnemann, C. J. McKay, et al. Phys. Rev. C 82, 044317 (2010)4. Computation of spectroscopic factors with the coupled-cluster method, O. Jensen, G. Hagen, T. Papenbrock,

D. J. Dean, and J. S. Vaagen, Phys. Rev. C 82, 014310 (2010). 5. Closed-shell properties of 24O with ab initio coupled-cluster theory, Ø. Jensen, G. Hagen, M. Hjorth-Jensen, and

J. S. Vaagen, Phys. Rev. C(R), 83, 021305 (2011).6. Towards open-shell nuclei with coupled-cluster theory, G. R. Jansen, M. Hjorth-Jensen, G. Hagen, and T.

Papenbrock Phys. Rev. C 83, 054306 (2011)

Submitted articles

1. Ab initio computation of circular quantum dots, M. Pedersen Lohne, G. Hagen, M. Hjorth-Jensen, S. Kvaal, and F. Pederiva, Phys. Rev. B (2011) submitted, arXiv:1009.4833v1.

2. Three-Body Forces and Shell Structure in Calcium Isotopes, J. D. Holt, T. Otsuka, A. Schwenk, and T. Suzuki arXiv:1009.5984, submitted to Phys. Rev. Lett.

3. Chiral Three-Nucleon Forces and Bound Excited States in Neutron-Rich Oxygen Isotopes, J. D. Holt, A. Schwenk, T. Otsuka, and T. Suzuki,Submitted to Phys. Rev. C

4. Role of many-body processes and three-nucleon forces in nuclear pairing, J. D. Holt, J. Menendez, and A. Schwenk, Submitted to Phys. Rev. C

Talks1. “Coupled-cluster approach to medium-mass and neutron-rich nuclei”, Nuclear Structure 2010 (NS2010), Clark-Kerr Campus, U. C. Berkeley,

CA, August 8th – August 13th, 2010. (GH)2. “Coupled-cluster approach to medium-mass and neutron-rich nuclei”, Frontiers In NUclear STructure, Astrophysics and Reactions (FINUSTAR

3), Rhodes, Greece, August 23 to August 27, 2010. (GH)3. “Probing the driplines with ab‐initio Coupled‐Cluster theory”, ECT Workshop on Limits of existence of Light nuclei, ECT, Trento, Italy, October

25, 2010. (GH)4. “The nuclear many-body problem”, Second Uio-MSU-ORNL-UT School on Topics in Nuclear Physics: Many-Body Theory, Connections to

Experiment and Nuclear Astrophysics (five lectures), January 3-7, 2011, Michigan State University. (GH)5. “Towards ab initio structure and reactions in neutron rich nuclei with coupled‐cluster theory”, ECT workshop on Effective theories and the

nuclear many-body problem, ECT, Trento, Italy, March 7-11, 2011. (GH) 6. “Towards ab initio structure and reactions in neutron rich nuclei with coupled‐cluster theory”, invited seminar talk at CEA, CEA, Saclay, May 13,

2011. (GH)7. “Towards a unified description of nuclear structure and reactions with coupled-cluster theory”, FUSTIPEN Topical Meeting «Structure and

reactions at the drip lines», May 24-25, 2011, GANIL, Caen, France. (GH) 8. “Ab initio calculations of weakly bound nuclei”, The first International Conference on Advances in Radioactive Isotope Science (ARIS - 2011),

Leuven, Belgium, June 1, 2011. (GH)9. "Nuclear structure at the neutron drip-line", Gordon Research Conferences, Intersections Between Structure and Reactions: Pushing the

Frontiers of Nuclear Science, Colby-Sawyer College, New London, June 14th 2011. (GH)10. “Ab initio Coupled Cluster Calculations of Nuclei: Status and Outlook”, annual UNEDF meeting, Henry Center, East Lansing, June 20th, 2011. 11. “Model independent computation of atomic nuclei: status and perspectives,” “The INT at 20 years: The Future of Nuclear Physics and its

Intersections,” Seattle, WA, July 1-2, 2010. (TP)12. “Ab initio nuclear structure calculations,” EMMI workshop “Strongly coupled systems,” GSI, Darmstadt, Germany, November 15-17, 2010. (TP)13. “Frontiers in nuclear structure theory,” Physics Colloquium, Technische Universität Darmstadt, Darmstadt, Germany, December 3, 2010. (TP)14. “Toward model-independent nuclear structure computations,” Physics seminar, Universität Frankfurt, Frankfurt, Germany, February 24, 2011.

