Quantum many-body problems 1 David J. Dean ORNL Outline I.Perspectives on RIA II.Ab initio coupled-cluster theory applied to 16 O A.The space and the Hamiltonian B.The CC method C.Ground- and excited state implementation: CCSD D.Triples Corrections: CR-CCSD(T) III.Conclusions

Quantum many-body problems 2 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. RIA Nuclei Interactions VNN, V3N Chiral EFT Microscopic nuclear structure theory Applications: beyond SM astrophysics Motivate, compare to experiments Energy density functional Nuclei and RIA

Quantum many-body problems 3 The challenge of neutron rich nuclei ‘normal’‘rare’ Half-lifeLong/stableShort SizeCompactHalos ContinuumUnimportantImportant Shell closuresNormal magic numbers New magic numbers Scattering Cross section ‘Normal’Enhanced

Quantum many-body problems 4 The opportunity: connection of nuclear structure and reaction 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 problems 5 Quantum many-body problem is everywhere Molecular scale: conductance through molecules. Delocalized conduction orbitals Molecules Nanoscale ~10 nm 2-d quantum dot in strong magnetic field. The field drives localization. Nuclei ~5 fm 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 many-body problems 6 Begin with a bare NN (+3N) Hamiltonian Basis expansions: Choose the method of solution Determine the appropriate basis Generate H eff Nucleus4 shells7 shells 4He4E49E6 8B4E85E13 12C6E114E19 16O3E149E24 Oscillator single-particle basis states Many-body basis states Steps toward solutions Bare (GFMC) Basis expansion method basis H eff

Quantum many-body problems 7 Morten Hjorth-Jensen (Effective interactions) Maxim Kartamychev (3-body forces in nuclei) Gaute Hagen (effective interactions; to ORNL in May) The ORNL-Oslo-MSU collaboration on 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 V 3N ) Thomas Papenbrock David Bernholdt (Computer Science and Mathematics) Trey White, Kenneth Roche (Computational Science) Research Plan -- Excited states (!) -- Observables (!) -- Triples corrections (!) -- Open shells -- V 3N -- 50<A< Reactions -- TD-CCSD??

Quantum many-body problems 8 Choice of model space and the G-matrix Q-Space P-Space +… = + h p G ph intermediate states CC-ph h p Use BBP to eliminate w-dependence below fermi surface.

Quantum many-body problems 9 In the near future: similarity transformed H Advantage: no parameter dependence in the interaction Current 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, (2000)

Quantum many-body problems 10 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.

Quantum many-body problems 11 Coupled Cluster Theory Correlated Ground-State wave function Correlation operator Reference Slater determinant Energy Amplitude equations Nomenclature Coupled-clusters in singles and doubles (CCSD) …with triples corrections CCSD(T); Dean & Hjorth-Jensen, PRC69, (2004); Kowalski, Dean, Hjorth-Jensen, Papenbrock, Piecuch, PRL92, (2004)

Quantum many-body problems 12 T 1 amplitudes from: Derivation of CC equations Note T 2 amplitudes also come into the equation.

Quantum many-body problems 13 T 2 amplitudes from: An interesting mess. But solvable…. Nonlinear terms in t2 (4 th order)

Quantum many-body problems 14 Correspondence with MBPT 2 nd order 3 rd order

Quantum many-body problems 15 + all diagrams of this kind (11 more) 4 th order [replace t(2) and repeat above 3 rd order calculation] + all diagrams of this kind (6 more) 4 th order A few more diagrams

Quantum many-body problems 16 Ground state of helium: Idaho-A We also investigated: Need for similarity transformed H eff (undearway).

Quantum many-body problems 17 Calculations for 16 O Reasonably converged Idaho-A: No Coulomb Coulomb: +0.7 MeV Expt result is -8.0 MeV/A (N3LO gives MeV/A)

Quantum many-body problems 18 Nuclear Properties Also includes second-order corrections from the two-body density.

Quantum many-body problems 19 Calculated ground-state energyExperiment (E/A) 15 O (PR-EOMCCSD) O (CCSD) O (PA-EOMCCSD) Very preliminary investigations in 16 O neighbors (N3LO, 6 shells, not converged)

Quantum many-body problems 20 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 k212 NSingle particle basis states 4 He (unknowns) 16 O (unknowns) Matrix element memory (Gbyte) , , ,976176k ,112345k51 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)

Quantum many-body problems 21 Correcting the CCSD results by non-iterative methods Goal: Find a method that will yield the complete diagonalization result in a given model space How do we obtain the triples correction? How do our results compare with ‘exact’ results in a given model space, for a given Hamiltonian? “Completely Renormalized Coupled Cluster Theory” P. Piecuch, K. Kowalski, P.-D. Fan, I.S.O. Pimienta, and M.J. McGuire, Int. Rev. Phys. Chem. 21, 527 (2002)

Quantum many-body problems 22 Completely renormalized CC in one slide CC generating functional T(A) = model correlation Leading order terms in the triples equation Different choices of  will yield slightly different triples corrections

Quantum many-body problems 23 Extrapolations: EOMCCSD level Wloch, Dean, Gour, Hjorth-Jensen, Papenbrock, Piecuch, PRL, submitted (2005)

Quantum many-body problems 24 Relationship between diagonalization and CC amplitudes “Connected quadruples” “Disconnected quadruples” CCSD CR-CCSD(T)

Supercomputing of the 1960s 1964, CDC6600, first commercially successful 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 ORNL Precursor 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

Quantum many-body problems 28 Conclusions 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; V 3N beginning Time-dependent CC theory for reactions?

Quantum many-body problems 29 Figure 1: The landscape and the models. COUPLED-CLUSTER METHODS We gratefully acknowledge the support by DOE (MSU, UT, ORNL), NSF (MSU), Alfred P. Sloan Foundation (MSU), Research Council of Norway (UO), and ORNL LDRD (FY02/03) and SEED (FY05/06) funds.