1. Exotic nuclear structures and dynamics in HFB theory The 9 th Japan-China Joint Nuclear Physics Symposium, Osaka University, 2015 Junchen Pei ( School of Physics, Peking University ) Collaborators: F.R. Xu, W. Nazarewicz, G. Fann, Y.N. Zhang, Y. Zhu, X.Y. Xiong, N. Fei, Y. Shi, M. Kortelainen, P. Schuck

2. superheavies Contents Stable nuclei Weakly bound nuclei Nuclear Landscape  Developments of coordinate-space HFB approach  Studies of weakly bound nuclear structures and collective excitations  2 nd -order superfluid Thomas-Fermi method  Developments of new Skyrme forces

3. Weakly bound nuclei Weakly bound effects ： halo/deformed halo, soft modes, Borromean, BEC-BCS pairing ， shell evolution ， clustering, continuum coupling, Thomas-Ehrman shift, other threshold effects… Deformed coordinate-space Hartree-Fock-Bogoliubov To solve the self-consistency problem, simultaneously describe the deformations, pairing, halo, continuum together Problems ： HO basis has been widely used in nuclear physics, it is efficient and has excellent mathematical properties, but insufficient for weakly bound systems

4. 2D-Coordinate-space HFB 2D coordinate-space HFB with axial-symmetry: HFB-AX Continuum discretization—Dense quasiparticle spectrum(high resolution) Lattice space flexible—not limited by deformation Hybrid parallel(Tianhe)—Large space Pei et al. PRC 78, 064306 (2008) Pei et al. PRL 102, 192501(2009) Pei et al. PRA 82, 021603(R)2010 Pei et al. PRC 84, 024311(2011) Pei et al. PRC 87, 051302(R)2013 Methods ： 2D B-spline Galerkin operators; Lattice direct diagonalization; Broyden optimized iteration; MPI+OpenMP Hybrid parallel

5. Quasiparticle continuum and resonances Dense quasiparticle spectrum ∝ L 3  Based on stabilization method ， we obtain the widths of quasiparticle resonances Stabilization has been widely applied in atomic physics and chemistry; has been used for nuclear single-particle resonances. S.G. Zhou, J. Meng, and E.G Zhao, JPB 2009 Change the box size  resonance widths Pei et al., PRC 84, 024311(2011)

6. 3D-Coordinate-space HFB Toward the 3D coordinate-space HFB ： a big challenge Conventional methods such as finite-difference, B-splines, are too expensive Collaborate with applied math, comput. chemists to apply the multi- wavelets techniques, with multi- resolutions, and it is very efficient Scaling wavelets at V n Multi- wavelets https://github.com/m-a-d-n-e-s-s JPEG picture compression quality=10 27KB quality=20 41KB quality=60 102KB Multi-wavelets Wavelets

7. Representation of wavefunctions

8. 3D-Coordinate-space HFB 3D coordinates-space HFB: MADNESS-HFB Benchmark Trapped Cold Fermi gases HO basis compare MADNESS-HFB potential applications to complex superfluid systems includes: weakly-bound nuclei, fission, cold atoms, neutron stars No good quantum numbers, fully broken symmetries

9. 3D-HFB solver  Diagonalization with initial wfs  Updating basis This procedure is independent of basis choices and the number of eigen- functions to be solved are reduced. This is very flexible to study two-body terms, state-dependent Hamiltonian, non- local potential. Poisson equation are solved by direct integral

10. MADNESS applications in nuclear structures Parallel computing strategy  MPI: node to node communication  Distributed arrays and FUTURES  POSIX threads: multi-threading within one node threads per node: 10+main+MPI server = 12  Load-balance: map tree to parallel hash table Main MPI Threading Pool API 1. 2. 3. 4. …… WorldTaskQueue ThreadPool Task dependencies: managed by Futures 1. 3. 2…… Globally distributed wfs

11. Remarks of Coordinate-space HFB Advantages: Results are independent of basis (basis numbers, deformations, frequency) (an issue also in ab initio calculations) Can describe densities at large distances with halo asympotics Exotic shapes and large deformations can be described. Continuum can be accurately discretized (for broad resonances) Adapted to other physical systems: cold atomic condensates… Can be extended to describe dynamics Computing issues Parallel calculations are necessary for non-spherical cases 2D: Distribution MPI scheme with good quantum numbers 3D: Global distribution of wavefunctions on all nodes: complicated Future computers: increase the local capability with accelerators

12. Coordinate-space HFB describe weakly bound nuclei Deformed halo ： core-halo deformation decoupling ， in principle various structures exist Expt. deformed halos ： 11 Be, 31 Ne, 37 Mg, but 2n deformed halos not found yet Need self-consistent description ： decoupling ， halo orbital components ， continuum enhancement ， pairing anti-halo Spherical haloVarious deformed halos S.G. Zhou, J. Meng, P. Ring, and E.G. Zhao, PRC(R)(2010)

13. Coordinate-space HFB describe weakly bound nuclei “egg-like deformed halo” Systematic calculations ： not completely decoupling ； egg-like halo mostly significant ； deformation and high level density suppress halo Based on Tianhe-1 calculations 。 Precise HFB solutions are essential "Evolution of surface deformations of weakly bound nuclei in the continuum" Pei*, Zhang, and Xu, PRC 87, 051302(2013)(R)

