Ab initio no-core approach to the properties of light nuclei

Ab initio no core approach
to the properties of light nuclei
James P. Vary
Iowa State University
Mardi Gras Nuclear Physics Workshop
February 18-20, 2009
 Overview
 N-N and N-N-N potentials
 ab initio quantum many-body theory
 Applications - comparisons with experiment
 Extrapolating to exact results - Infinite matrix
 Conclusions/Outlook
QCD
Theory of strong interactions
χEFT
Chiral Effective Field Theory
Big Bang
Nucleosynthesis
& Stellar Reactions
r,s processes
& Supernovae
www.unedf.org
How do you build the first part of the bridge?
Ans: Start with QCD -> make an effective field theory
Chiral Effective Field Theory
•
•
Chiral symmetry of QCD (mu≈md≈0), spontaneously broken with pion as the
Goldstone boson
Systematic low-momentum expansion in (Q/Λχ)n ; Λχ≈ 1 GeV, Q≈100 MeV
– Power-counting
– Chiral perturbation theory (χPT)
•
Describe pion-pion, pion-nucleon and inter-nucleon interactions at low energies
– Nucleon-nucleon sector - S. Weinberg (1991)
•
•
Worked out by van Kolck, Kaiser, Meissner, Epelbaum, Machleidt…
Leading order (LO)
– One-pion exchange
•
•
•
3N interaction appears at next-to-next-to-leading order
(N2LO)
4N interaction appears at N3LO order
Consistency between NN, 3N and 4N terms
– NN parameters enter in the 3N terms etc.
•
•
Low-energy constants (LEC) need to be fitted to experiment
N3LO is the lowest order where a high-precision fit
to NN data can be made
– Entem and Machleidt (2002) N3LO NN potential
•
Only two new 3N and no new 4N low-energy constants
up to N3LO (CD & CE)
CE
CD
Is there another starting point that can
be used to calibrate our progress?
Ans: Inverse scattering applied to NN data,
then tuned to fit properties of light nuclei
JISP16 NN interaction:
J-matrix Inverse Scattering Potential
tuned with phase-shift-equivalent
unitary transformations
to the binding energy of 16O
 High quality fit to np scattering data (chisq/dof = 1.05)
 High quality fit to Deuteron gs properties
 Finite rank separable in each NN channel in oscillator basis
 Highly non-local, soft and rapidly convergent in nuclear apps
 High quality description of nuclei through the p-shell
 Subroutines and documentation: nuclear.physics.iastate.edu
A.M. Shirokov, J.P. Vary, A.I. Mazur and T.A. Weber,
“Realistic Nuclear Hamiltonian: Ab exitu approach,”
Phys. Letts. B 644, 33(2007), ArXiv nucl-th/0512105
ab initio No-Core Hamiltonian Methods
Problem Statement
Solve eigenvalue problem in large enough basis to converge
H = Trel + VNN + V3N + •••
H "i = E i "i
#
"i = $ Ani %n
n= 0
Diagonalize { %m H %n
}
Employ eigenvectors to calculate experimental observables
!
ˆ$
Transition Rate (i " k ) # $k %
i
2
Why?
Test fundamental strong interaction forces in nature
!Test fundamental symmetries - standard model and beyond
Predict cross sections for nuclear energy applications
What are the basic elements to solving the problem?
•
•
•
Adopt a realistic NN (and 3N) interaction & renormalize as needed - retain
induced many-body interactions
Adopt the 3-D Harmonic Oscillator (HO) for the single-nucleon basis states,
α, β,…
Evaluate the nuclear Hamiltonian, H, in basis space of HO (Slater)
determinants (manages the bookkeepping of anti-symmetrization)
"n = [a#+ ••• a$+ ]n 0
•
Diagonalize resulting sparse many-body H in this “m-scheme” where
n = 1,2,...,1010 or more!
•
Evaluate
! observables and compare with experiment
Comments:
• Straightforward but computationally demanding => new algorithms/computers
• Requires convergence assessments and extrapolation tools
!
• Achievable for nuclei up to A=16 (40) today with largest computers available
Foundations of ab initio No-Core Methods
depend on renormalization [RN] of H
No-Core Shell Model (NCSM): RN = Lee-Suzuki-Okamoto
No-Core Full Configuration method (NCFC): Initial H, RN=Vlowk, SRG, UCOM
Common features:
• Both are ab-initio
• Both are No-Core
• Both exactly preserve all symmetries of H
• Both use a many-body basis constructed from harmonic
oscillator single-particle states
=> basis space depends on (Nmax, h" )*
Major distinction:
NCSM uses Heff calculated for each (Nmax, h")
!
