no core ci calculations for light nuclear systems · int extreme computing workshop, june 2011 no...
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INT Extreme Computing workshop, June 2011
No Core CI Calculationsfor light nuclear systems
Pieter [email protected]
Iowa State University
SciDAC project – UNEDFspokespersons: Rusty Lusk (ANL), Witek Nazarewicz (ORNL/UT)http://www.unedf.org
PetaApps awardPIs: Jerry Draayer (LSU), Umit Catalyurek (OSU)Masha Sosonkina, James Vary (ISU)
INCITE award – Computational Nuclear StructurePI: James Vary (ISU)
NERSC CPU time
INT Extreme Computing, June 7 2011 – p.1/43
SciDAC/UNEDF – Uniform description of nuclear structure
Universal Nuclear EnergyDensity Functional thatspans the entire mass tablebased on ab initio calculations
Greens Function MonteCarlo (Pieper et al, ANL)
No-Core ConfigurationInteraction calculations
Coupled Cluster(Papenbrock et al, ORNL)
http://www.unedf.org
spokespersons:R. Lusk (ANL)W. Nazarewicz (ORNL/UT)
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Functional Theory
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INT Extreme Computing, June 7 2011 – p.2/43
INT Extreme Computing, June 7 2011 – p.3/43
Configuration Interaction Methods
Expand wave function in basis states |Ψ〉 =∑
ai|ψi〉
Express Hamiltonian in basis 〈ψj |H|ψi〉 = Hij
Diagonalize Hamiltonian matrix Hij
Complete basis −→ exact result
caveat: complete basis is infinite dimensional
In practicetruncate basisstudy behavior of observables as function of truncation
Computational challenge
construct large (1010 × 1010) sparse symmetric real matrix Hij
use Lanczos algorithmto obtain lowest eigenvalues & eigenvectors
INT Extreme Computing, June 7 2011 – p.4/43
Many-Body Basis Space
Expand wave function in basis states |Ψ〉 =∑
ai|ψi〉
Many-Body basis states |ψi〉
Slater Determinants of single-particle states |φ〉
|ψ〉 = |φ1〉 ⊗ . . .⊗ |φA〉
single-particle basis stateseigenstates of SU(2) operatorsL
2, S2, J
2 = (L + S)2, and Jz
w. quantum numbers |φ〉 = |n, l, s, j,m〉
radial wavefunctions:Harmonic Oscillator
M -scheme: many-body basis states eigenstates of Jz
Jz|ψ〉 = M |ψ〉 =A
∑
i=1
mi|ψ〉
Alternatives: LS-scheme, Total-J-scheme, Symplectic basis, . . .
INT Extreme Computing, June 7 2011 – p.5/43
Truncation Schemes
Nmax truncationtruncation on the total numberof H.O. oscillator quantaabove minimal configurationfor that nucleus
NMIN=0
NMAX=6
configuration
“6h!” configuration for 6Li
allows for exact seperationof Center-of-Mass motionand intrinsic motion
Alternative truncation schemesFCI – Full ConfigurationInteraction – truncation onsingle-particle basis only
100 1 k 10 k 100 k 1 M 10 M 100 M
basis space dimension
-32
-30
-28
-26
-24
-22
-20
grou
nd s
tate
ene
rgy
4He, FCI truncation4He, N
max truncation
6Li, FCI truncation6Li, N
max truncation
4He, NCFC result6Li, NCFC result
Importance Sampling, Monte Carlo Sampling, Symplectic, . . .
