quantum many-body problems 1 david j. dean ornl outline i.perspectives on ria ii.ab initio...

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


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


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’.


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


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


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??


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.


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)


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.


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)


T1 amplitudes from:T1 amplitudes from: 0exp)exp( THTai

Derivation of CC equationsDerivation of CC equations

Note T2 amplitudes also come into the equation.


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


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


+ 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


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)


Nuclear Properties

)(

ReaaeL TT

Also includes second-ordercorrections from the two-bodydensity.


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)


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)


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)


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


Extrapolations: EOMCCSD level

Wloch, Dean, Gour, Hjorth-Jensen, Papenbrock, Piecuch, PRL, submitted (2005)


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


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?

quantum many-body problems 1 david j. dean ornl outline i.perspectives on ria ii.ab initio...

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