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

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

Coordinate-space HFB Approach -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


SciDAC Project

Chemistry

CS, Math, Physics, Chemistry, Materials, Energy, Industry

SupernovaeClimate

BurningMaterials

QCD

Coordinate-space HFB Approach

Computing Facilities

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K-Computer Tianhe-1A

Jaguar


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)

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UNEDF Project

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UNEDF.org

Universal Nuclear Energy Density Functional


Supercomputing for nuclear physics

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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.


Challenges in nuclear physics

SuperheaviesRIB facilities offer unprecedented opportunities to access unstable nuclei

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Physics of drip-line nuclei

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New collective excitation mode New collective excitation modeJ. Dobaczewski, et al, Prog. Nucl. Part. Phys. 59, 432, 2007


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)

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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


HFB Theory

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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.:HFB G.S.:

BCS G.S.:

The general HFB equation(or BdG) 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


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-iΓ/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

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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


2D Coordinate-space HFB

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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)

J.Pei et al., Phys. Rev. C 78, 064308(2008) J.Pei et al., Eur. Phys. J. A (2009)


2D Coordinate-space HFB

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Benchmarking

M. Stoitsov et al, J. Phys. Conf. Ser. 180, 012802 (2009)M. Stoitsov et al, J. Phys. Conf. Ser. 180, 012802 (2009)

N. Nikolov et al. Phys. Rev. C 83, 034305 (2011)N. Nikolov et al. Phys. Rev. C 83, 034305 (2011)

A Show Case in SciDAC 2009

HFB-AX

110Zr


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.

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J.Pei et al., JPCS, 2012


3D MADNESS-HFB

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API 1. do A2. do B if A3. do C 4. do D if A

WorldTaskQueue ThreadPoolTask dependencies: m

anaged by Futures

1.3.2.4.

Main MPI

Multi-resolution ADaptive Numerical Scientific Simulation

CS infrastructure:

Task-oriented: MPI, Global arrays, multi-threaded, futures (asynchronous computation), loadbalance

• Multi-resolution:


Multi-Wavelets

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Truncation

Decomposition

Wavelet space at level n

Scaling function expansion: expensive

Transformation


3D MADNESS-HFB calculations

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J.C. Pei et al, JPCS, (in press), 2012


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


Continuum to observables

Transform from the p-representations to the integral in terms of energy


Quasiparticle continuum contribution

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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


Quasiparticle resonances

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Continuum level density:Continuum level density:

Phase shift: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


Quasiparticle resonances

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HFB resonances widths (KeV) in Ni90, compared to the exact Gamow HFB solutions


Unitarity Fermi gases

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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))


Imbalanced Fermi gases

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New phases: phase separation (in situ)

Experiments performed with highly elongated traps

A good chance for looking for FFLO (one-dimensional limit)

2 22 20 0

1( , ) (1 exp( ( ) / ))

2 r zU r z U r zw Uw

Liao, et al, Nature 467, 567-569 (2010)

Spin-up density

Spin-down density

polarized density

Science 311(2006)503


Expected exotic FFLO pairing

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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.

0( ) niq xn

n

x C e r rr


FFLO superfluid phases

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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)


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.

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Collaborators

Thanks to them:W. NazarewiczM.StoitsovN. SchunckM. KortelainenJ.DukelskyA. KruppaA. KermanJ. SheikhA. BulgacM. ForbesG. BertschJ. Dobaczewski

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G. FannR. HarrisonJ. HillD. GalindoJ. JiaN. NikolovP. StevensonH. Nam

F.R. XuY. Shi


Thanks for your attention!

[email protected]

Documents

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