INT Extreme Computing, June 7 2011 – p.6/43
Intermezzo: Center-of-Mass excitations
Use single-particle coordinates, not relative (Jacobi) coordinatesstraightforward to extend to many particleshave to seperate Center-of-Mass motion from intrinsic motion
Add Lagrange multiplier to Hamiltonian
Hrel −→ Hrel + ΛCM
(
HH.O.CM −
3
2~ω
)
with Hrel = Trel + Vrel the relative Hamiltonianseperates CM excitations from CM ground state |ΦCM 〉
Center-of-Mass wave function factorizes forH.O. basis functions in combination with Nmax truncation
|Ψtotal〉 = |φ1〉 ⊗ . . .⊗ |φA〉
= |ΦCenter-of-Mass〉 ⊗ |Ψintrinsic〉
whereHrel|Ψj, intrinsic〉 = Ej|Ψj, intrinsic〉
INT Extreme Computing, June 7 2011 – p.7/43
Configuration Interaction Methods
Expand wave function in basis states |Ψ〉 =∑
ai|ψi〉
Express Hamiltonian in basis 〈ψj |H|ψi〉 = Hij
H = Trel + ΛCM
(
HH.O.CM −
3
2~ω
)
+∑
i<j
Vij +∑
i<j<k
Vijk + . . .
Pick your favorite potentialArgonne potentials: AV8, AV18(plus Illinois NNN interactions)Bonn potentialsChiral NN interactions (plus chiral NNN interactions). . .JISP16 (phenomenological NN potential). . .
INT Extreme Computing, June 7 2011 – p.8/43
Configuration Interaction Methods
Expand wave function in basis states |Ψ〉 =∑
ai|ψi〉
Express Hamiltonian in basis 〈ψj |H|ψi〉 = Hij
large sparse symmetric matrix
Obtain lowest eigenvaluesusing Lanczos algorithm
Eigenvalues:bound state spectrumEigenvectors:nuclear wavefunctions
Use wavefunctions to calculate observables
Challenge: eliminate dependence on basis space truncation
INT Extreme Computing, June 7 2011 – p.8/43
CI calculation – convergence
Expand wave function in basis states |Ψ〉 =∑
ai|ψi〉
Express Hamiltonian in basis 〈ψj |H|ψi〉 = Hij
Diagonalize sparse real symmetric matrix Hij
Variational: for any finite truncation of the basis space,eigenvalue is an upper bound for the ground state energy
Smooth approach to asymptotic valuewith increasing basis space=⇒ extrapolation
to infinite basis
Convergence: independence ofbasis space parameters
different methods(NCFC, CC, GFMC, DME, . . . )
using the same interactionshould give same resultswithin numerical errors
-16
-12
-8
-4
0
bind
ing
ener
gy p
er n
ucle
on [
MeV
]
No-Core Full ConfigurationCoupled Cluster Singles Doubles
SRG evolved N3LO 2-body interactions
8He
16O
40Ca
λ = 1.5
λ = 2.5
λ = 2.5
λ = 1.5
λ = 1.5
λ = 2.5
PRELIMINARY
INT Extreme Computing, June 7 2011 – p.9/43
Intermezzo: Extrapolation Techniques
Challenge: achieve numerical convergence for no-core Full Configuationcalculations using finite model space calculations
Perform a series of calculations with increasing Nmax truncation(while keeping everything else fixed)
Extrapolate to infinite model space −→ exact resultsbinding energy: exponential in Nmax
ENbinding = E∞binding + a1 exp(−a2Nmax)
use 3 or 4 consecutive Nmax values to determine E∞binding
use ~ω and Nmax dependenceto estimate numerical error bars
Maris, Shirokov, Vary, Phys. Rev. C79, 014308 (2009)
need at least Nmax = 8 for meaningfull extrapolations
INT Extreme Computing, June 7 2011 – p.10/43
Intermezzo: Extrapolation Techniques
Challenge: achieve numerical convergence for no-core Full Configuationcalculations using finite model space calculations
Perform a series of calculations with increasing Nmax truncation(while keeping everything else fixed)
Extrapolate to infinite model space −→ exact results
0 5 10 15 20 25 30 35 40 45hw
-30
-25
-20
grou
nd s
tate
ene
rgy
Nmax
= 4N
max= 6
Nmax
= 8N
max= 10
Nmax
= 12N
max= 14
extrapolatedN
max= 16
extrapolatedα D thresholdexpt.