NCFC uses H independent of basis space
* N ma x is num ber of oscillator qu anta in the basis
abo ve the minim um for that nuc le
!us
Challenge
Exponential increase in Matrix Dimension (D)
(D)
Opportunities
 Memory/cpu time grows only as D3/2
 Algorithm development (UNEDF/SciDAC funding)
 Petaflop machines starting up (Japan, US-NSF, US-DOE, …)
 Improved understanding of H & RN (Chiral EFT, others)
 Developing methods for extrapolating D->inf (Nmax->inf)
P. Maris, J.P. Vary and A. Shirokov, PRC to appear, ArXiv:0808.3420
ab initio NCSM
Effective Hamiltonian for A-Particles
Lee-Suzuki-Okamoto Method plus Cluster Decomposition
P. Navratil, J.P. Vary and B.R. Barrett,
Phys. Rev. Lett. 84, 5728(2000); Phys. Rev. C62, 054311(2000)
C. Viazminsky and J.P. Vary, J. Math. Phys. 42, 2055 (2001);
K. Suzuki and S.Y. Lee, Progr. Theor. Phys. 64, 2091(1980);
K. Suzuki, ibid, 68, 246(1982);
K. Suzuki and R. Okamoto, ibid, 70, 439(1983)
Preserves the symmetries of the full Hamiltonian:
Rotational, translational, parity, etc., invariance
r r
( pi " p j )2
HA = Trel + V = #[
+ Vij ] + VNNN
2mA
i< j
A
Select a finite oscillator basis space (P-space) and evaluate an
a - body cluster effective Hamiltonian:
H eff = P [Trel + V a (N max ,h")] P
Guaranteed to provide exact answers as a " A or as P " 1 .
!
ab initio NCSM with χEFT Interactions
•
•
Presently the only method capable to apply the χEFT NN+NNN interactions to p-shell nuclei
Importance of NNN interactions for describing nuclear structure and transition rates
P. Navratil, V.G. Gueorguiev,
J. P. Vary, W. E. Ormand
and A. Nogga,
PRL 99, 042501(2007);
ArXiV: nucl-th 0701038.
Extensions and work in progress
•
•
•
Better determination of the NNN force itself, feedback to χEFT (LLNL, OSU, MSU, TRIUMF)
Implement Vlowk & SRG renormalizations (Bogner, Furnstahl, Maris, Perry, Schwenk &
Vary, NPA 801, 21(2008); ArXiv 0708.3754)
Response to external fields - bridges to DFT/DME/EDF (SciDAC/UNEDF)
- Axially symmetric quadratic external fields - in progress
- Triaxial and spin-dependent external fields - planning process
•
•
•
Cold trapped atoms (Stetcu, Barrett, van Kolck & Vary, PRA 76, 063613(2007); ArXiv
0706.4123) and applications to other fields of physics (e.g. quantum field theory)
Effective interactions with a core (Lisetsky, Barrett, Navratil, Stetcu, Vary)
Nuclear reactions & scattering (Forssen, Navratil, Quaglioni, Shirokov, Mazur, Vary)
ν-12C cross section
proportional to
Gamow-Teller transition
Exp
N3LO + 3NF
A.C.Hayes, P. Navratil, J.P. Vary,
PRL 91, 012502 (2003);
nucl-th/0305072
N3LO only
⇒ First successful description
of the GT data requires 3NF
Nmax
⇒Non-local NN interaction
from inverse scattering
also successful
JISP16
ab initio NCSM - comparison of Chiral EFT with JISP16
J.P. Vary, P. Maris, A. Negoita, P. Navratil, V.G. Gueorguiev, W. E. Ormand, A. Nogga, A. Shirokov and S. Stoica,
in Exotic Nuclei and Nuclear/Particle Astrophysics (II), Proceedings of the Carpathian Summer School of Physics 2007,
L. Trache and S. Stoica, Editors, AIP Conference Proceedings 972, 49(2008).
How collective modes are sensitive to the Hamiltonian:
Giant Quadrupole Resonance in 12C
E2 fragmenation with
JISP16 is intermediate
that of Chiral NN &
Chiral NN + 3NF
Greater fragmentation
of E2 strength found
including Chiral 3NF
99% of Isoscalar EWSR
Obtained below 55 MeV
96%
Isospin weighting:
(1-T)B(E2)
86%
Inelastic α - 12C scattering
Now turn to NCFC where one
attains convergence directly or through extrapolation
P. Maris, J.P. Vary and A. Shirokov, PRC to appear, ArXiv:0808.3420
P. Maris, J.P. Vary and A. Shirokov,
PRC to appear, ArXiv:0808.3420
A-uses 4 successive Nmax points
B-uses 3 successive Nmax points
P. Maris, J.P. Vary and A. Shirokov,
PRC to appear, ArXiv:0808.3420
Extend the exponential extrapolation to all states
6Li
RMS difference
Eabs 802keV
Eex
387keV
P. Maris, J.P. Vary and A. Shirokov,
PRC to appear, ArXiv:0808.3420
12C
- At the heart of matter
The first excited 0+ state of 12C, the “Hoyle
state”, is the key state of 12C formation in the
triple-alpha fusion process that occurs in stars.