14. Odd-A weakly bound nuclei Xiong, Pei, Zhang, Zhu, 2015(accepted by Chin.Phys.C) Possible Larkin-Ovchinnikov pairing

15. Coordinate-space HFB describe weakly bound nuclei neutron emission rates Bohr-Wheeler statistical model continuum Neutron emission rates with finite-temperature deformed coordinate-space HFB Different Picture ： Evaporation equilibrium due to neutron gas in FT-HFB; statistical models depends a level density parameter Our approach "Microscopic description of neutron emission rates in compound nuclei", Yi Zhu ( 朱怡 ) and J. C. Pei ( 裴俊琛 )*, Phys. Rev. C 90, 054316(2014). Only in coordinate- space HFB

16. Exotic excitations in weakly bound nuclei New excitation modes and dynamics ： core-halo motion ， new physics of soft modes Continuum Pairing Deformation Collective Excitation Halo Challenge ： self-consistency （ spherical available ） Aim ： fully self-consistent deformed continuum QRPA Pygmy Soft modes Giant resonance

17. Collective excitation descriptions Collective excitation based on coordinate space HFB+ FAM-QRPA Fully self-consistent deformed continuum QRPA Conventional QRPA matrix form FAM-QRPA iterative solution Need calculate (A klmn, B klmn ), Too costly, can not treat large deformed coordinate space Tianhe-1A hybrid parallel calculation different excitation frequency: MPI ； on each node: OpenMP T. Nakatsukasa, PRC 2007 P. Avogadro and T. Nakatsukasa, PRC 2011

18. Exotic excitations in weakly bound nuclei dipole modes has been studied extensively soft monopole modes(less incompressibility of halo) issues ： collective coherence of halo excitation ， the role of continuum in excitation Breath modes of halo nuclei impressibility Khan, et al, PRL 2012 Vandebrouck, et al, PRL 2014 Emergent soft monopole modes in weakly bound deformed nuclei, J.C.Pei, M, Kortelainen, Y.N.Zhang, F.R.Xu, PRC 90,051304(2014)(R) 68Ni

19. Self-consistent Deformed Continuum-FQRPA Transform from B-spline to Gauss-Legendre lattice Solve the complex Poisson equation Time-odd terms are included Be careful of spurious states Large coordinate space is essential Zr100 (mix pairing, surface pairing) FAM-QRPA test of 40Mg

20. Exotic excitations in weakly bound nuclei Non-resonant continuum; sensitive to pairing interaction Not so collective Further studies of multipole modes are in progress

21. Coordinate-space HFB development High-Energy spectrum (Thomas-Fermi) Demonstrate the coordinate-space HFB treatment of continuum is accurate Hybrid-HFB is very useful: coordinate-space HFB+Thomas-Fermi  3D-HFB too costly ？ the Hybrid-HFB 2 nd -Thomas-Fermi(2015) "Quasiparticle continuum and resonances in the Hartree-Fock-Bogoliubov theory", J. C. Pei, A. T. Kruppa, and W. Nazarewicz, Phys. Rev. C 84, 024311(2011). Hybrid-HFB

22. 2 nd -order superfluid Thomas-Fermi  2 nd Thomas-Fermi is very complex for superfluid systems  We derived the terms with effective mass and spin-orbit, based on Green function method, could be used for general superfluid systems

23. Some results of non-sc calculations J.C. Pei, Na Fei, Y.N. Zhang, P. Schuck, arXiv: 1509.02616 Different contribution terms 0TF underestimate pairing at surface

24. Hybrid self-consistent HFB calculations Hybrid HFB iterations with: 2 nd -order TF for continuum, low-energy HFB solutions, and deep-hole states Hybrid HFB can reduce HFB computing costs and still maintain the accuracy 2 nd -TF will be more useful in cold atomic gases and neutron star crusts, which have large pairing fields. J.C. Pei, Na Fei, Y.N. Zhang, P. Schuck, arXiv: 1509.02616

25. New parameterization of Skyrme force  Skyrme interaction (1956) can be seen as a very low-momentum effective potential with a two-body part v ij and three-body part v ijk. 3-body term in Skyrme force: Density dependency is an open question γ ranges from 1/6 to 1 γ=1/6 in SLy4, SkM*, SkP; 0.25 in SkIx γ=1/3 in Gogny, Bsk1 UNEDF0=0.32, UNEDF1=0.27 Important for saturation properties

26. Present work - high-order terms Arguments:  A single density dependence might be too simplistic;  Consider the higher order density-dependent terms and correlations  Impact on high-density region and isovector properties Lee-Yang, 1957 : Fermi systems with repulsive cores

27. New fittings of Skyrme force Fitting binding energies, charge radii, EOS in the s-wave channel Simulated annealing method for parameter fitting X.Y. Xiong, J.C. Pei, ∗ and W.J. Chen, arXiv:1509.07633

28. Effects of higher-order terms X.Y. Xiong, J.C. Pei, ∗ and W.J. Chen, arXiv:1509.07633

29. Summary Coordinate-space HFB for deformed weakly bound systems have a lot of advantages and extensions. However the computation is still very costly. 2 nd superfluid TF has been extended to general systems. Hybrid HFB is useful for large systems to reduce computing cost High-order density dependencies can impact dense nuclear matter properties Thank you for your attention!