6Li JISP16
0 4 8 12 16N
max
-32
-31
-30
-29
-28variational upperboundextrapolation at fixed hwα D thresholdexpt.
6Li JISP16
INT Extreme Computing, June 7 2011 – p.10/43
CI calculations – main challenges
0 2 4 6 8 10 12 14N
max
100
101
102
103
104
105
106
107
108
109
1010
M-s
chem
e ba
sis
spac
e di
men
sion
4He6Li8Be10B12C16O19F23Na27Al
0 2 4 6 8 10N
max
100
103
106
109
1012
1015
num
ber
of n
onze
ro m
atri
x el
emen
ts
16O, dimension2-body interactions3-body interactions4-body interactionsA-body interactions
Single most important computational issue:exponential increase of dimensionality with increasing H.O. levels
Additional computational issue:sparseness of matrix / number of nonzero matrix elements
INT Extreme Computing, June 7 2011 – p.11/43
High-performance computing
Hardwareindividual desk- and lap-topslocal linux clustersNERSC (DOE)
10,000,000 CPU hours for ISU collaborationLeadership Computing Facilities (DOE)INCITE award – Computational Nuclear Structure (PI: J. Vary, ISU)
28,000,000 CPU hours on Cray XT5 at ORNL15,000,000 CPU hours on IBM BlueGene/P at ANL
grand challenge award at Livermore (Jurgenson, Navratil, Ormand)
applied for CPU time at NCSA (NSF) – Blue Waters (IBM)
SoftwareLanczos algorithm – iterative methodto find lowest eigenvalues and eigenvectors of sparse matriximplemented in Many Fermion Dynamics
parallel F90/MPI/OpenMP CI code for nuclear physics
INT Extreme Computing, June 7 2011 – p.12/43
MFDn – 2-dimensional distribution of matrix
Real symmetric matrix: store only lower (or upper) triangle
Store Lanczos vectors distributed over all processors
In principle, we can deal with arbitrary large vectorseven if we cannot store an entire vector on a single processor
largest dimension: 8 billion, 32 GB / vector in single precision
INT Extreme Computing, June 7 2011 – p.13/43
MFDn – load-balancing
Lexico-graphical enumeration of basis states on d procs
Round-robin distribution of basis states over d procs
Almost perfect load balancing
However, no (apparent) structure in sparse matrixmulti-level blocking scheme to locate nonzero’s (Sternberg 2008)
Under development: distribute groupds of basis states over d procsin order to retain part of the natural structure of the matrix
INT Extreme Computing, June 7 2011 – p.14/43
Strong force between nucleons
Strong interaction in principle calculable from QCD
Use chiral perturbation theory to obtain effective A-bodyinteraction from QCD Entem and Machleidt, Phys. Rev. C68, 041001 (2003)
controlled power series expansionin Q/Λχ with Λχ ∼ 1 GeVnatural hierarchyfor many-body forces
VNN ≫ VNNN ≫ VNNNN
in principleno free parameters
in practice a fewundetermined parameters
renormalization necessaryLee–Suzuki–OkamotoSimilarity Renormalization Group +... +... +...
+...
