1. konstrukcja mikroskopowego jądrowego funkcjonału gęstości
Projekt UNEDF
konstrukcja mikroskopowego jądrowego funkcjonału gęstości
Witold Nazarewicz (Tennessee-Warszawa-Paisley)
UW, Warszawa, 19 marca 2008
Wstęp
Jądrowa teoria DFT
Projekt UNEDF
Strategia obliczeniowa
Perspektywy

2. Introduction

3. Nuclear Structure

4. number of nuclei ~ number of processors!

5. Nuclear Structure: the interaction
Effective-field theory (χPT)
potentials
Vlow-k: can it describe low-energy observables?
Quality two- and three-nucleon interactions exist
Not uniquely defined (local, nonlocal)
Soft and hard-core
The challenge is:
to understand their origin
to understand how to use them in nuclei
Bogner, Kuo, Schwenk, Phys. Rep. 386, 1 (2003)
N3LO: Entem et al., PRC68, (2003)
Epelbaum, Meissner, et al.

6. Ab initio: GFMC, NCSM, CCM (nuclei, neutron droplets, nuclear matter)
Quantum Monte Carlo (GFMC) 12C
No-Core Shell Model C
Coupled-Cluster Techniques O
Faddeev-Yakubovsky
Bloch-Horowitz …
Input:
Excellent forces based on the phase shift analysis
EFT based nonlocal chiral NN and NNN potentials
deuteron’s shape
GFMC: S. Pieper, ANL
1-2% calculations of A = 6 – 12 nuclear energies are possible
excited states with the same quantum numbers computed
The nucleon-based description works to <0.5 fm

7. Coupled Cluster Theory
David J. Dean, "Beyond the nuclear shell model” Physics Today, vol. 60, pg. 48 (November, 2007).

8.
G. Hagen et al.

10. Nuclear DFT

11. Density Functional Theory (introduced for many-electron systems)
The Hohenberg-Kohn theorem states that the ground state electron density minimizes the energy functional:
an universal
functional
P. Hohenberg and W. Kohn, “Inhomogeneous Electron Gas”, Phys. Rev. 136, B864 (1964)
M. Levy, Proc. Natl. Acad. Sci (USA) 76, 6062 (1979)
The original HK proof applies to systems with nondegenerate ground states. It proceeds by reductio ad absurdum, using the variational principle. A more general proof was given by Levy.
The minimum value of E is the ground state electronic energy
Since F is a unique functional of the charge density, the energy is uniquely defined by 
Electron density is the fundamental variable
proof of the Hohenberg-Kohn theorem is not constructive, hence the form of the universal functional F is not known
Since the density can unambiguously specify the potential, then contained within the charge density is the total information about the ground state of the system. Thus what was a 4N(3N)-variable problem (where N is the number of electrons, each one having three Cartesian variables and electron spin) is reduced to the four (three) variables needed to define the charge density at a point.

12. Density Functional Theory
Kohn-Sham equations
W. Kohn and L.J. Sham, "Self-Consistent Equations Including Exchange and Correlation Effects,” Phys. Rev. 140, A1133 (1965)
Takes into account shell effects
The link between T and r is indirect, via the orbitals f
The occupations n determine the electronic configuration
Orbitals f form a complete set. The occupations n are given by the Pauli principle (e.g., n=2 or 0). The variation of the functional can be done through variations of individual s.p. trial functions with a constraint on their norms. It looks like HF, but is replaced by E[r].
Kohn-Sham equation
Kohn-Sham potential (local!)
has to be evaluated
approximately

13. Mean-Field Theory ⇒ Density Functional Theory
Nuclear DFT
two fermi liquids
self-bound
superfluid
mean-field ⇒ one-body densities
zero-range ⇒ local densities
finite-range ⇒ gradient terms
particle-hole and pairing channels
Has been extremely successful. A broken-symmetry generalized product state does surprisingly good job for nuclei.

