Post on 04-Aug-2020
Atomic nuclei: Computational challenges in an ab initio approachp g pp
David J. DeanOak Ridge National Laboratory
IGCMS SeminariApril 2009
Collaboration: G. Hagen (ORNL), T. Papenbrock (ORNL/UT)M. Hjorth-Jensen (Oslo), P. Piecuch (MSU); K. Roche (ORNL)
Funding: gDOE/SC Nuclear Physics; ORNL LDRD (FY02/FY03) & SEED (FY05/FY06) SciDAC-II; ASCR/NNSAComputing: NERSC/LBL and NCCS/ORNL
The Physics
QCD
EFT}}
Nuclear Structure
Applications in astrophysics, defense, energy, and medicine
Landscape and consequences
Present and next Generation Radioactive Ion Beam facilities(multi $100M investments world wide)
“[C]ountries throughout theworld are aggressively pursuingrare-isotope science, often as their highest priority in nucleartheir highest priority in nuclearscience, attesting to the significance accorded internationallyto this exciting area of research”NAS RISAC ReportNAS RISAC Report
Future U.S. FRIB basedon a heavy-ion linac driver
hi h i ita high priority-- NAS RISAC Report-- NSAC RIBT Report-- NSAC 2007 LRP
Key nuclear physics questions
• What is the nature of the nuclear force that bindsprotons and neutrons into stable nuclei and rareisotopes?Wh t i th i i f i l tt i l l i?• What is the origin of simple patterns in complex nuclei?
• What is the nature of neutron stars and dense nuclear matter?• What is the origin of the elements in the cosmos?• What are the nuclear reactions that drive stars and
How are we going to describe nuclei that we cannot measure?
stellar explosions?
cannot measure?Robust and predictive nuclear theoryNeed for nuclear data to constrain theoryWe are after the Hamiltonian
bare intra-nucleon Hamiltonianenergy density functional
This talk:This talk: Walk through some of thecomputational aspects of the problem…
What’s the vision?Calculate at least to mass 100+ nuclei with ab initio techniques.
WHY?WHY? Nuclear Discovery: 60 years
Old Moore’s law figure
Moore’s lawMoore’s Law
• Discovery of new nuclei is slow (ND law)• Discovery of new nuclei is slow (ND law)• Computational capability follows a power law (Moore’s law)
• TODAY: 1.3 Pflops (sustained)!!• ab initio (from V2+V3+ forces) gives us a viable platform toab initio (from V2+V3+…forces) gives us a viable platform to make predictions.
• exa-scale computing is in the DOE/SC 10 year plan.
Physics of nuclei and computing
• 1980s: Time-dependent Hartree FockKoonin, Strayer, Davies, Bonche,…Umar
• Early 1990s: Auxilliary Field Monte CarloKoonin Dean Langanke Ormand Alhassid JohnsonKoonin, Dean, Langanke, Ormand, Alhassid, Johnson
• 2000s to present: ab inito takes hold• GFMC (Pieper, Wiringa, Calson, Pandharipande, Lusk)• NCSM (Barrett, Vary, Ormand, Navratil, Ng)• Coupled-clusters (Dean, Hjorth-Jensen, Hagen, Piecuch,
Papenbrock, Bernholdt)Papenbrock, Bernholdt)
• Late 2000s: quest for Universal Nuclear Energy Density Functional• Nazarewicz, More’, Fann,… (UNEDF)
With Skyrme’s force, each iteration of the Hartree-Fock equations for 208Pb takes lessthan 30 seconds on the Univac 1008. Vautherin & Brink, PLB32, 149 (1970)
Begin with a NN (+3N) Hamiltonian
From the interaction to solving the nuclear many-body problem
Begin with a NN (+3N) Hamiltonian
cmkji
kjiNji
jiN
A
i i
i TrrrVrrVm
H
,,61,
21
2 321
2 Bare (GFMC)(Local only, Av18plus adjusted 3-body)
B i i
kjijii im 622 1 plus adjusted 3 body)
Basis expansion(explore forces)Basis expansions:
• Determine the appropriate basis• Generate Heff in that basis• Use many-body technique to solve problem
( p )
• Use many-body technique to solve problem
NucleusNucleus 4 shells 4 shells 7 shells7 shells
4He4He 4E44E4 9E69E6Oscillatorsingle-particle
Substantial progress inmany-body developments• GFMC; AFDMC
N C h ll d l4He4He 4E44E4 9E69E6
8B8B 4E84E8 5E135E13
12C12C 6E116E11 4E194E19
basis states
Many body
• No Core shell model (not a model)
• Coupled-cluster theory• UCOM,…
16O16O 3E143E14 9E249E24
Many-body basis states
,• AFMC
Green’s Function Monte Carlo
Idea:1. Determine accurate approximate wave function via variation of the
energy (The high-dimensional integrals are done via Monte Carlo integration).
