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

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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 2 Coordinate-space HFB Approach

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

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

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Developments in Computing Facilities On the way from petascale(10 15 ) to exascale(10 18 ) 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*Voltage 2 +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) -5- Coordinate-space HFB Approach

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

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Supercomputing for nuclear physics Coordinate-space HFB Approach -7- Examples Ab inito structure in light nuclei Ab inito study of nuclear reactions Microscopic calculations of nuclear fission Mass table calculations for new DFT fittings 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.

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

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

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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) Coordinate-space HFB Approach -10- S.G. Zhou, PRC, 2010 Model Implications Understand effective nuclear interactions with extreme large isospin Improving nuclear model prediction precision for and nuclear astrophysics Model Implications Understand effective nuclear interactions with extreme large isospin Improving nuclear model prediction precision for and nuclear astrophysics

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HFB Theory Coordinate-space HFB Approach -11- 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 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 HFB G.S.: BCS G.S.: The general HFB equation(or BdG) HFB includes generalized quasi-particle correlations; while BCS is a special quasiparticle transformation only on conjugate states. JP et al. PRC, 2011

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-12- HFB Quasiparticle spectrum 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 -i Γ/2 (widths); and purely scattering states are integrated over the contour. 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 Coordinate-space HFB Approach

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Coordinate-space HFB Coordinate-space HFB Approach -13- Advantages: weakly-bound systems and large deformations The HO basis has a Gaussian form exp(-ar 2 ) that decays too fast, while the density distribution decays exponentially exp(-kr). Bound states, continuum and embedded resonances are treated on an equal footing; L 2 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

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2D Coordinate-space HFB Coordinate-space HFB Approach -14- 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)

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

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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. Coordinate-space HFB Approach -16- J.Pei et al., JPCS, 2012

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

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

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

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-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

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

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Quasiparticle continuum contribution Coordinate-space HFB Approach -22- Continuum contribution to densities (from 30 to 60 MeV) Box solutions Local-density approximation 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 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

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

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

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Unitarity Fermi gases Coordinate-space HFB Approach -25- Unitary limit: two body s-wave scattering length diverges: a s →±∞ System is strongly correlated and its properties do not dependent on the value of scattering length a s, 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))

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

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Expected exotic FFLO pairing Coordinate-space HFB Approach -27- 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.

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FFLO superfluid phases Coordinate-space HFB Approach -28- 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) 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)

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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 -29-

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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 Coordinate-space HFB Approach -30- G. Fann R. Harrison J. Hill D. Galindo J. Jia N. Nikolov P. Stevenson H. Nam F.R. Xu Y. Shi

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

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