(TP)15. “Model-independent approaches to nuclear structure,” Annual NuSTAR meeting, GSI, Darmstadt, Germany, March 2-4, 2011. (TP)16. “Model-independent nuclear structure computations,” HISKP Colloquium, Universität Bonn, Bonn, Germany, May 12 2011 (TP)17. “Toward model-independent nuclear structure computations,” ECT* workshop on “Nuclear many-body open quantum systems,” Trento, Italy,

June 7 2011. (TP) 18. “Three-Nucleon Forces for Neutron-Rich, Medium-Mass Nuclei”, International Nuclear Physics Conference Vancouver, BC, Canada; July 4,

2010 (JH)19. “Three-Nucleon Forces for Medium Mass Nuclei Towards the Driplines“, Pan-American Advanced Studies Institute on Rare Isotopes

Joao Pessoa, Brazil; August 1, 2010 (JH)20. “Three-Nucleon Forces and the Evolution of Nuclear Structure Towards the Driplines”, ECT* Workshop: “The Limits of Existence of Light

Nuclei” Trento, Italy; October 27, 2010 (JH)21. “Three-Nucleon Forces and the Evolution of Nuclear Structure in Exotic Nuclei”, Nuclear Theory / EMMI Seminar; Technical University

Darmstadt, Darmstadt, Germany; November 29, 2010.(JH)22. “Exploring Neutron-Rich Exotic Nuclei with Three-Nucleon Forces”, Nuclear Physics Seminar Instituto de Fisica Teorica Universidad Autonoma

de Madrid, Madrid, Spain; February 2, 2011 (JH)23. “Three-Body Forces and Shell Structure in Exotic Nuclei”, ECT* Workshop: “Effective Theories and The Nuclear Many-Body Problem”

Trento, Italy; March 7, 2011 (JH)24. “Three-Nucleon Forces and the Evolution of Nuclear Structure in Exotic Nuclei”, The Fifth LACM-EFES-JUSTIPEN Workshop

Oak Ridge, TN; March 17, 2011. (JH)

Main accomplishments since last meeting

Science:

1. Long article on nuclear structure with NN interactions from chiral EFT

2. Quenching of spectroscopic factors for proton removal in neutron oxygen isotopes (Phys. Rev. Lett. 2011).

3. Computation of one-nucleon overlap functions for nuclear reactions and scattering (unpublished)

4. Computation of intrinsic densities with application to oxygen isotopes (unpublished)

5. Towards open shell nuclei with CCM extension to A±2: 18O and 26F

6. Coupled-cluster theory for nuclear matter. Implemented exact Pauli operator and HF + MBPT(2).

7. Constraining EDFs from ab-initio CC theory (See J. Holt’s talk)

Computing / algorithm developments:

1. Implementation of J-coupled code for computing one-nucleon overlaps and spectroscopic factors

2. Implementation of two-particle attached EOM-CCSD in J-coupled scheme.

Benchmarks in light nuclei:

Coupled-cluster meets few-body benchmarks for4He. Recent IT-NCSM and UMOA calculations of16O agree with CCM. R. Roth et al, arXiv:1105.3173 (2011)Fujii et al, PRL 103, 182501(2009)

Benchmarks in light nuclei:

Coupled-cluster meets few-body benchmarks for4He. Recent IT-NCSM and UMOA calculations of16O agree with CCM. R. Roth et al, arXiv:1105.3173 (2011)Fujii et al, PRL 103, 182501(2009)

Saturation of N3LO (NN only) in medium mass nuclei

G. Hagen, T. Papenbrock, D. J. Dean, M.Hjorth-Jensen, Phys. Rev. C 82, 034330 (2010).

Overlap functions and spectroscopic factors

Microscopic definition of Spectroscopic Factors (SF): Elastic scattering, capture and transfer of a nucleon on/to a target nucleus with mass A is given by the overlap function

The norm of the overlap functions defines the spectroscopic factor:

Asymptotic properties of the one-nucleon overlap function:Outside the range of the interaction the overlap function is proportional to a single-particle wave function.

Microscopic definition of Spectroscopic Factors (SF): Elastic scattering, capture and transfer of a nucleon on/to a target nucleus with mass A is given by the overlap function

The norm of the overlap functions defines the spectroscopic factor:

Asymptotic properties of the one-nucleon overlap function:Outside the range of the interaction the overlap function is proportional to a single-particle wave function.

Quenching of SFs for proton removal in the oxygens

•SF is a useful tool to study the role correlations toward the dripline.•We find a significant quenching of the SFs due to enhanced correlations coming from coupling to the scattering continuum.