2. Refine wave function and energy via projection with Green’s functionfunction
Vi t ll t th d Virtually exact method. Limited to certain forms of Hamiltonians. Computational expense increases dramatically with A due to
sampling of spin/isospin sampling. Possible extensions to heavy nuclei via AFMDC
Basis expansion techniques• Generate a single-particle basis (e.g., spherical oscillator)• Transform problem into a large, sparse eigenvalue problem• Worry about ‘down-folding’ of the interaction• Solve with Krylov space techniques
CMkji
kjiNji
jiN
A
i i
i TrrrVrrVm
H
,,61,
21
2 321
2
)()(),()()(),(|| 21212*
1*
2121 rrrrVrrrdrdrsrrVpqrspq srqp
pqrs
rsqppq
qp aaaarsVpqaaqtpH41
aaaarsVpqaaqtpH ff1
pqrsrsqp
pqqp aaaarsVpqaaqtpH eff4
The a+ and a are creation and anihilation operators
General many-body problem for fermions(basis expansions)
particles are spin ½ fermionsmany-body wave function is fully anti-symmetricy y y y certain quantum numbers will be conserved
for nuclei: total angular momentumtotal paritytotal parity‘isospin’ (analogous to spin)‘isospin projection, Tz= (N-Z)/2
H ilt i ill b l ti i ti ( ll ) Hamiltonian will be non-relativistic (usually)We (usually) work in second quantization
particlesandstatesparticlesinglewithspaceFock AN01010010
particles.andstatesparticlesingle with spaceFock
aaaaaaaaaaaa
aaaaAN
1000011021
Aaaa
aaaaaaaaaaaa
0 3
Specific example: 2 particles in 4 states
113
012
10012
10101
11000
aaI
aaI0
1
3
323
214
01014
01103
10012
I
aaI
aaI1 4
534
424
00115
01014
aaI
aaI2 5
!states particle-single ofnumber
particles;ofnumber
NNn
Scaling: Number of basis states 13107.1)100,10(!!
!,
xCnnN
NnNC
Oops. These are HUGE numbers
139106100,1000 xC
PROBLEM : How to deal with such large dimensions???
Correlated wave function representation
W h l t t f t t th t t t d Hilb tWe have a complete set of states that span our truncated Hilbert space:
IJ
N
JIII
;11
“mean field” Uncorrelated state of lowest energy
I0
mean field Uncorrelated state of lowest energy.
11000
0 jibaabij
iaai aaaabaabb
1p-1h 2p-2h … np-nh1p 1h 2p 2h … np nh(implicit summation assumed)
;11N
Problem II: How do we solve for the correlated many-body wave function?
;10
Diagonal contributions to the Hamiltonian matrix
Here we apply Wick’s theorem to the one body termHere we apply Wick s theorem to the one-body termand the diagonal contributions of the two-body term.
