1. 1 Nuclear DFT: Questions and Challenges Witold Nazarewicz (Tennessee) First FIDIPRO-JSPS Workshop on Energy Density Functionals in Nuclei October 25-27, 2007, Keurusselka, (Jyvä s kylä) FINLAND Introduction Recent/relevant examples Questions and challenges Perspectives

2. Introduction

4. number of nuclei ~ number of processors!

5. Density Functional Theory (introduced for many-electron systems) The Hohenberg-Kohn theorem states that the ground state electron density minimizes the energy functional: P. Hohenberg and W. Kohn, “Inhomogeneous Electron Gas”, Phys. Rev. 136, B864 (1964) M. Levy, Proc. Natl. Acad. Sci (USA) 76, 6062 (1979) an universal functional 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.

6. W. Kohn and L.J. Sham, "Self-Consistent Equations Including Exchange and Correlation Effects,” Phys. Rev. 140, A1133 (1965) Density Functional Theory Kohn-Sham equations Takes into account shell effects The link between T and  is indirect, via the orbitals  The occupations n determine the electronic configuration Orbitals  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[  ]. Kohn-Sham equation Kohn-Sham potential (local!) has to be evaluated approximately

7. 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, e xc (  ), 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.

8. Mean-Field Theory ⇒ Density Functional Theory 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. Nuclear DFT two fermi liquids self-bound superfluid

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

10. 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

11. Not all terms are equally important. Some probe specific observables pairing functional Construction of the functional Perlinska et al., Phys. Rev. C 69, 014316 (2004) Most general second order expansion in densities and their derivatives p-h densityp-p density

12. A remark: physics of exotic nuclei is demanding Interactions Poorly-known spin-isospin components come into play Long isotopic chains crucial Interactions Many-body Correlations Open Channels 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 (Intruders and the islands of inversion) New collective modes; polarization effects

13. Nuclear DFT Global properties, global calculations * Global DFT mass calculations: HFB mass formula:  m~700keV Taking advantage of high-performance computers M. Stoitsov et al. S. Goriely et al., ENAM’04

14. S. Cwiok, P.H. Heenen, W. Nazarewicz Nature, 433, 705 (2005) Mass differences, global calculations Stoitsov et al., PRL 98, 132502 (2007)

15. Global calculations of ground-state spins and parities for odd-mass nuclei L. Bonneau, P. Quentin, and P. Möller, Phys. Rev. C 76, 024320 (2007)

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

17. Isospin dynamics important! High-spin terminating states Zdunczuk et al.,Phys.Rev. C71, 024305(2005) Stoitcheva et al., Phys. Rev. C 73, 061304(R) (2006) Excellent examples of single- particle configurations Weak configuration mixing Spin polarization; probing time- odd terms! Experimental data available

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

19. Different deformabilities! Shell effects in metastable minima seem to be under control P.H. Heenen et al., Phys. Rev. C57, 1719 (1998) Important data needed to fix the deformability of the NEDF: absolute energies of SD states absolute energies of HD states Advantages: large elongations weak mixing with ND structures Example: Surface Symmetry Energy Microscopic LDM and Droplet Model Coefficients: P.G. Reinhard et al. PRC 73, 014309 (2006)

20. Beyond Mean Field nuclear collective dynamics, correlation energy Large-scale GCM+proj correlation energy calculations M. Bender et al.: PRC 73, 034322 (2006); PRL 94, 102503 (2005) PN Projected HFB equations M. Stoitsov et al.: PRC75, 014308 (2007)

21. Questions and challenges

22. Intrinsic-Density Functionals J. Engel, Phys. Rev. C75, 014306 (2007) Generalized Kohn-Sham Density-Functional Theory via Effective Action Formalism M. Valiev, G.W. Fernando, cond-mat/9702247 B.G. Giraud, B.K. Jennings, and B.R. Barrett, arXiv:0707.3099 (2007); B.G. Giraud, arXiv:0707.3901 (2007) How to extend DFT to finite, self-bound systems?

23. What are the missing pieces? What is density dependence? (ph and pp channels) Spin-isospin sector (e.g., tensor) Momentum dependence of the effective mass? Induced interaction Isovector and isoscalar

24. How to root nuclear DFT in a microscopic theory? ab-initio - DFT connection NN+NNN - EDF connection (via EFT+RG)

25. Ab-initio - DFT Connection 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 12 C, 16 O and 40,48,60 Ca (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) Negele and Vautherin: PRC 5, 1472 (1972) isospin UNEDF Pack Forest meeting

26. 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

27. DFT Mass Formula (can we go below 500 keV?) 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 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, 054311(2005)

28. http://orph02.phy.ornl.gov/workshops/lacm08/unedf.html Nicolas Schunck http://www.phy.ornl.gov/theory/papenbro/jan08.htm Thomas Papenbrock

29. J. Dobaczewski and J. Dudek, Phys. Rev. C52, 1827 (1995) M. Bender et al., Phys. Rev. C65, 054322 (2002) H. Zdunczuk, W. Satula and R. Wyss, Phys. Rev. C71, 024305 (2005) very poorly determined Can be adjusted to the Landau parameters Important for all I>0 states (including low-spin states in odd-A and odd-odd nuclei) Impact beta decay Influence mass filters (including odd-even mass difference) Limited experimental data available How to parameterize time-odd pieces?