Ø. Jensen, G. Hagen and M. Hjorth-Jensen, Phys. Rev. C(R) 83, 021305 (2011)Ø. Jensen, G. Hagen, Hjorth-Jensen, Brown, Gade, in press Phys. Rev. Lett. (2011) arXiv:1104.1552

For threshold effects in SFs see N. Michel et al Phys. Rev. C(R) 75, 031301 (2007)

•SF is a useful tool to study the role correlations toward the dripline.•We find a significant quenching of the SFs due to enhanced correlations coming from coupling to the scattering continuum.

Ø. Jensen, G. Hagen and M. Hjorth-Jensen, Phys. Rev. C(R) 83, 021305 (2011)Ø. Jensen, G. Hagen, Hjorth-Jensen, Brown, Gade, in press Phys. Rev. Lett. (2011) arXiv:1104.1552

For threshold effects in SFs see N. Michel et al Phys. Rev. C(R) 75, 031301 (2007)

Asymptotic normalization coefficients

Overlap function for a bound A + 1 nucleus:We use an SRG evolved interaction with cutoff 2.66 fm-1. The CCSD ground state energy for 16O in N = 11 major shells is 140.52 MeV.One neutron overlap functions for the bound J = 1/2+ state in 17O with theground state of 16O. For 17O we get E(1/2+) = -3.83 MeV.

Overlap function for a bound A + 1 nucleus:We use an SRG evolved interaction with cutoff 2.66 fm-1. The CCSD ground state energy for 16O in N = 11 major shells is 140.52 MeV.One neutron overlap functions for the bound J = 1/2+ state in 17O with theground state of 16O. For 17O we get E(1/2+) = -3.83 MeV.

Elastic scattering of neutrons on 16O with CCM

(Preliminary)

Overlap functions provides a simple and intuitive picture of reactions

Left figure: One neutron overlap functions for various J = 1=2+ scatteringstates in 17O using SRG evolved interaction with cutoff = 2.66fm-1Right figure: By matching the known asymptotic form of the overlap functionsto scattering solutions we can extract low-energy elastic scattering phase shifts.

Overlap functions provides a simple and intuitive picture of reactions

Left figure: One neutron overlap functions for various J = 1=2+ scatteringstates in 17O using SRG evolved interaction with cutoff = 2.66fm-1Right figure: By matching the known asymptotic form of the overlap functionsto scattering solutions we can extract low-energy elastic scattering phase shifts.

Intrinsic densities from microscopic CCM

Computing intrinsic densitiesThe one-body density matrix for the A and A-1 nuclei are computed in thecoupled-cluster formalism

and the local laboratory density is

CC wave function factorizes and the CoM wave function is a Gaussian with fixed width (G. Hagen et al Phys. Rev. Lett. 103, 062503 (2009))

We can get the internal density by a deconvolution of the local laboratorydensity (B. G. Giraud, Phys. Rev. C 77, 014311 (2008))

Computing intrinsic densitiesThe one-body density matrix for the A and A-1 nuclei are computed in thecoupled-cluster formalism

and the local laboratory density is

CC wave function factorizes and the CoM wave function is a Gaussian with fixed width (G. Hagen et al Phys. Rev. Lett. 103, 062503 (2009))

We can get the internal density by a deconvolution of the local laboratorydensity (B. G. Giraud, Phys. Rev. C 77, 014311 (2008))

Matter and charge radii in the 21,22,23,24O

isotopes

Binding energies of 21,22,23,24O with respect to the ground state of 21O.Point matter and charge radii for computed from intrinsic densitiesFor the nucleon-nucleon (NN) interaction we used the SRG evolved N3LOnucleon-nucleon interaction with momentum cutoffs = 3.6, 3.8, 4.0fm-1.

Binding energies of 21,22,23,24O with respect to the ground state of 21O.Point matter and charge radii for computed from intrinsic densitiesFor the nucleon-nucleon (NN) interaction we used the SRG evolved N3LOnucleon-nucleon interaction with momentum cutoffs = 3.6, 3.8, 4.0fm-1.

Matter and charge radii in the 21,22,23,24O

isotopes

Binding energies of 21,22,23,24O with respect to the ground state of 21O.Point matter and charge radii for computed from intrinsic densitiesFor the nucleon-nucleon (NN) interaction we used the SRG evolved N3LOnucleon-nucleon interaction with momentum cutoffs = 3.6, 3.8, 4.0fm-1.

