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1Quantum many-body problems
David J. DeanORNL
OutlineI. Perspectives on RIAII. Ab initio coupled-cluster theory applied to 16O
A. The space and the HamiltonianB. The CC methodC. Ground- and excited state implementation: CCSDD. Triples Corrections: CR-CCSD(T)
III. Conclusions
OutlineI. Perspectives on RIAII. Ab initio coupled-cluster theory applied to 16O
A. The space and the HamiltonianB. The CC methodC. Ground- and excited state implementation: CCSDD. Triples Corrections: CR-CCSD(T)
III. Conclusions
2Quantum many-body problems
Scientific Thrusts:• How do complex systems emerge from simple ingredients (interaction question)?• What are the simplicities and regularities in complex systems (shell/symmetries question)?• How are elements produced in the Universe (astrophysics question)?Broad impact of neutron-rich nuclei:• Nuclei as laboratories for tests of ‘standard model physics’.• Nuclear reactions relevant to astrophysics. • Nuclear reactions relevant to Science Based Stockpile Stewardship.• Nuclear transportation, safety, and criticality issues.
Energydensity
functional
Motivate,compare toexperiments
Applications:beyond SMastrophysics
Microscopicnuclear
structuretheory
InteractionsVNN, V3N
Chiral EFT
RIANuclei
Nuclei and RIA
4Quantum many-body problems
The opportunity: connection of nuclear structure andreaction theory and simulation to the Rare Isotope Accelerator
• RIA: $1 billion facility, near-term number-three priority (first priority in Nuclear Physics)• CD-0 approved, moving to CD-1
• OSTP Report: directs DOE and NSF to ‘generate a roadmap for RIA’.
5Quantum many-body problems
Quantum many-body problem is everywhereQuantum many-body problem is everywhere
Molecular scale: conductancethrough molecules.
Delocalized conduction orbitals
MoleculesMoleculesNanoscaleNanoscale
~10 nm
2-d quantum dotin strong magnetic field.
The field drives localization.
Nuclei Nuclei
~5 fm
• Studies of the interaction• Degrees of freedom• Astrophysical implications
• Studies of the interaction• Degrees of freedom• Astrophysical implications
Quantum mechanics plays a role when the size of the object is of the same order as the interaction length.
Common properties• Shell structure• Excitation modes• Correlations• Phase transitions• Interactions with external probes
Quantum mechanics plays a role when the size of the object is of the same order as the interaction length.
Common properties• Shell structure• Excitation modes• Correlations• Phase transitions• Interactions with external probes
6Quantum many-body problems
Begin with a bare NN (+3N) HamiltonianBegin with a bare NN (+3N) Hamiltonian
Basis expansions:• Choose the method of solution • Determine the appropriate basis• Generate Heff
Basis expansions:• Choose the method of solution • Determine the appropriate basis• Generate Heff
kji
kjiNji
jiN
A
i i
i rrrVrrVm
H
,,6
1,
2
1
2 321
22
Nucleus 4 shells 7 shells
4He 4E4 9E6
8B 4E8 5E13
12C 6E11 4E19
16O 3E14 9E24
Oscillatorsingle-particle basis states
Many-body basis states
Steps toward solutionsSteps toward solutions
Bare (GFMC)
Basis expansion
method
basis Heff
7Quantum many-body problems
Morten Hjorth-Jensen (Effective interactions) Maxim Kartamychev (3-body forces in nuclei) Gaute Hagen (effective interactions; to ORNL in May)
The ORNL-Oslo-MSU collaborationon nuclear many-body problems
The ORNL-Oslo-MSU collaborationon nuclear many-body problems
Piotr Piecuch (CC methods in chemistry and extensions) Karol Kowalski (to PNNL) Marta Wloch Jeff Gour
David Dean (CC methods for nuclei and extensions to V3N) Thomas Papenbrock David Bernholdt (Computer Science and Mathematics) Trey White, Kenneth Roche (Computational Science)
Research Plan-- Excited states (!)-- Observables (!)-- Triples corrections (!)-- Open shells-- V3N
-- 50<A<100 -- Reactions-- TD-CCSD??
8Quantum many-body problems
Choice of model space and the G-matrixChoice of model space and the G-matrix
Q-Space
P-Space
Fermi
Fermi
+…
=
+
h
p
G
ph intermediatestates
CC-phh
p
)~(~)~(
GQtQ
QVVG
aitab
ijt pqrs
rsqppq
qposc aaaarsGpqaaqtpH4
1
Use BBP to eliminate w-dependencebelow fermi surface.
