Computing nuclei and cold atoms

Solving HFB equation for superfluid Fermi systems in non-spherical coordinate-spaceJunchen Pei School of Physics, Peking University14 2012.4

ContentsBackground: UNEDF project and supercomputingHFB description for nuclei Coordinate-space Hartree-Fock-Bogoliubov solvers Continuum effects in weakly-bound nucleiSome studies of cold atoms Larkin-Ovchinnikov superfluid phaseSummary and to do list 2Coordinate-space HFB ApproachSciDAC Project 3Coordinate-space HFB Approach

ChemistryCS, Math, Physics, Chemistry, Materials, Energy, Industry

SupernovaeClimateBurningMaterialsQCDComputing Facilities -4-Coordinate-space HFB Approach

K-ComputerTianhe-1AJaguar

Developments in Computing FacilitiesOn 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) -5-Coordinate-space HFB ApproachUNEDF Project -6-Coordinate-space HFB Approach

UNEDF.org

Universal Nuclear Energy Density FunctionalSupercomputing for nuclear physics Coordinate-space HFB Approach -7- ExamplesAb inito structure in light nucleiAb inito study of nuclear reactionsMicroscopic calculations of nuclear fissionMass table calculations for new DFT fittingsThree components of modern science: experiments, theory, computingIn nuclear physics, an extremely large of configuration space is involved and conventionally approximations (truncations) are made due to computing limitations.

Challenges in nuclear physics

SuperheaviesRIB facilities offer unprecedented opportunities to access unstable nuclei -8-Coordinate-space HFB Approach2012/4/218ProgressPhysics of drip-line nucleiCoordinate-space HFB Approach -9-

New collective excitation modeJ. Dobaczewski, et al, Prog. Nucl. Part. Phys. 59, 432, 20072012/4/21Progress9Physics of drip-line nucleiPygmy resonances:Rich physics: neutron skin, symmetry energy, equation of state of neutron star, neutron capture in r-processDeformed Halo

Island beyond drip-line (e.g., neutron star)Coordinate-space HFB Approach -10-

S.G. Zhou, PRC, 2010 Model ImplicationsUnderstand effective nuclear interactions with extreme large isospin

Improving nuclear model prediction precision for and nuclear astrophysicsHFB 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

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, 20112012/4/21Progress11

-12-HFB Quasiparticle spectrumIn the complex k plane

These resonances are observable in experiments; widths depend very much on the pairing interaction; unique in HFB theory beyond BCS descriptionContinuum 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/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 statesOutgoing boundary condition: difficult for deformed casesCoordinate-space HFB ApproachCoordinate-space HFB Coordinate-space HFB Approach -13- Advantages: weakly-bound systems and large deformationsThe 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 spaceVery 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 technqiues2D 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-XT4Very 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)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 2009HFB-AX110ZrHybrid parallel calculationMPI+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, 20123D MADNESS-HFB Coordinate-space HFB Approach -17-

API1. do A2. do B if A3. do C 4. do D if AWorldTaskQueueThreadPoolTask dependencies: managed by Futures1.3.2.4.MainMPI Multi-resolution ADaptive Numerical Scientific SimulationCS infrastructure: Task-oriented: MPI, Global arrays, multi-threaded, futures (asynchronous computation), loadbalanceMulti-resolution:Multi-Wavelets Coordinate-space HFB Approach -18-

TruncationDecomposition

Wavelet space at level n Scaling function expansion: expensive

Transformation3D MADNESS-HFB calculations Coordinate-space HFB Approach -19-

J.C. Pei et al, JPCS, (in press), 2012 -20-High energy continuum statesHigh-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 spectrumHF energyDerivatives of effective mass, spin-orbit terms omitted for high energy statesCoordinate-space HFB Approach -21-Continuum to observables

Transform from the p-representations to the integral in terms of energyCoordinate-space HFB ApproachQuasiparticle continuum contribution Coordinate-space HFB Approach -22-Continuum contribution to densities (from 30 to 60 MeV) Box solutions Local-density approximationWhat 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

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 Zhou SG et al, J. Phys. B, 20092012/4/21Progress23Quasiparticle resonances Coordinate-space HFB Approach -24-

HFB resonances widths (KeV) in Ni90, compared to the exact Gamow HFB solutions Unitarity Fermi gases Coordinate-space HFB Approach -25-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 (~0.4): Ideally suited for DFT description (A. Bulgac, PRA 76, 040502(2007), T. Papenbrock, PRA 72, 041603(2005))

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 densitypolarized densityScience 311(2006)503Expected 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.

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)

2012/4/21Progress28SummaryCoordinate-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-CollaboratorsThanks to them:W. NazarewiczM.StoitsovN. SchunckM. KortelainenJ.DukelskyA. KruppaA. KermanJ. SheikhA. BulgacM. ForbesG. BertschJ. Dobaczewski

Coordinate-space HFB Approach -30-G. FannR. HarrisonJ. HillD. GalindoJ. JiaN. NikolovP. StevensonH. Nam

F.R. XuY. ShiCoordinate-space HFB Approach -31-Thanks for your attention!

peij@pku.edu.cn