2N Force 3N Force 4N Force
Q0LO
Q2NLO
Q3NNLO
Q4N LO3
INT Extreme Computing, June 7 2011 – p.15/43
Similarity Renormalization Group – NN interaction
SRG evolution Bogner, Furnstahl, Perry, PRC 75 (2007) 061001
drives interaction towards band-diagonal structureSRG shifts strength between 2-body and many-body forces
Initial chiral EFT Hamiltonianpower-counting hierarchy A-body forces
VNN ≫ VNNN ≫ VNNNN
key issue: preserve hierarchy of many-body forces
INT Extreme Computing, June 7 2011 – p.16/43
Improve convergence rate by applying SRG to N3LO
−30
−25
−20
−15
Gro
und-
Stat
e E
nerg
y [M
eV]
20 25 30 35 40h⁄ Ω [MeV]
−30
−25
−20
Nmax
= 2N
max = 4
Nmax
= 6N
max = 8
Nmax
= 10
Nmax
= 12
10 15 20 25 30h⁄ Ω [MeV]
4He
N3LO (500 MeV)
λ = 3.0 fm−1
λ = 2.0 fm−1
λ = 1.5 fm−1
Bogner, Furnstahl, Maris, Perry, Schwenk, Vary, NPA801, 21 (2008), arXiv:0708.3754
INT Extreme Computing, June 7 2011 – p.17/43
Effect of three-body forces
0 2 4 6 8 10 12 14N
max cut
λ = 1.0 fm−1
λ = 1.2 fm−1
λ = 1.5 fm−1
λ = 1.8 fm−1
λ = 2.0 fm−1
λ = 2.2 fm−1
0 2 4 6 8 10 12 14N
max cut
0 2 4 6 8 10 12 14N
max cut
−34−33−32−31−30−29−28−27−26−25−24−23−22−21−20
Gro
und-
Stat
e E
nerg
y [M
eV]
NA3max
= 40
NA2max
= 300
NN+NNNNN+NNN-induced
6Li
h- Ω = 20 MeV
NN-only
(Jurgenson, Navratil, Furnstahl, PRC83, 034301 (2011), arXiv:1011.4085)
Induced 3NF significantly reduce dependence on SRG parameter
N2LO 3NFbinding energy in agreement with experimentmay need induced 4NF?
Calculations for A = 7 to 12 in progress (LLNL)
INT Extreme Computing, June 7 2011 – p.18/43
Do we really need 3-body interactions?
0
2
4
6
8
E (
MeV
)
NN NN + NNN Exp JISP16chiral eff. interaction
3+
1+
0+
1+
2+
3+
2+2+
4+
2+10
BSpectrum of 10B
with chiral 2- and3-body forcesat Nmax = 6
nonlocal 2-bodyinteraction JISP16at Nmax = 8
Vary, Maris, Negoita, Navratil, Gueorguiev, Ormand, Nogga, Shirokov, and Stoica,in “Exotic Nuclei and Nuclear/Particle Astrophysic (II)”, AIP Conf. Proc. 972, 49 (2008);N3LO+3NF from Navratil, Gueorguiev, Vary, Ormand, and Nogga, PRL 99, 042501 (2007);
for JISP16 see Shirokov, Vary, Mazur, Weber, PLB 644, 33 (2007)
INT Extreme Computing, June 7 2011 – p.19/43
Phenomeological NN interaction: JISP16
A.M. Shirokov, J.P. Vary, A.I. Mazur, T.A. Weber, PLB 644, 33 (2007)
J-matrix Inverse Scattering Potential tuned up to 16O
finite rank seperable potential in H.O. representation
fitted to available NN scattering data
use unitary transformations to tune off-shell interaction tobinding energy of 3He
low-lying spectrum of 6Li (JISP6, precursor to JISP16)
binding energy of 16O
good fit to a range of light nuclear properties
very soft potential compared to other NN potentials
nonlocal potential (by construction)
details available at
http://nuclear.physics.iastate.edu/
INT Extreme Computing, June 7 2011 – p.20/43
Ground state energy Be-isotopes with JISP16
6 8 10 12 14total number of nucleons A
-70
-60
-50
-40
-30
grou
nd s
tate
ene
rgy
[M
eV]
variational boundNCFC w. JISP16experiment
Be isotopes