1222212121121111 aaaaaaaaaaaaH
11121212122121121211 4
141 VaaaaaaaaVH
121221 4
1 V
23
41
31
43
23
43123412432134212143124321341234 VVVVVVVV
Two-body contributions to the Hamiltonian matrix
123434342121123416 41
41 VaaaaaaaaVH Hamiltonian matrix now
‘mixes’ bare eigenstates
H0
41
12342143123461 aaaaaaaaVH
mixes bare eigenstates
1313311312
12341213121221
1141
41
21
VV
VVV
341212124343341261 4
141 VaaaaaaaaVH
2134141441
1313311312
1141
21
24
VV
VV
231414143232231443 41
41 VaaaaaaaaVH
343443
2323322314
21
21
41
V
VV
142323234141142334 41
41 VaaaaaaaaVH
3434433412 2
141 VV
121313134121121312 41
41 VaaaaaaaaVH
Solve the eigen problem
• Generate the Hamiltonian matrix and diagonalize (Lanczos)• Yields eigenvalues and eigenvectors of the problem
IJ HUIHJU
I
I IU
0aaaabaabb jibaabij
iaai
surfaceFermitheaboverunsurface Fermi thebelowrun ,
baji
j
surfaceFermitheaboverun ,ba
Theorists agree with each other
No core shell model
Idea: Solve the A-body problem in a harmonic oscillator basis.1. Take K single particle orbitals2. Construct a basis of Slater determinants3. Express Hamiltonian in this basis4. Find low-lying states via diagonalization
Get eigenstates and energies Symmetries like center-of-mass treated exactly No restrictions regarding Hamiltonian No restrictions regarding Hamiltonian
Number of configurations and resulting matrix very large: There are
ways to distribute A nucleons over K single-particle orbitals.ways to distribute A nucleons over K single particle orbitals.
Progress: from structure to reactions in the same framework
n- scattering
Nollett et al., PRL 99, 022502 (2007)
NCSM clustersNCSM clusters
Navratil et al., PRC73, 065801 (2006)
Coupled Cluster Theory: ab initio in medium mass nuclei
TexpCorrelated Ground-State Correlation Reference Slater
wave function operator determinant
TTTT Energy 321 TTTT
f
ai
iaai aatT
1 THTE exp)exp(
Energy
Amplitude equations
f
f
f
abij
ijbaabij
a
aaaatT
2 0exp)exp( HTHT abij
abij
Amplitude equations
fab
• Nomenclature• Coupled-clusters in singles and doubles (CCSD)• with triples corrections CCSD(T);…with triples corrections CCSD(T);
Dean & Hjorth-Jensen, PRC69, 054320 (2004); Kowalski et al., PRL 92, 132501 (2004); Wloch et al., PRL94, 212501 (2005) Gour et al., PRC (2006); Hagen et al, PLB (2006); PRC 2007a, 2007b
Am
The many-body wave function in cluster amplitudes
AA
kk
AT TTe1
)(,)(
kji
abcijk
abcijk
ji
abij
abij
i
ai
ai tTtTtT 321 ,,
cbabaa
a,b,…
i,j,…
22422
theoryeapproximat ,;ryexact theo ,
A
A
CCSDTTT
NmNm
3353
321
224221
3
2
uouoA
uouoA
nnnnCCSDTTTTTm
nnnnCCSDTTTm
Relationship between shell model and CC amplitudes
2
11
1 TTB
TB
3
2122
121 TTB
CCSD
422
311233
11161 TTTTB CR-CCSD(T)
41
212
221344 24
121
21 TTTTTTTB
“Disconnected quadruples”
“Connected quadruples”
These equations are sort of messy in their full blown formulation
CCSD T1, T2 amplitudesCCSD T1, T2 amplitudes
Non-linearCoupledCoupledAlgebraicTensor-tensor multiplies
Solution of Coupled-Cluster equation
System of non-linear coupled algebraic equations solve by iteration Basic Numerical Operation:
n=number of neutrons and protons
N=number of basis states
oldold
new
klabtijcdtcdklVijabt
),(),(),(),(
Solution tensor memory (N-n)**2*n**2
Interaction tensor memory
Nndc
nlkoldold j
,1,,1,
),(),(),(
Interaction tensor memory N**4
Operations count scaling
• Many terms like this• Cast into a matrix-matrix
multiply algorithm O(n**2*N**4) O(n**4*N**4) with 3-body O(n**3*N**5) at CCSDT
multiply algorithm• Parallel Issue: block sizes of V and t• Petascale target problem
100 N 1000( )
• n=100; N=1000
Verification and Validation (V&V)
Doing the problem right. – VerifyDoing the right problem. – Validate
Ab initio in medium mass nuclei4HeHe
16OHagen, Dean, Hjorth-Jensen,Papenbrock, Schwenk, PRC76, 044305 (2007)
Error estimate: << 1% < 1% 1%
p ( )