9Quantum many-body problems
In the near future: similarity transformed H
2/12/111
0
1 ;
PPPQPPHPPH
kkPHeQe
QPkEkH
eff
PPQQ
k
P
Advantage: no parameter dependence in the interactionCurrent status• Exact deuteron energy obtained in P space• Working on full implementation in CC theory. • G-matrix + all folded-diagrams+…• Implemented, new results cooking….stay tuned.
K. Suzuki and S.Y. Lee, Prog. Theor. Phys. 64, 2091 (1980)P. Navratil, G.P. Kamuntavicius, and B.R. Barrett, Phys. Rev. C61, 044001 (2000)
10Quantum many-body problems
Fascinating Many-body approach:Coupled Cluster Theory
Fascinating Many-body approach:Coupled Cluster Theory
Some interesting features of CCM:• Fully microscopic• Size extensive: only linked diagrams enter • Size consistent: the energy of two non-interacting fragments computed separately is the same as that computed for both fragments simultaneously• Capable of systematic improvement• Amenable to parallel computing
Computational chemistry: 100’s of publications in any year(Science Citation Index) for applications and developments.
11Quantum many-body problems
Coupled Cluster TheoryCoupled Cluster Theory
TexpCorrelated Ground-State
wave functionCorrelation
operatorReference Slater
determinant
321 TTTT
f
f
f
f
abij
ijbaabij
ai
iaai
aaaatT
aatT
2
1 THTE exp)exp(
0exp)exp( THTabij
EnergyEnergy
Amplitude equationsAmplitude equations
• Nomenclature• Coupled-clusters in singles and doubles (CCSD)• …with triples corrections CCSD(T);
Dean & Hjorth-Jensen, PRC69, 054320 (2004); Kowalski, Dean, Hjorth-Jensen,Papenbrock, Piecuch, PRL92, 132501 (2004)
12Quantum many-body problems
T1 amplitudes from:T1 amplitudes from: 0exp)exp( THTai
Derivation of CC equationsDerivation of CC equations
Note T2 amplitudes also come into the equation.
13Quantum many-body problems
T2 amplitudes from:T2 amplitudes from: 0exp)exp( THTabij
An interesting mess. But solvable….An interesting mess. But solvable….
Nonlinear terms in t2(4th order)
)()()()( jifijfijfijP
14Quantum many-body problems
Correspondence with MBPTCorrespondence with MBPT
jiba
abij
ijababij
tijabE
Dijabt
)1(
)1(
2
2nd order
jiba
abij
kc
acik
cd
cdij
kl
abkl
abij
tijabE
tcjkbabPijPtcdab
tijklt
)2(
)1()()()1(2
1
)1(2
1)2(
3
3rd order
15Quantum many-body problems
+ all diagrams of this kind (11 more) 4th order[replace t(2) and repeat above 3rd order calculation]
+ all diagrams of this kind (6 more) 4th order
jiba
abij
Q
klcd
cdjl
abik
klcd
bdkl
acij
klcd
abkl
cdij
klcd
dblj
acik
abij
NtijabE
ttcdklijPttcdklabP
ttcdklttcdklabPijPNt
);3(
)1()1()(2
1)1()1()(
2
1
)1()1(4
1)1()1()()(
2
1;3
4
A few more diagramsA few more diagrams
17Quantum many-body problems
Calculations for 16OCalculations for 16O
Reasonably converged
• Idaho-A: No Coulomb• Coulomb: +0.7 MeV• Expt result is -8.0 MeV/A
(N3LO gives -6.95 MeV/A)
18Quantum many-body problems
Nuclear Properties
)(
ReaaeL TT
Also includes second-ordercorrections from the two-bodydensity.
19Quantum many-body problems
Calculated ground-state energy Experiment (E/A)
15O 6.0864 (PR-EOMCCSD) 7.4637 16O 6.9291 (CCSD) 7.9762 17O 6.6505 (PA-EOMCCSD) 7.7507
Very preliminary investigations in 16O neighbors(N3LO, 6 shells, not converged)
20Quantum many-body problems
Moving to larger model spaces always requires innovation
• CCSD code written for IBM using MPI. • Performs at 0.18 Tflops on 100 processors RAM. • Requires further optimization for new science.
