1. 形变弱束缚核的理论进展 Junchen Pei ( 裴俊琛 ) School of Physics, Peking University State Key Laboratory of Nuclear Physics and Technology RIBLL1 合作会议, 兰州， 2013.8

2. Contents Recent progress: weakly-bound deformed nuclei Method progress: deformed coordinate-space HFB approach Results: Exotic shapes and continuum effects and MADNESS Summary HFB solvers and Continuum effects-------J.C. Pei -2-

3. Principles of theory progress The descriptive precision of theoretical approaches depends very much the degrees of freedom we assumed in solving nuclear many-body problems The history we observed is that in what perspective to chose the degree of freedom in developing theoretical tools is a topic closely related to the development of computing capabilities. HFB solvers and Continuum effects-------J.C. Pei -3-  For example: ab inito no-core full configuration (A~14) For heavier nuclei, we have to freeze out more degrees of freedom: HFB  Progress: the continuum degree of freedom; from 2D to 3D without restriction on symmetries from single configuration to configuration mixing

4. Expected exotic structures Exotic nuclear structures appear as a threshold effect: halo and cluster Pairing anti-halo versus continuum coupling effect Continuum resulted in phase-space decoupling in quasiparticle spectrum 2-n halos still have puzzles due to 3-body and continuum effects HFB solvers and Continuum effects-------J.C. Pei -4- Molecular-like Prof. Ye’s group Rotational band Spherical halo extensive studies pygmy dipole res. Egg-like Halo recently predicted dipole, quardpole res. Not likely exist Various core-halo deformation decoupling situations F. Nunes, NPA, 2005 Pei, Zhang, Xu, 2013 Misu, Nazarewicz, NPA, 1997 Zhou SG et al, PRC(R)2010

5. Continuum effects Continuum coupling: pairing induced; depend on different angular- momentum Weakly-bound states coupled with continuum: resonances Near-threshold non-resonant continuum is important for halos High energy continuum: Thomas-Fermi approximation (Pei et al. 2011) Enhance stability; enhance halo HFB solvers and Continuum effects-------J.C. Pei -5-

6. Continuum effects 6He: momentum distribution with continuum Breakup, elastic scattering, transfer… HFB solvers and Continuum effects-------J.C. Pei -6- S.N. Ershov, et al, PRL 1998 J. Al-Khalili & F. Nunes, JPG 2003

7. Continuum coupling in HFB theory -7--7- HFB is superior to BCS for describing weakly-bound systems where continuum coupling becomes essential The general HFB-DFT equation (or BdG) is the theoretical tool of choice for describing medium-mass and heavier nuclei and also for cold atomic gases. (ab inito for A ≤ 14) Hartree-Fock-Bogoliubov includes generalized quasi-particle correlations; while BCS is a special quasiparticle transformation only on conjugate states. HFB solvers and Continuum effects-------J.C. Pei Pei et al. PRC, 2011

8. Continuum coupling in HFB theory Current progress: particularly difficult for deformed nuclei Coordinate-space HFB takes an advantage for describing weakly-bound systems and large deformations HFB solvers and Continuum effects-------J.C. Pei -8-  Diagonalization on single-particle basis  Direct diagonalization on coordinate-space lattice  Outgoing boundary condition: difficult for deformed cases ※ H. Oba, M. Matsuo, PRC, 2009, self-consistent calculations are still missing  Bound states, continuum and embedded resonances are treated on an equal footing; L 2 discretization leads to a very large configuration space  Computing resources and capabilities are increasing exponentially

9. Deformed coordinate-space HFB approach Breaking geometric symmetry is very expensive: parallel calculations 2D axial symmetric HFB based on B-splines, finite-difference method 3D MADNESS-HFB with Multi-wavelets techniques and sophisticated parallel techniques -9--9-  J.C.Pei., M. V. Stoitsov, G. I. Fann, W. Nazarewicz, N. Schunck, and F. R. Xu, PRC, 2008 (HFB-AX: Much faster and be able to calculate heavy nuclei and cold atoms)  J.C.Pei, G.I. Fann, R.J. Harrison, W. Nazarewicz, J. Hill, D. Galindo, J. Phys. Conf. Ser., 2012 HFB solvers and Continuum effects-------J.C. Pei