40CaCa
1063 many-body basis states
Inclusion of full TNF in CCSD: F-Y comparisons in 4He
0 Solution at CCSD and CCSD(T) levelsSolution at CCSD and CCSD(T) levels involve roughly 67 more diagrams…..
E 28 24 M V -1
100
|
2-body only
<E>=-28.24 MeV +/- 0.1MeV (sys)
10-2
10-1
E /
EC
CSD
| 0-body 3NF
1-body 3NF
estimated triples corrections
10-3
10
| ΔE
/
2-body 3NFestimated triples corrections
Challenge: do we really need the full(1) (2) (3) (4) (5)10
-4residual 3NF
Challenge: do we really need the full 3-body force, or just its density dependent terms? Hagen, Papenbrock, Dean, Schwenk, Nogga, Wloch, Piecuch
PRC76, 034302 (2007)
Utilize the nuclear total spin symmetry to push further
Implemented a CCSD J-coupled (Hagen) code for heavier nuclei:• Scaling at CCSD goes from O(no
2nu4) to O(no
4/3nu8/3)
• Can do up to 14 complete major shells on a single node• Can do up to 14 complete major shells on a single node. • CCSDT Gold standard for these heavier nuclei (developing)• Enables specified calculations for heavy nuclei• The large model spaces mean that we can approach BARE interactions!The large model spaces mean that we can approach BARE interactions!
• Must start with a spherical HF basis
• Is it technically feasible to go further? (YES)• Does size extensivity work in the nuclear case? (YES)• Opens some interesting doors for future research….
Ca isotopes from bare chiral NN potential at NCa isotopes from bare chiral NN potential at N33LO (no 3LO (no 3--body yet)body yet)
1081 Mb b i !!1081 Mb basis states!!
3Chiral NN potential at N3LO underbinds by ~1MeV/nucleon. (Size extensivity at its best.)
NucleusNucleus E / A [MeV]E / A [MeV]44HeHe 1.08 1.08 (0.73(0.73FYFY))1616OO 1.251.2540404040CaCa 0.840.844848CaCa 1.271.274848NiNi 1.211.21
Ca isotopes from bare chiral NN potential at NCa isotopes from bare chiral NN potential at N33LO (no 3LO (no 3--body yet)body yet)
1081 Mb b i !!1081 Mb basis states!!
Chiral NN potential at N3LO underbinds by ~1MeV/nucleon. (Size extensivity at its best.)
Including triples corrections
CCSD(T) h t 4H b b t 0 2 K VCCSD(T) over shoots 4He by about 0.2 KeVCCSD(T) close to the F-Y result for 4He (E/A = 0.7 MeV)
Triples corrected results
E ti t b t 0 3 d 0 4 M V/A ill b i i fEstimate: between 0.3 and 0.4 MeV/A will be missing from binding energies across medium mass nuclei
There is some room for the 3-body force (perturbative?) 40Ca may be slightly over bound with just 2-body interaction
Oxygen chain results: Is 28O bound??
28O b d i h• 28O bound with respectto 4n by about 1 MeV
• Results not fully converged(N=13)(N=13)
• Experiment would be difficult!