Binding energies of 21,22,23,24O with respect to the ground state of 21O.Point matter and charge radii for computed from intrinsic densitiesFor the nucleon-nucleon (NN) interaction we used the SRG evolved N3LOnucleon-nucleon interaction with momentum cutoffs = 3.6, 3.8, 4.0fm-1.

Going beyond closed-shell nuclei. Low-lying states in 18O and 26F (Preliminary)

Two-particle attached coupled-cluster works very well for low-lying states in open-shell nuclei like 18O and 26F.Our results for 26F seem to suggest a more compressed spectrum as compared to USDA/USDB calculations. G. Jansen, M. Hjorth-Jensen, G. Hagen, T. Papenbrock, Phys. Rev. C 83, 054306 (2011).

Two-particle attached coupled-cluster works very well for low-lying states in open-shell nuclei like 18O and 26F.Our results for 26F seem to suggest a more compressed spectrum as compared to USDA/USDB calculations. G. Jansen, M. Hjorth-Jensen, G. Hagen, T. Papenbrock, Phys. Rev. C 83, 054306 (2011).

Low-lying states in neutron rich Potassium isotopes

Proton separation energies 40,48,52,54,60Ca.Low lying states in Potassiumisotopes calculated using PA/PR-EOMCCSD with “bare" chiral interactions. Model space consists of 15 major harmonic oscillator shellswith fixed oscillator frequencyhw = 30MeV.

Proton separation energies 40,48,52,54,60Ca.Low lying states in Potassiumisotopes calculated using PA/PR-EOMCCSD with “bare" chiral interactions. Model space consists of 15 major harmonic oscillator shellswith fixed oscillator frequencyhw = 30MeV.

Coupled-cluster theory for nuclear matter(Preliminary)

Implemented exact Pauli operator in relative-center of mass coordinates Implemented Hartree-Fock and MBPT(2) for nuclear matter in relative and Center-of-mass coordinates. Code has been validated and verified using Argonne-V18 and vlow-k.

Particle-particle and hole-hole ladders are in progress. Particle-hole channels will be implemented using exact and angle average Pauli operator

Implemented exact Pauli operator in relative-center of mass coordinates Implemented Hartree-Fock and MBPT(2) for nuclear matter in relative and Center-of-mass coordinates. Code has been validated and verified using Argonne-V18 and vlow-k.

Particle-particle and hole-hole ladders are in progress. Particle-hole channels will be implemented using exact and angle average Pauli operator

Progress for years 4-5Proposed work Status

Interface with DFT: response of energy and density to external potentials

See J. Holts talk

Spectroscopic factors in J-coupled scheme with continuum coupling

Completed oxygen isotopes

Computing intrinsic ground state densities Completed : 21-24O

Lorentz integral transform (w/ Sonia Bacca and Nir Barnea)

In progress

Role of three-nucleon forces in medium-mass nuclei (w/A. Schwenk and S. Bacca)

In progress

Coupled-cluster approach to nuclear matter(w/ Gustav Baardsen UiO)

In progress

Project is reasonably well on track

Computational status

Data organization

• Interaction spread across processors

• cluster amplitudes stored locally

• oxygen-16 in 20 shells ~ 1500 processor hours per model space

• oxygen-28 with continuum: total cost of about 100,000 CPU hours on Jaguar.

Challenge

• Number of j-coupled matrix elements of interactions fluctuate strongly for given sets of quantum numbers

• load balancing non-trivial: Calculation of estimated computational cost distribution of data

• scaling (load-balancing) up to few thousand processors

• Working on optimizing code with OpenMP.

Future plans (remainder of year 5 and outlook)

1. Interface with DFT, and toward ab initio reactions

1. Compute phaseshifts for elastic scattering of neutrons on selected closed shell nuclei using one-nucleon overlap functions.

2. Constraining EDFs from ab-initio coupled-cluster theory

3. Develop spherical EOM-CCSD code for general spin and compute dipole response function using Lorentz integral transform.

2. Role of three-nucleon forces in medium-mass nuclei

Outlook / SciDAC-3 ?

1. Ab initio computation of 48Ca with 3NF as possible highlight

2. Validation and verification of nuclear forces from Chiral EFT with uncertainty quantification.

3. Exploring Magic Numbers at the extreme using nuclear coupled-cluster

4. One- and two-nucleon overlap functions from coupled-cluster theory

5. Time dependent coupled-cluster approach to reactions

6. Nuclear matter (PhD student Gustav Baardsen from UiO)