10. Hybrid parallel calculations for large boxes -10- MPI+OpenMP hybrid parallel calculations Works well in Tianhe-1A; Tianhe-2 is coming next year! J.C. Pei et al., JPCS 402, 012035(2012) Large boxes calculations are crucial for describing density diffuseness, large deformation and discretized continuum From 20 fm to 30 fm, the estimated computing cost increased by 40 times. HFB solvers and Continuum effects-------J.C. Pei

11. Egg-like halo structure 38Ne, (a) neutron density; (b) neutron pairing density About 2 neutrons in the halo beta2 = 0.24, beta2_pair=0.48 Mainly contributed by near-threshold continuum HFB solvers and Continuum effects-------J.C. Pei -11- New exotic “egg”-like halo structure found; accurate approach is essential J.C. Pei, Y.N. Zhang, F.R. Xu, PRC (rapid communications) 87, 051302(2013) Isovector deformation: pygmy quardpole resonances

12. Near-Threshold Continuum Different box calculations to distinguish resonances and continuum states Near threshold non-resonant continuum is responsible for halo and surface deformations No halo in 40 Mg since no near-threshold continuum contributions (N=28) HFB solvers and Continuum effects-------J.C. Pei -12-

13. Phase-space decoupling Non-resonant continuum states gradually grows and decouples in heavy nuclei sparse negative-parity level density is crucial; deformed halos in medium-mass nuclei is possible, e.g. in 110 Ge Core-halo decoupling is related to the phase space decoupling in quasiptaricle spectrum HFB solvers and Continuum effects-------J.C. Pei -13-

14. Coordinate-space vs. Basis calculations Surface asympotics; energy spectrum HFB solvers and Continuum effects-------J.C. Pei -14- Basis HFB has problem in broad resonances Y.N. Zhang, J.C. Pei, F.R. Xu, in preparation, 2013

15. Density and pairing density HFB solvers and Continuum effects-------J.C. Pei -15- 38Ne

16. Quasiparticle spectrum near thershold HFB solvers and Continuum effects-------J.C. Pei -16- pairingparticle

17. Box size dependence of energies Total energy is not sensitive to box size, however…. HFB solvers and Continuum effects-------J.C. Pei -17- Y.N. Zhang, J.C. Pei, F.R. Xu, in preparation, 2013

18. Stability Peninsula (continuum and deformation) Interesting! HFB solvers and Continuum effects-------J.C. Pei -18- Y.N. Zhang, J.C. Pei, F.R. Xu, in preparation, 2013

19. 3D MADNESS-HFB -19- 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: HFB solvers and Continuum effects-------J.C. Pei

20. Multi-Wavelets -20-  Truncation  Decomposition Wavelet space at level n Scaling function expansion: expensive  Transformation HFB solvers and Continuum effects-------J.C. Pei

21. MADNESS-HFB strategy HFB solvers and Continuum effects-------J.C. Pei -21- Step-1: Step-2:

22. MADNESS-HFB performance -22- S. Bonger et al., CPC, 2013 HFB solvers and Continuum effects-------J.C. Pei J.C. Pei et al, JPCS, 2012

23. MADNESS-SHF+BCS solver Benchmark for the triaxial deformed nucleus Mo110: Next step: HFB calculations is very expensive. HFB solvers and Continuum effects-------J.C. Pei -23- J.C. Pei, G. Fann, W. Nazarewicz, et al. in preparation, 2013

24. Summary Coordinate-space HFB has a very good opportunity to explore new exotic structures and continuum effects, with the development of supercomputing facilities To be done: It will be interesting to look for continuum effects in excited states, such as Pygmy dipole and quardpole resonances based on deformed coordinate-space QRPA, and it is on track. MADNESS-HFB for nuclear calculations -24- HFB solvers and Continuum effects-------J.C. Pei