30. DFT for odd-A and odd-odd nuclei For J>0 odd-T fields come into play For J>0 nuclear density is usually triaxial Many quasi-particle states need to be considered Pairing has to be treated carefully as blocking can result in a phase transition In odd-odd nuclei, residual p-n interaction appears Good agreement with even-even masses can be misleading N. Schunck et al. K. Rutz, M. Bender, P. -G. Reinhard and J. A. Maruhn Phys. Lett. B468, 1 (1999) HFB+SLy4, full blocking, alpha chains in SHE S. Ćwiok, W. Nazarewicz, and P. H. Heenen Phys. Rev. Lett. 83, 1108 (1999) Recent work by M. Zalewski et al. on SO, tensor, odd-T fields, and self-consistency odd-T≠0

31. How to restore broken symmetry in DFT? J. Dobaczewski et al., Phys. Rev. C (2007); arXiv:0708.0441 The transition density matrices contains complex poles. Some cancellation appears if the ph and pp Hamiltonians are the same The projection operator cannot be defined uniquely Problems with fractional density dependence Projected DFT yields questionable results when the pole appears close to the integration contour. This often happens when dealing with PESs see also:M. Bender, T. Duguet, D. Lacroix, in preparation. S. Krewald et al.,Phys. Rev. C 74, 064310 (2006).

32. 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, 040502 (2007) See also: Density-functional theory for fermions in the unitary regime T. Papenbrock Phys. Rev. A72, 041603 (2005) Density functional theory for fermions close to the unitary regime A. Bhattacharyya and T. Papenbrock Phys. Rev. A 74, 041602(R) (2006)

33. Perspectives

34. Young talent Focused effort Large theoretical collaborations UNEDF (US+collaborators) EFES (JP) in works: ARTHENS, EURONS-2 networks (EU) Data from terra incognita High-performance computing Interaction with computer scientists What is needed/essential?

35. 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 1Teraflop=10 12 flops 1peta=1000 tera

36. 36 Example: Large Scale Mass Table Calculations Science scales with processors  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)  The even-even calculations define 250,754 configurations in odd-A and odd-odd nuclei assuming 0.5 MeV threshold for the blocking candidates  Using 10,000 processors - about 25 CPU hours Even-Even Nuclei Jaguar@ Odd and odd-odd Nuclei M. Stoitsov, HFB+LN mass table, HFBTHO

37. Summary A. Staszczak, J. Dobaczewski, W. Nazarewicz, in preparation Bimodal fission in nuclear DFT nucl-th/0612017 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.

38. Backup

39.  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 Example: 3-D Adaptive Multiresolution Method for Atomic Nuclei (taking advantage of jpeg200 technology) PTG Exact2D-HO3D-HOWavelets (Prec.: 10 -3 ) -39.740 042-39.740 041-39.740 021-39.740 -18.897 703-18.897 675-18.897 059-18.898 -0.320 521-0.314 751-0.256 288-0.320 N State (2-cosh) WaveletHO basis1-cosh 11/2 (+)-22.1542-22.0963-22.2396 21/2 (-)-22.1540-22.0961- 31/2 (+)-9.1011-9.0200-9.2115 41/2 (-)-9.0927-9.0121- 53/2 (-)-9.0926-9.0121-9.2115 61/2 (+)-9.0913-9.0107- 73/2 (+)-9.0912-9.0107-9.2115 81/2 (-)-9.0857-9.0051- 91/2 (+)-1.6541-1.6008-1.6225 101/2 (-)-1.4472-1.3944- G. Fann, N. Schunck, et al.

40. The origin of SO splitting can be attributed to 2-body SO and tensor forces, and 3-body force R.R. Scheerbaum, Phys. Lett. B61, 151 (1976); B63, 381 (1976); Nucl. Phys. A257, 77 (1976); D.W.L. Sprung, Nucl. Phys. A182, 97 (1972); C.W. Wong, Nucl. Phys. A108, 481 (1968); K. Ando and H. Bando, Prog. Theor. Phys. 66, 227 (1981); R. Wiringa and S. Pieper, Phys. Rev. Lett. 89, 182501 (2002) The maximum effect is in spin-unsaturated systems Discussed in the context of mean field models: Fl. Stancu, et al., Phys. Lett. 68B, 108 (1977); M. Ploszajczak and M.E. Faber, Z. Phys. A299, 119 (1981); J. Dudek, WN, and T. Werner, Nucl. Phys. A341, 253 (1980); J. Dobaczewski, nucl-th/0604043; Otsuka et al. Phys. Rev. Lett. 97, 162501 (2006); Lesinski et al., arXiv:0704.0731,… …and the nuclear shell model: T. Otsuka et al., Phys. Rev. Lett. 87, 082502 (2001); Phys. Rev. Lett. 95, 232502 (2005) Example: Spin-Orbit and Tensor Force (among many possibilities) j<j< j>j> FF 2, 8, 20 Spin-saturated systems j<j< j>j> FF 28, 50, 82, 126 Spin-unsaturated systems

41. Importance of the tensor interaction far from stability [523]7/2 [411]1/2 Proton emission from 141 Ho