14. Nuclear Local s.p. Densities and Currents
isoscalar (T=0) density
isovector (T=1) density
isoscalar spin density
isovector spin density
current density
spin-current tensor density
kinetic density
kinetic spin density
+ analogous p-p densities and currents

15. Construction of the functional
Perlinska et al., Phys. Rev. C 69, (2004)
p-h density
p-p density
Most general second order expansion in densities and their derivatives
pairing
functional
Not all terms are equally important. Some probe specific observables

16. Justification of the standard Skyrme functional: DME
In practice, the one-body density matrix is strongly peaked around r=r’. Therefore, one can expand it around the mid-point:
The Skyrme functional was justified in such a way in, e.g.,
Negele and Vautherin, Phys. Rev. C5, 1472 (1972); Phys. Rev. C11, 1031 (1975)
Campi and Bouyssy, Phys. Lett. 73B, 263 (1978)
… but nuclear EDF does not have to be related to any given effective two-body force!
Actually, many currently used nuclear energy functionals are not related to a force
DME and EFT+RG

17. S. Cwiok, P.H. Heenen, W. Nazarewicz
Nuclear DFT: Large-scale Calculations
S. Cwiok, P.H. Heenen, W. Nazarewicz
Nature, 433, 705 (2005)
Stoitsov et al., PRL 98, (2007)
Global DFT mass calculations: HFB mass formula: m~700keV
Taking advantage of high-performance computers

18. Global calculations of ground-state spins and parities
for odd-mass nuclei
L. Bonneau, P. Quentin, and P. Möller, Phys. Rev. C 76, (2007)

19. Global calculations of the lowest 2+ states
Terasaki et al.
Sabbey et al.
Bertsch et al.
QRPA

20. A remark: physics of exotic nuclei is demanding
Interactions
Many-body
Correlations
Open
Channels
Interactions
Poorly-known spin-isospin components come into play
Long isotopic chains crucial
Open channels
Nuclei are open quantum systems
Exotic nuclei have low-energy decay thresholds
Coupling to the continuum important
Virtual scattering
Unbound states
Impact on in-medium Interactions
Configuration interaction
Mean-field concept often questionable
Asymmetry of proton and neutron Fermi surfaces gives rise to new couplings
New collective modes; polarization effects

21. UNEDF Project

22. http://www.scidac.gov 1peta=1015 flops SciDAC focus areas include:
Scientific Challenge Codes
Computing Systems and Mathematical Software
Collaboratory Software Infrastructure
Scientific Computing Hardware Infrastructure
Scientific Computing Software Infrastructure
1peta=1015 flops

23. SciDAC 2 Project: Building a Universal Nuclear
Energy Density Functional
Understand nuclear properties “for element formation, for properties of stars, and for present and future energy and defense applications”
Scope is all nuclei, with particular interest in reliable calculations of unstable nuclei and in reactions
Order of magnitude improvement over present capabilities
Precision calculations
Connected to the best microscopic physics
Maximum predictive power with well-quantified uncertainties
[See
by Bertsch, Dean, and Nazarewicz]

24. Other SciDAC Science at the Petascale Projects
Physics (Astro): Computational Astrophysics Consortium: Supernovae, Gamma Ray Bursts, and Nucleosynthesis, Stan Woosley (UC/Santa Cruz) [$1.9 Million per year for five years]
Physics (QCD): National Computational Infrastructure for Lattice Gauge Theory, Robert Sugar (UC/Santa Barbara) [$2.2 Million per year for five years]
Physics (Turbulence): Simulations of Turbulent Flows with Strong Shocks and Density Variations, Sanjiva Lele (Stanford) [$0.8 million per year for five years]
Physics (Petabytes): Sustaining and Extending the Open Science Grid: Science Innovation on a PetaScale Nationwide Facility, Miron Livny (U. Wisconsin) [$6.1 Million per year for five years]

25. Universal Nuclear Energy Density Functional
15 institutions
~50 researchers
physics
computer science
applied mathematics
foreign collaborators
Annual budget $3M

30. Accomplishments, Year-1, DFT
Physics
functionals benchmarked
masses (even-even and odd-A/BCS)
2+ excitations (in CSE, GCM, QRPA)
functional developed for finite superfluid system
odd-A HFB calculations initiated
-dependence of EDF studied with DME
projected-DFT studied
problems identified
cure proposed
Computing
HFB solvers benchmarked
time, memory requirements, parameter variations
HFB solvers improved
HFODD (improvement in computation time, odd-A capability added, Lipkin-Nogami capability added)
HFBTHO (blocking added)
ev8 (blocking added)
New HFB solvers on the way
TD_SLDA (time dependent)
HFB_PTG (improved asymptotics)
Wavelets
benchmarked in a one-body case (no SO)
SO advanced