• Systematic deviation fromSystematic deviation fromdata (but fairly constant)
PRELIMINARY
Nuclear ‘scale separation’in weakly bound nuclei
TRIUMF/GSI (2006)T1/2 ≈ 8.6 ms
8He
ANL (2004) ANL/GANIL (2007)
d f 11Li / 7 10 8… and mass of 11Li: m/m=7·10-8
nuclear radius (fm)
gyCoupling of nuclear structure and reaction theory
(microscopic treatment of open channels)
Open QSOpen QS
Ene
rg
Important interdisciplinaryaspects…(see recent ECT*workshop on subject)
Correlation
p j )
Correlationdominated n ~ n
Sn=0
S
Closed QS
Sn
Closed QSClosed QSNeutron number
Introduction of Continuum basis states (Gamow, Berggren)Continuum shell models
(many including: Michel, Rotureau, Volya, Ploszajczak, Liotta, Nazarewicz,…)
Progress: ab initio weakly bound and unbound nuclei
CCSD He Chain ResultsCCSD He Chain Resultsp
N3LO Vlowk (=1.9 fm-1)
n
6He gs spin (smaller space)6He gs spin (smaller space)Naïve filling =1.4CCSD =0.6CCSD(T) =0.6CCSDT 1 2 3 =0 2
Challenge: include 3-body forceHagen, Dean, Hjorth-Jensen, Papenbrock, Phys. Lett. B 656, 169 (2007)
CCSDT-1,2,3 =0.2CCSDT =0.04
Strategy:Strategy:Resources:Resources:Opportunity:Opportunity:
Nuclear Coupled Cluster Theory Perspectives
Strategy: Develop and deploy all
necessary elements to calculate nuclear masses, excitation energies and
Strategy: Develop and deploy all
necessary elements to calculate nuclear masses, excitation energies and
Resources: CC theory implementations
described in this talk: CCSD, CCSD(T), CCSDT-1, V3-CCSD(T) Gamow-based
Resources: CC theory implementations
described in this talk: CCSD, CCSD(T), CCSDT-1, V3-CCSD(T) Gamow-based
Opportunity: Capitalize on CC
developments to realize an ab initio foundation for nuclear structure (including
Opportunity: Capitalize on CC
developments to realize an ab initio foundation for nuclear structure (including excitation energies, and
transition properties Develop ‘shell-model’ like
effective interactions from CC theory
excitation energies, and transition properties
Develop ‘shell-model’ like effective interactions from CC theory
V3 CCSD(T), Gamow based CCSD(T), CCSDT, (some under development in J-scheme)
Excited state calculations
V3 CCSD(T), Gamow based CCSD(T), CCSDT, (some under development in J-scheme)
Excited state calculations
nuclear structure (including the shell model) and reactions with calculations reaching into heavy nuclei
Develop the CC technology
nuclear structure (including the shell model) and reactions with calculations reaching into heavy nuclei
Develop the CC technology y Leverage ultrascale
computing and nuclear theory expertise to solve a wide range interesting
y Leverage ultrascale
computing and nuclear theory expertise to solve a wide range interesting
Effective (and now BARE) interaction expertise
Software development at scale
Effective (and now BARE) interaction expertise
Software development at scale
p gyto include powerful tools for investigating the relationship between ab inito approaches and DFT
p gyto include powerful tools for investigating the relationship between ab inito approaches and DFT
problems within the CC framework
Develop ‘one-off’ problems for nurturing of post-docs and students (NRC grant)
problems within the CC framework
Develop ‘one-off’ problems for nurturing of post-docs and students (NRC grant)
Dynamic and promising interface of nuclear theory and computational science
Broad collaboration base
Dynamic and promising interface of nuclear theory and computational science
Broad collaboration base
Enable future experimental directions through ab initio predictions of nuclear properties in uncharted regions
Enable future experimental directions through ab initio predictions of nuclear properties in uncharted regions and students (NRC grant)and students (NRC grant)
Longstanding partnerships with DOE (NP and ASCR), Oslo/CMA among others
Longstanding partnerships with DOE (NP and ASCR), Oslo/CMA among others
regionsregions
Outcome: A cross-cutting theoryfor understanding and building nuclei from the ground up