8 480 --- 604k 212
N Single particle basis states
4He
(unknowns)
16O
(unknowns)
Matrix element
memory (Gbyte)
4 80 1792 24,960 0.165
5 140 4000 77,880 1.5
6 224 7,976 176k 10.1
7 336 14,112 345k 51
• Problem size increases by about a factor of 5 for each major oscillator shell• Number of unknowns by a factor of 2 ( for each step in N)• Number of unknowns by a factor of 20 (for 4 times the particles)
21Quantum many-body problems
Correcting the CCSD results by non-iterative methodsCorrecting the CCSD results by non-iterative methods
Goal: Find a method that will yield the completediagonalization result in a given model space
How do we obtain the triples correction?
How do our results compare with ‘exact’ results in a givenmodel space, for a given Hamiltonian?
“Completely Renormalized Coupled Cluster Theory”P. Piecuch, K. Kowalski, P.-D. Fan, I.S.O. Pimienta, andM.J. McGuire, Int. Rev. Phys. Chem. 21, 527 (2002)
22Quantum many-body problems
Completely renormalized CC in one slideCompletely renormalized CC in one slide
AT
N
n
ATA
CCe
eEH1
0
CC generating functionalT(A) = model correlation
ACC EE 00
0
then
if
abcijk
abcijk
ijkabc M
,
~
36
1
CCSDabcijk
ijkabc HM
)(~ CCSDTe
3)( ~~ TT CCSD
Pe
Leading order terms in the triples equation
Different choices of will yieldslightly different triples corrections
23Quantum many-body problems
Extrapolations: EOMCCSD level
Wloch, Dean, Gour, Hjorth-Jensen, Papenbrock, Piecuch, PRL, submitted (2005)
24Quantum many-body problems
Relationship between diagonalization and CC amplitudes
“Connected quadruples”
“Disconnected quadruples”
41
212
221344
311233
2122
11
24
1
2
1
2
16
12
1
TTTTTTTB
TTTTB
TTB
TB
CCSDCR-CCSD(T)
Supercomputing of the 1960s
1964, CDC6600, first commerciallysuccessful supercomputer; 9 MFlops
1965: The ion-channeling effect, one of the first materials physics discoveries made using computers, is key to the ion implantation used by current chip manufacturers to "draw" transistors with boron atoms inside blocks of silicon.
1969, CDC7600, 40 Mflops
1969, early days of the internet. 1967, Computer simulations used to calculate radiation dosages.
NERSC: IBM/RS6000 (9.1 TFlops peak)
Office of Science Computing Today
CRAY-X1 at ORNLPrecursor to NLCF (300 TF, FY06)2003, Turbulent flow in Tokomak Plasmas
2001, Dispersive waves in magnetic reconnection
Today’s science on today’s computers
Type IA supernova explosion(BIG SPLASH)
Fusion Stellarator
Accelerator design
Atmospheric models
Multiscale model of HIV
Materials: Quantum Corral
Structure of deutrons and nuclei
Reacting flow science
28Quantum many-body problems
Conclusions and perspectivesConclusions and perspectives
• Solution of the nuclear many-body problems requires extensive use of computational and mathematical tools. Numerical analysis becomes extremely important; methods from other fields (chemistry, CS) invaluable.
• Detailed investigation of triples corrections via CR-CCSD(T) indicates convergence at the triples level (in small space) for both He and O calculations. Promising result.
• Excited states calculated for the first time using EOMCCSD in O-16
• O-16 nearly converged at 8 oscillator shells (shells, not Nhw).
• Continuing work on the interaction; efforts to move to similarity transformed G are underway. • Open shell systems is the next big algorithmic step; EOMCCSD for larger model spaces underway; Ca-40 underway; V3N beginning• Time-dependent CC theory for reactions?
• Solution of the nuclear many-body problems requires extensive use of computational and mathematical tools. Numerical analysis becomes extremely important; methods from other fields (chemistry, CS) invaluable.
• Detailed investigation of triples corrections via CR-CCSD(T) indicates convergence at the triples level (in small space) for both He and O calculations. Promising result.
• Excited states calculated for the first time using EOMCCSD in O-16
• O-16 nearly converged at 8 oscillator shells (shells, not Nhw).
• Continuing work on the interaction; efforts to move to similarity transformed G are underway. • Open shell systems is the next big algorithmic step; EOMCCSD for larger model spaces underway; Ca-40 underway; V3N beginning• Time-dependent CC theory for reactions?
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