Nmax
= 12
Nmax
= 14
Nmax
= 10
Nmax
= 8
INT Extreme Computing, June 7 2011 – p.21/43
7Be – Ground state properties
0 5 10 15 20 25 30 35 40basis space frequency hω (MeV)
-38
-36
-34
-32
-30
bind
ing
ener
gy (
MeV
)
Nmax
= 6N
max = 8
Nmax
= 10N
max = 12
extrapolatedN
max = 14
extrapolated4He +
3He
experiment
7Be
JISP16
0 5 10 15 20 25 30 35 40basis space frequency hω (MeV)
1.5
2.0
2.5
3.0
prot
on r
ms
radi
us (
fm)
Nmax
= 6N
max = 8
Nmax
= 10N
max = 12
Nmax
= 14
7Be JISP16
Binding energy converges monotonically,with optimal H.O. freuqency around ~ω = 20 MeV to 25 MeV
Ground state about 0.7 MeV underbound with JISP16
Proton point radius does not converge monotonically
INT Extreme Computing, June 7 2011 – p.22/43
7Be – Excited states
0 expt. 10 15 20 25 30 35 40hω (MeV)
0
4
8
12
exci
tatio
n en
ergy
(M
eV)
Nm
= 10N
m = 12
Nm
= 14
7/2
5/2
7/2
5/2
3/2
1/2
JISP167Be
0 expt. 10 15 20 25 30 35 40hω (MeV)
-1
0
1
2
3
4
Mag
netic
mom
ent JISP16
7Be
5/2
5/2
1/2
7/2
3/2 (g.s.)
Excitation energy of narrow statesconverge rapidlyagree with experiments
Broad resonances depend ~ω
Magnetic momentswell converged
2-body currents neededfor agreement with data(meson-exchange currents)
INT Extreme Computing, June 7 2011 – p.23/43
7Be – Proton density
Intrinsic density – center-of-mass motion taken outw. Cockrell, PhD student ISU
0 1 2 3 4r (fm)
0
0.05
0.1
prot
on d
ensi
ty ρ
(r)
Nm
= 2N
m = 6
Nm
= 10N
m = 14
JISP167Be
hω = 12.5 MeV
hω = 20.0 MeV
0 2 4 6r (fm)
10-5
10-4
10-3
10-2
10-1
prot
on d
ensi
ty ρ
(r)
Nm
= 2N
m = 6
Nm
= 10N
m = 14
fit: a exp(-b r)
JISP167Be
hω = 12.5 MeV
hω = 20.0 MeV
Slow build up of asymptotic tail of wavefunction
Proton density appears to converge more rapidly at ~ω = 12.5 MeVthan at 20 MeV because long-range part of wavefunction is betterrepresented with smaller H.O. parameter
INT Extreme Computing, June 7 2011 – p.24/43
7Be – Proton radius
0 2 4 6r (fm)
0
0.2
0.4
0.6
0.8
r4ρ(
r)
Nm
= 2N
m = 6
Nm
= 10N
m = 14
JISP167Be
hω = 12.5 MeV
hω = 20.0 MeV
0 4 8 12 16N
max
2.0
2.2
2.4
2.6
2.8
3.0
prot
on p
oint
RM
S ra
dius
(fm
)
hω = 8.0 MeVhω = 9.0 MeVhω = 10.0 MeVhω = 12.5 MeVhω = 15.0 MeVhω = 17.5 MeVhω = 20.0 MeV
7Be
JISP16
Calculation one-body observables 〈i|O|j〉 ∼∫
O(r) r2 ρij(r) dr
RMS radius: O(r) = r2
Slow convergence of RMS radius due toslow build up of asymptotic tail
Ground state RMS radius in agreement with data
INT Extreme Computing, June 7 2011 – p.25/43
7Be – Quadrupole moment
4 8 12 16N
max
-8
-7
-6
-5
-4
-3
-2
-1
0
quad
rupo
le m
omen
t
hω = 8.0 MeVhω = 9.0 MeVhω = 10.0 MeVhω = 12.5 MeVhω = 15.0 MeVhω = 17.5 MeVhω = 20.0 MeV
7Be JISP16
Ground state quadrupole moment in agreement with data
Optimal basis space around ~ω = 10 MeV to 12 MeV
Similar slow convergence for E2 transitions
INT Extreme Computing, June 7 2011 – p.26/43
7Be – Rotational band
E2 observables suggest rotational structure for 3
2, 1
2, 7
2, 5
2states
0 5 10 15 20 25 30 35 40hω (MeV)
-10
-8
-6
-4
-2
0
Qua
drup
ole
mom
ent
JISP16
7Be
rotational band?