31. Can dynamics be incorporated directly into the functional?
Example: Local Density Functional Theory for Superfluid Fermionic Systems: The Unitary Gas, Aurel Bulgac, Phys. Rev. A 76, (2007)
See also:
Density-functional theory for fermions
in the unitary regime
T. Papenbrock
Phys. Rev. A72, (2005)
Density functional theory for fermions
close to the unitary regime
A. Bhattacharyya and T. Papenbrock
Phys. Rev. A 74, (R) (2006)

32. Why us?

33. There is a zoo of nuclear functionals on the market
There is a zoo of nuclear functionals on the market. What makes us believe we will make a breakthrough?
Solid microscopic foundation
link to ab-initio approaches
limits obeyed (e.g., unitary regime)
Unique opportunities provided by coupling to CS/AM
Comprehensive phenomenology probing crucial parts of the functional
different observables probing different aspects
Stringent optimization protocol providing not only the coupling constants but also their uncertainties (theoretical errors)
Unprecedented international effort
Conclusion: we can deliver a well theoretically founded EDF, based on as much as possible ab intio input at this point in time

34. Building Connections: Bridging Approaches
Ab-initio - Ab-initio
Ab-initio - DFT
low-density neutron matter
finite nuclei (DME)
DFT - DFT
Increased coherence

35. Ab-initio - DFT Connection
UNEDF
Pack Forest meeting
One-body density matrix is the key quantity to study
“local DFT densities” can be expressed through (x,x’)
Testing the Density Matrix Expansion and beyond
UNEDF Homework
Introduce external potential
HO for spherical nuclei (amplitude of zero-point motion=1 fm)
2D HO for deformed nuclei
Density expressed in COM coordinates
Calculate x,x’) for 12C, 16O and
40,48,60Ca (CC)
Perform Wigner transform to relative and c-o-m coordinates q and s
Extract , J, 
Analyze data by comparing with results of DFT calculations and low-momentum expansion studies.
Go beyond I=0 to study remaining densities (for overachievers)
isospin
Negele and Vautherin: PRC 5, 1472 (1972)

36. Density Matrix Expansion for RG-Evolved Interactions
S.K. Bogner, R.J. Furnstahl et al.
see also:
EFT for DFT
R.J. Furnstahl
nucl-th/070204

37. DFT Mass Formula (can we go below 500 keV?)
How to optimize the search? SVD can help…
Fitting theories of nuclear binding energies
G. F. Bertsch, B. Sabbey and M. Uusnakki,
Phys. Rev. C71, (2005)
P.G. Reinhard 2004
need for error and covariance analysis
(theoretical error bars in unknown regions)
a number of observables need to be considered
(masses, radii, collective modes)
different terms sensitive to particular data
only data for selected nuclei should be used

38. How many parameters are really needed?
Spectroscopic (s.p.e.)
Global (masses)
Bertsch, Sabbey, and Uusnakki
Phys. Rev. C71, (2005)
Kortelainen, Dobaczewski, Mizuyama,
Toivanen, arXiv:
New optimization strategy and protocol needed

39. 41 participants

40. The bottom line From J. Dobaczewski
Spectroscopic-quality energy density functional  correct description of positions and evolution of single-particle levels.
Single-particle levels  correct description of one-particle separation energies with all polarization effects included.
Within the EDF method, shape and spin polarization effects in doubly-magic nuclei are relatively small – much smaller than deviations from data.
Dependence of total and single-particle energies on coupling constants is very linear.
No fits without error estimates and error propagation!
Extensions beyond the simple Skyrme functionals are mandatory.
From J. Dobaczewski

41. Computational Strategy

44. From Ian Thompson

45. (n+AXi) at energy Eprojectile Computational Workflow
Trans. Density (r)
Ground state Excited state Continuum states
(n,Z)
VNN, VNNN…
Veff
Structure Model Classes: HF, RPA, CI, CC, … Codes: many
Transition Density
Codes: trdens
Folding
Transition Potential V(r)
local; Later: non-local, exchange
Decay of 
Coupled Channels Codes: fresco, nrc
Hauser-Feshbach
Residue A-m
Data Bank
Voptical, n+A scattering
equil emission
preeq emission
(n+AXi) at energy Eprojectile Computational Workflow
Key:
Code Modules
User Inputs/Outputs
Exchanged Data
Need Clarification
From Ian Thompson