Due to its role in astrophysics and the fact that
carbon is central to life, some refer to this as
one of the “holy grails” of nuclear theory.
Many important unsolved problems of the Hoyle state:
Microscopic origins of the triple-alpha structure are unsolved
Breathing mode puzzle - experiments disagree on sum rule fraction
Laboratory experiments to measure the formation rate are very
difficult - resulting uncertainties are too large for predicting
the 12C formation rate through this state that dictates the size
of the iron core in pre-supernova stars
Conclusion: Need ab initio solutions of the Hoyle state with no-core
method that accurately predicts the ground state binding energy
==> parameter free predictions for the Hoyle state
Ab-initio methods now sought for their predictions
Applications focused on recent & planned experiments
Argonne National Lab
Texas A&M University
JYFL-Jyvaskyla
KVI - Groningen
Initial results for 14F
ground state energy
Note similar convergence
trends with 12C
Experimental binding energy = 85.42 MeV
NCFC binding energy = 85.5 +/- 2 MeV
Experimental binding energy = ?
NCFC binding energy = 71.8 +/- 2 MeV
Computational resources: 3 hours on 30,628 processors (Jaguar)
NCFC with JISP16
1
2
1
2
Exotic
Nuclei
13
O (2-)
75.556
75.6 (2.2)
77.7?
69.1
14F
(2-)
?
70.2 (3.5)
71.8 (2.4)
61.4
14B
(2-)
85.42
83.7 (3.1)
85.5 (2.0)
76.0
ab initio theory of nuclei is successful so far
with aid of terascale computers
Quantum Chromodynamics
Effective Field Theory
Theory of Effective Operators
No Core Shell Model
Similarity Renormalization Group or Vlowk
No Core Full Configuration
(coming soon->needs 3NF)
Comparison with experiments
Electroweak properties
sensitively determine the low-energy constants
CD ~ -1 & CE ~ -0.4
Previous puzzles solved - 6Li quadrupole moment, neutrino-12C cross section,
some M1 and E2 transition rates, ground state spin of 10B, total binding energies, . . .
Many challenges remain - Hoyle states in 12C, cluster excited states, GT’s in A=14,
giant resonances, heavier nuclei, precision 3N & 4N interactions, . . .
⇒ Motivation for Improved Theory
and Petascale Machines
Jaguar 1.3 petaflops
Additional physics developments/collaborators
 Symplectic basis space - Draayer, Dytrych, Sviratcheva, Bahri
 Renormalization with a core - Lisetskiy, Barrett, Navratil, Stetcu, Kruse
 Alternative converging sequences - Forssen, Navratil
 A = 12 GT transitions - Forssen, Navratil, Experimentalists
 Extensions to reactions/scattering - Navratil, Quaglioni, Forssen, Shirokov, Mazur
 Chiral VNN+V3N with NCSM - Navratil, Ormand
 Chiral VNN+V3N with SRG/Vlowk+NCFC - Maris, Bogner, Schwenk, Furnstahl
 Trapped fermion systems - Stetcu, Barrett, van Kolck
 Comparison of CC with FCI - Maris, Hagen, Papenbrock, Dean
 UNEDF/SciDAC external fields - Maris, Furnstahl, Bogner
 UNEDF/SciDAC realistic basis - Negoita, Maris, Shirokov
 UNEDF/SciDAC ab initio scattering - Mazur, Shirokov
 Infinite matter applications - Bogner, Furnstahl, Shirokov, Coester, Negoita
Additional applied math & computer science
developments/collaborators
 Search & per-processor efficiencies - Sosonkina, Laghave, Sharda (Ames Lab,ISU)
 Mat-Vec & blocking strategies - Ng, Yang, Sternberg (LBNL)
 Cluster and web-site development - Aronnax (ISU)
 Compressed basis representation - Cockrell (ISU)
 I/O improvements - Hai Ah Nam (ORNL), Sosonkina, Laghave (Ames Lab, ISU)
Conclusions and Outlook
 Established need for 3N potentials with Chiral NN potentials
 Chiral EFT N3LO+3NF & JISP16 are suitable interactions
for light nuclei up through 16-O (40-Ca under investigation)
 No-Core FC with realistic NN interactions can be extrapolated
to obtain converged energies & uncertainties for low-lying states
 Many experimental observables need investigation
 Current efforts - nuclei with tunable external fields (DFT/DME)
 Future plans - accelerate convergence using a symplectic basis
 Future plans - explore exotic nuclei with these frameworks
 Future plans - reactions/scattering with these frameworks
 Leadership class computing facilities - “discovery engines”
Nuclear Physics Calculator:
ab initio NCSM demonstration project
nuclear.physics.iastate.edu
Select the NCSM application
Enter your email address
Select the number of neutrons and protons
Select the Nmax
Select the oscillator energy, h"
Results file with the JISP16 interaction
will be emailed to you
! in a few minutes
Note that this is a demonstration project
and will evolve as funding permits