3/2
5/2
7/2
5/2
0 10 20 30 40basis space frequency hω (MeV)
10-1
100
101
102
B(E
2) t
o g.
s. 3
/2-
JISP167Be1/2
-
5/2-
5/2-
7/2-
rotational band?
Q(J) =3
4− J(J + 1)
(J + 1)(2J + 3)Q
1
2
0
B(E2; i→ f) =5
16π
(
Q1
2
0
)2(
Ji,1
2; 2, 0
∣
∣
∣Jf ,
1
2
)2
INT Extreme Computing, June 7 2011 – p.27/43
8Be – Spectrum positive parity
0 5 10 15 20 25 30 35 40hω (MeV)
0
5
10
15
20
exci
tatio
n en
ergy
(M
eV)
dotted: Nmax
= 8 solid: Nmax
= 12
2
4
13
2
4
JISP168Be
Pairs of isospin 0 and 1states with J = 2, 1, and 3
Evidence of continuumstates (J = 0 and 2)at Nmax = 12
Rotational band
expt. calc rotor
E4/E2 3.75 3.40 3.33
Ground state above 2α threshold: radius not converged
Quadrupole moments 2+ and 4+ not converged, nor B(E2)’s,but in qualitative agreement with rotational structure
INT Extreme Computing, June 7 2011 – p.28/43
Results with JISP16 for 12C
0 5 10 15 20 25 30 35 40
-100
-90
-80
-70
Nmax
= 2
Nmax
= 4
Nmax
= 6
Nmax
= 8
extrapolatedN
max=10
extrapolatedexpt.
Calculations for Nmax = 10 underway (D = 8 billion)using 100,000 cores on JaguarPF (ORNL) under INCITE award
INT Extreme Computing, June 7 2011 – p.29/43
Spectrum of 12C with JISP16 – work in progress
0 10 20 30 40basis space hω (MeV)
0
5
10
15
20
25
exci
tatio
n en
ergy
(M
eV)
spectrum 12C with JISP16 at Nmax = 8 (solid) and 10 (crosses)
(2+, 0)
(1+, 0)
(1+, 1)
(2+, 1)
(0+, 0)
(2+, 0)
(0+, 0)
(4+, 0)
so-called Hoyle state missing
(0+, 1)
Pos. parity states in agreement with data, except for Hoyle state
Electromagnetic transitions in progressrotational 2+ and 4+ states, significantly enhanced B(E2)optimal basis ~ω for Q and B(E2) around ~ω = 12.5 MeV
Neutrino and pion scattering calculations in progressINT Extreme Computing, June 7 2011 – p.30/43
Density of 12C with JISP16
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Radius (fm)
Den
sity
(fm
− 3)Carbon 12 Proton Densities
Pieper GFMCNmax 8, hw=12.5, JISP16Experimental Data
GFMC: AV18 + IL7, on BlueGene/P using 131,072 cores (INCITE)“More scalability, Less pain”, Lusk, Pieper, and Butler, SciDAC review 17, 30 (2010)
JISP16 density at Nmax = 8, ~ω = 12.5 MeVINT Extreme Computing, June 7 2011 – p.31/43
Scientific Discovery – unstable nucleus 14F
Maris, Shirokov, Vary, arXiv:0911.2281 [nucl-th], Phys. Rev. C81, 021301(R) (2010)
0 2 4 6 8Nmax
0
5
10
Exc
itat
ion
ener
gy (
MeV
)
Extrapolation B Exp.2
-
1-(?)
3-(?)