46. Connections to computational science
1Teraflop=1012 flops
1peta=1015 flops (next 2-3 years)
1exa=1018 flops (next 10 years)
Jaguar Cray XT4 at ORNL
No. 2 on Top500
11,706 processor nodes
Each compute/service node contains 2.6 GHz dual-core AMD Opteron processor and 4 GB/8 GB of memory
Peak performance of over 119 Teraflops
250 Teraflops after Dec.'07 upgrade
600 TB of scratch disk space

47. Example: Large Scale Mass Table Calculations
Science scales with processors
M. Stoitsov
HFB+LN mass table, HFBTHO
Even-Even Nuclei
The SkM* mass table contains 2525 even-even nuclei
A single processor calculates each nucleus 3 times (prolate, oblate, spherical) and records all nuclear characteristics and candidates for blocked calculations in the neighbors
Using 2,525 processors - about 4 CPU hours (1 CPU hour/configuration)
All Nuclei
9,210 nuclei
599,265 configurations
Using 3,000 processors - about 25 CPU hours

48. Perspectives

49. What is needed/essential?
Young talent
Focused effort
Large collaborations
Data from terra incognita
RNB facilities provide strong
motivation!
High-performance computing
UNEDF

50. Radioactive Ion Beam Facilities Timeline
HIE-ISOLDE
ISOLDE
ISAC-II
ISAC-I
SPIRAL2
SPIRAL
FAIR
SIS
RIBF
RARF
NSCL
HRIBF
In Flight
ISOL
Fission+Gas Stopping
Beam on target
FRIB
2000
2005
2010
2015
2020

51. Conclusions
Exciting science; old paradigms revisited
Interdisciplinary (quantum many-body problem, cosmos,…)
Relevant to society (energy, medicine, national security, …)
Theory gives the mathematical formulation of our understanding and predictive ability
New-generation computers provide unprecedented opportunities
Large coherent international theory effort is needed to make a progress
Guided by data on short-lived nuclei, we are embarking on a comprehensive study of all nuclei based on the most accurate knowledge of the strong inter-nucleon interaction, the most reliable theoretical approaches, and the massive use of the computer power available at this moment in time. The prospects look good.
Thank You

52. Backup

53. Weinberg’s Laws of Progress in Theoretical Physics
From: “Asymptotic Realms of Physics” (ed. by Guth, Huang, Jaffe, MIT Press, 1983)
First Law: “The conservation of Information” (You will get nowhere by churning equations)
Second Law: “Do not trust arguments based on the lowest order of perturbation theory”
Third Law: “You may use any degrees of freedom you like to describe a physical system, but if you use the wrong ones, you’ll be sorry!”
D. Furnstahl, INT Fall’05

54. Local Density Approximation (LDA)
for the exchange+correlation potential:
One performs many-body calculations for an infinite system with a constant density 
The resulting energy per particle is used to extract the xc-part of the energy, exc(r), which is a function of 
The LDA of a finite system with variable density (r) consists in assuming the local xc-density to be that of the corresponding infinite system with density  =(r):
The formalism can be extended to take spin degrees of freedom, by introducing spin-up and spin-down densities (‘Local Spin Density’ formalism, LSD or LSDA)
The LDA is often used in nuclear physics (Brueckner et al., Negele). It states that the G-matrix at any place in a finite nucleus is the same as that for nuclear matter at the same density, so that locally one can calculate G-matrix as in nuclear matter calculation.
Pairing extension: DFT for semiconductors; also: the Hartree-Fock-Bogoliubov (HFB) or Bogoliubov-de Gennes (BdG) formalism.

55. Normalized Q.P. Spectrum
Nicolas Schunck, Oak Ridge, 2008
SLy4 + Time Odd Fields (Gauge Invariance)
SLy4
Experiment

56. Example: 3-D Adaptive Multiresolution Method for Atomic Nuclei (taking advantage of jpeg200 technology) N
State
(2-cosh)
Wavelet
HO basis
1-cosh 1
1/2 (+) 2
1/2 (-) - 3 4 5
3/2 (-) 6 7
3/2 (+) 8 9 10
Comparison between adaptive 3-D multi-wavelet (iterative) and direct 2D and 3D harmonic oscillator (HO) bases (direct)
One-body test potential (currently no spin-orbit)
Pöschl-Teller-Ginocchio (PTG): analytical solutions
2-cosh : simulates fission process; HO expansion expected to work poorly at large separations
PTG Exact
2D-HO
3D-HO
Wavelets (Prec.: 10-3)
-0.320
G. Fann, N. Schunck, et al.

57. Prog. Part. Nucl. Phys. 59, 432 (2007)