Solving HFB equation for superfluid Fermi systems in non-spherical coordinate-space

1. Solving HFB equation for superfluid Fermi systems in non-spherical coordinate-space 第14届核结构会议，湖州， 2012.4 Junchen Pei School of Physics, Peking University

2. Contents • Background: UNEDF project and supercomputing • HFB description for nuclei Coordinate-space Hartree-Fock-Bogoliubov solvers Continuum effects in weakly-bound nuclei • Some studies of cold atoms Larkin-Ovchinnikov superfluid phase • Summary and to do list Coordinate-space HFB Approach

3. SciDAC Project CS, Math, Physics, Chemistry, Materials, Energy, Industry Climate Chemistry Supernovae Burning QCD Materials Coordinate-space HFB Approach

4. Computing Facilities K-Computer Tianhe-1A Jaguar Coordinate-space HFB Approach

5. Developments in Computing Facilities • On the way from petascale(1015) to exascale(1018) k-computer(10petaflops), Jaguar(1.75PF to 20 PF Titan) • Architectures: to address a number of important problems that require even bigger computations Why Multi-core+GPU accelerators in Titan: power Power = Capacitance*Frequency*Voltage2+Leakage will be 50-100 MegaWatt for exascale computer! Moving data is expensive(data locality) Hybrid parallel computing(MPI+OpenMP, pthreads, Cuda) New numerical software(Lapack, Scalapack, Plasma, Magma) Coordinate-space HFB Approach

6. UNEDF Project Universal Nuclear Energy Density Functional UNEDF.org Coordinate-space HFB Approach

7. Supercomputing for nuclear physics Three components of modern science: experiments, theory, computing In nuclear physics, an extremely large of configuration space is involved and conventionally approximations (truncations) are made due to computing limitations. • Examples • Abinito structure in light nuclei • Abinito study of nuclear reactions • Microscopic calculations of nuclear fission • Mass table calculations for new DFT fittings Coordinate-space HFB Approach

8. Superheavies Challenges in nuclear physics RIB facilities offer unprecedented opportunities to access unstable nuclei Coordinate-space HFB Approach

9. Physics of drip-line nuclei • New collective excitation mode J. Dobaczewski, et al, Prog. Nucl. Part. Phys. 59, 432, 2007 Coordinate-space HFB Approach

10. Physics of drip-line nuclei • Pygmy resonances: Rich physics: neutron skin, symmetry energy, equation of state of neutron star, neutron capture in r-process • Deformed Halo • Island beyond drip-line (e.g., neutron star) S.G. Zhou, PRC, 2010 Model Implications • Understand effective nuclear interactions with extreme large isospin • Improving nuclear model prediction precision for and nuclear astrophysics Coordinate-space HFB Approach

11. HFBG.S.: BCS G.S.: HFB Theory HFB includes generalized quasi-particle correlations; while BCS is a specialquasiparticle transformation only on conjugate states. • The general HFB equation(or BdG) Deep bound states in BCS become quasiparticle resonances in HFB theory due to continuum coupling. HFB is superior to BCS for describing weakly-bound systems where continuum coupling becomes essential JP et al. PRC, 2011 Coordinate-space HFB Approach

12. HFB Quasiparticle spectrum • Diagonalization on single-particle basis • Direct diagonalization on box lattice (B-spline) • Very expensive; dense discretized continuum states • Outgoing boundary condition: difficult for deformed cases • In the complex k plane • These resonances are observable in experiments; widths depend very much on the pairing interaction; unique in HFB theory beyond BCS description • Continuum effects are crucial for description of drip-line nuclei; and also their excited states with QRPA. N.Michel, et al Identify resonance states that have the complex energy: E-iΓ/2 (widths); and purely scattering states are integrated over the contour. Coordinate-space HFB Approach

13. Coordinate-space HFB Advantages: weakly-bound systems and large deformations • The HO basis has a Gaussian form exp(-ar2) that decays too fast, while the density distribution decays exponentially exp(-kr). • Bound states, continuum and embedded resonances are treated on an equal footing; L2 discretization leads to a very large configuration space • Very accurate for describing weakly-bound nuclei with diffused density distributions; nuclear fission processes Infrastructures • 1D HFB code: HFBRAD is available • 2D HFB code: HFB-AX based on B-Spline techniques, beyond the capability of single-CPU computers, MPI becomes a must. • 3D HFB code: MADNESS-HFB with Multi-wavelets technqiues Coordinate-space HFB Approach

14. 2D Coordinate-space HFB • Techniques: B-spline Galerkin method, Broyden iteration, LAPACK, MPI +OpenMP • From 2006 summer at PKU HP cluster to 2008 spring in ORNL Cray-XT4 • Very precise and fast for weakly-bound nuclei and large deformation structures; benchmark for other calculations. J.Pei et al., Phys. Rev. C 78, 064308(2008) J.Pei et al., Eur. Phys. J. A (2009) Coordinate-space HFB Approach

15. 2D Coordinate-space HFB • A Show Case in SciDAC2009 • Benchmarking 110Zr HFB-AX M. Stoitsov et al, J. Phys. Conf. Ser. 180, 012802 (2009) N. Nikolov et al. Phys. Rev. C 83, 034305 (2011) Coordinate-space HFB Approach