2-
4-
??
1-0-
2-5
-
14F
1-
14B
3-
3-
2-
4-(?)
4-
2-
2-3
-
2-
3-
1-
5-
2-
4-
2-
1-
1-3
-
3-
4-
2-
1-0-
dimension 2 · 109
# nonzero m.e. 2 · 1012
runtime 2 to 3 hours on7,626 quad-core nodeson Jaguar (XT4)(INCITE 2009)
Predicted ground state energy: 72 ± 4 MeV (unstable)
Mirror nucleus 14B: 86 ± 4 MeV agrees with experiment 85.423 MeV
INT Extreme Computing, June 7 2011 – p.32/43
Predictions for 14F confirmed by experiments at Texas A&M
NCFC predictions (JISP16) in
close agreement with experiment
TAMU Cyclotron Institute
Experiment published: Aug. 3, 2010 Theory published PRC: Feb. 4, 2010
INT Extreme Computing, June 7 2011 – p.33/43
Petascale Early Science – Ab initio structure of Carbon-14
0 4 8basis space truncation N
max
103
106
109
1012
1015
dimension# nonzero’s NN# nonzero’s NNN# matrix elements
14C (Z= 6, N= 8)
0 4 8basis space truncation N
max
103
106
109
1012
1015
dimension# nonzero’s NN# nonzero’s NNN# matrix elements
14N (Z= 7, N= 7)
Chiral effective 2-body plus 3-body interactions at Nmax = 8
Basis space dimension 1.1 billion
Number of nonzero m.e. 39 trillion
Memory to store matrix (CRF) 320 TB
Total memory on JaguarPF 300 TB
ran on JaguarPF (XT5) using up to 36k 8GB processors (216k cores)after additional code-development for partial “reconstruct-on-the-fly”
INT Extreme Computing, June 7 2011 – p.34/43
Ab initio structure of Carbon-14 and Nitrogen-14
Maris, Vary, Navratil, Ormand, Nam, Dean, arXiv:1101.5124 [nucl-th]
2 4 6 8 expt 8 6 4 20
5
10
15
20ex
cita
tion
ener
gy (
MeV
)(3
+,0)
(1+,1)
(2+,1)
(2+,0)
(0+,1)
(1+,0)
N3LON3LO + 3NF
chiral 2-body plus 3-body forces (left) and 2-body forces only (right)
INT Extreme Computing, June 7 2011 – p.35/43
Origin of the anomalously long life-time of 14C
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
GT
mat
rix
elem
ent
no 3NF forceswith 3NF forces (c
D= -0.2)
with 3NF forces (cD
= -2.0)
s p sd pf sdg pfh sdgi pfhj sdgik pfhjl
shell
-0.1
0
0.1
0.2
0.3
0.2924
near-completecancellationsbetweendominantcontributionswithin p-shell
very sensitiveto details
Maris, Vary, Navratil,
Ormand, Nam, Dean,
arXiv:1101.5124 [nucl-th]
INT Extreme Computing, June 7 2011 – p.36/43
Neutrons in a trap: Why
Validate ab-initio DFT approachesagainst microscopic ab-initio calculations
compare Density Matrix Expansionand ab-initio NCFC calculations
using the same interactioncalculating the same observables for the same systems
Construct Universal Nuclear Energy Density Fuctionalconsistent with ab-initio calculations
Theoretical ’laboratory’ to exploreproperties of different nuclear interactionseffect of density and gradienton nuclear properties for different interactions
Model for neutron-rich systemsin particular those with closed shell protons (Oxygen, Calcium)
Essential in order to make meaningful comparison with other methods:quantify dependence on basis space truncation parameters
INT Extreme Computing, June 7 2011 – p.37/43
Validating ab-initio DME/DFT calculations
Bogner, Furnstahl, Kortelainen, Maris, Stoistov, Vary, in preparation
Simple model for interactionMinnesota potential
Ab-initio NCFC calculations for neutrons in H.O. potentialincluding numerical error estimates on all ’observables’
DFT using same NN interaction as NCFCHartree–FockDensity Matrix Expansion, Hartree–FockDensity Matrix Expansion, Brueckner–Hartree–Fock
DFT fit to NCFC results
Comparison for 8, 14, and 20 neutronstotal and internal energy per neutronrms radiusform factor F (q)
INT Extreme Computing, June 7 2011 – p.38/43
Minnesota potential – Total energy vs. radius
2.0 2.5 3.0radius [fm]
0.85
0.9
0.95
Eto
t/(N
4/3 hΩ
) NCFCHFPSABHFfit
2.0 2.5 3.0radius [fm]
0.8
0.85
0.9
Eto
t/(N
4/3 hΩ
) Minnesota potential
N = 8
N = 20
Neither HFnor DME/PSA HFin agreement with NCFC
DME BHF close to NCFCresults oftenwithin error estimates
Fit with volume termand surface term canreproduce NCFC data
Bogner, Kortelainen, Furnstahl, Stoistov, Vary, PM, work in progress
INT Extreme Computing, June 7 2011 – p.39/43
Internal energy vs. radius
2.0 2.5 3.0radius [fm]
0.4
0.42
0.44
Ein
t/(N
4/3 hΩ
)
NCFCHFPSABHFfit
2.0 2.5 3.0radius [fm]
0.38
0.4
0.42
Ein
t/(N
4/3 hΩ
)
N = 8
N = 20 Neither HFnor DME/PSA HFwithin variational bound
DME BHF close to NCFCresults oftenwithin error estimatesand within bounds
Fit with volume termand surface term canreproduce NCFC data
Variational upper bound on combination Eint + 1
2N mΩ2r2
INT Extreme Computing, June 7 2011 – p.40/43
Minnesota potential – density
0 1 2 3 4 5r [fm]
0
0.02
0.04
0.06
0.08
0.1
ρ [f
m−3
]
NCFCPSABHFfit
NN-only
Minnesota potential
N = 8hΩ = 10 MeV
0 1 2 3 4r [fm]
0
0.1
0.2
0.3
ρ [f
m−3
]
NCFCPSABHFfit
NN-only
Minnesota potential
N = 20
hΩ = 15 MeV
Agreement between DME/DFT calculations and NCFC
Density profile dominated by H.O. external fieldmodefied by NN interaction
Bogner, Kortelainen, Furnstahl, Stoistov, Vary, PM, work in progress
INT Extreme Computing, June 7 2011 – p.41/43
Eint vs. radius – more realistic potentials
1.5 2 2.5 3 3.5 4radius [fm]
0
10
20
30
40E
int/ A
[M
eV]
8 neutrons, JISP168 n, N3LO
SRG[λ=1.5]
20 neutrons, JISP1620 n, N3LO
SRG[λ=1.5]
neutron droplets in external fields
Results virtually identical for N3LO(λ = 1.5) and JISP16despite different results for nuclei (e.g. 186 vs. 144 MeV for 16O)presented at JUSTIPEN–EFES–Hokudai–UNEDF workshop, 2008, Hokkaido, Japan
INT Extreme Computing, June 7 2011 – p.42/43
Conclusions
MFDn: Scalable and load-balanced CI code for nuclear structurenew version under development, has run on 200k+ coreson Jaguar (ORNL) enabling largest model-space calculations
Significant benefits from collaboration between nuclear physicists,applied mathematicians, and computer scientists
JISP16, nonlocal phenomenological 2-body interactiongood description of light nuclei (more than just energies!)rapid convergence
prediction of new isotope, 14F
Understanding of the anomalously large lifetime of 14C
Validation of DFT/DME calculations (in progress)
Main challenge: construction and diagonalizationof extremely large (D > 1 billion) sparse matrices
Future developments: Taming the scale explosion
INT Extreme Computing, June 7 2011 – p.43/43