16. Hybrid parallel calculation • MPI+OpenMP (400 cores for one nucleus takes 1 hour) Computing different blocks on different nodes (MPI) Multi-thread computing within a node(OpenMP) • Very large box calculations for large systems, important for weakly-bound systems and cold atomic gases. J.Peiet al., JPCS, 2012 Coordinate-space HFB Approach

17. Main • MPI 3D MADNESS-HFB Multi-resolution ADaptive Numerical Scientific Simulation • Multi-resolution: • CS infrastructure: Task-oriented: MPI, Global arrays, multi-threaded, futures (asynchronous computation), loadbalance 1. do A 2. do B if A 3. do C 4. do D if A • ThreadPool • WorldTaskQueue • 1. • 3. • 2. • 4. • Task dependencies: managed by Futures • API Coordinate-space HFB Approach

18. Multi-Wavelets • Decomposition Wavelet space at level n Scaling function expansion: expensive • Transformation • Truncation Coordinate-space HFB Approach

19. 3D MADNESS-HFB calculations J.C. Pei et al, JPCS, (in press), 2012 Coordinate-space HFB Approach

20. High energy continuum states • High-energy approximation (local density approximation) • This approximation works for HFB-popov equation for Bose gas; works for Bogoliubov de Gennes equations for Fermi gas. J. Reidl, A. Csordas, R. Graham, and P.Szepfalusy, Phys. Rev. A 59, 3816 (1999) X.J. Liu, H. Hu, P.D. Drummond, Phys. Rev. A 76,043605 (2007) • We follow this approximation for Skyrme HFB. quasiparticle spectrum HF energy Derivatives of effective mass, spin-orbit terms omitted for high energy states Coordinate-space HFB Approach

21. Continuum to observables Transform from the p-representations to the integral in terms of energy Coordinate-space HFB Approach

22. Quasiparticle continuum contribution Continuum contribution to densities (from 30 to 60 MeV) • Box solutions • Local-density approximation What we see from the figures: Local-density approximation works well for high energy continuum contributions. Continuum mainly impacts the pp channel. Continuum contribution increases as nuclei towards drip line Coordinate-space HFB Approach

23. Quasiparticle resonances Continuum level density: Phase shift: Zhou SG et al, J. Phys. B, 2009 • Remarkable widths of HFB resonances in weakly-bound nuclei • Box solutions are very precise • Provided an alternative way to study quasparticle resonances Coordinate-space HFB Approach

24. Quasiparticle resonances HFB resonances widths (KeV) in Ni90, compared to the exact Gamow HFB solutions Coordinate-space HFB Approach

25. Unitarity Fermi gases • Unitary limit: two body s-wave scattering length diverges: as→±∞ System is strongly correlated and its properties do not dependent on the value of scattering length as, providing clear many-body physics picture • Bertsch parameter (ξ~0.4): • Ideally suited for DFT description (A. Bulgac, PRA 76, 040502(2007), T. Papenbrock, PRA 72, 041603(2005)) Coordinate-space HFB Approach

26. Imbalanced Fermi gases • New phases: phase separation (in situ) • Experiments performed with highly elongated traps • A good chance for looking for FFLO (one-dimensional limit) Science 311(2006)503 Spin-up density Spin-down density polarized density Liao, et al, Nature 467, 567-569 (2010) Coordinate-space HFB Approach

27. Expected exotic FFLO pairing • In imbalanced Fermi systems, pairing with none-zero momentum can happen: Flude-Ferrell-Larkin-Ovchinnikov (FFLO) Oscillation pairing gap is expected; Modulated densities (crystallized). • It exists in many theoretical calculations, but difficult to find. • Some signatures in heavy fermions systems. Radovan, et al. Nature 425, 51, 2003. Coordinate-space HFB Approach

28. FFLO superfluid phases J. Pei et al. Eur. Phys. A (2009) G. Bertsch et al., Phys. Rev. A (2009) J. Pei et al., Phys. Rev A (R)(2010) Coordinate-space HFB Approach

29. Summary • Coordinate-space HFB approach works successfully for weakly-bound systems, however, solving the equation in 2D is very expensive and 3D is extremely expensive. • Novel numerical techniques are under developing for full 3D DFT calculations. Supercomputing is another direction. • Important issues to address: microscopic nuclear fission, cold Fermi systems, neutrons star crusts. • A great basis for continued developing method beyond mean-field: deformed continuum QRPA for collective motions. Coordinate-space HFB Approach

30. Collaborators • Thanks to them: W. Nazarewicz M.Stoitsov N. Schunck M. Kortelainen J.Dukelsky A. Kruppa A. Kerman J. Sheikh A. Bulgac M. Forbes G. Bertsch J. Dobaczewski G. Fann R. Harrison J. Hill D. Galindo J. Jia N. Nikolov P. Stevenson H. Nam F.R. Xu Y. Shi Coordinate-space HFB Approach

31. Thanks for your attention! peij@pku.edu.cn Coordinate-space HFB Approach