1. HFB studies of exotic nuclear structures and dynamics 2015 PKU-CUSTIPEN Workshop Junchen Pei (School of Physics, PKU) 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 1.Developments of coordinate-space HFB 2.Weakly bound nuclei and excitations 3.Other issues: Skyrme forces Stable nuclei Weakly bound nuclei Nuclear Landscape

3. RNB facilitates provides great opportunities Extremely unstable nuclei can emerge exotic properties and dynamics due to threshold effects, continuum, weakly bound effects… [J. Erler et al., Nature 486, 509 (2012)] Known FRIB(2020) Predicted Background ： Key issues ： How to describe exotic nuclear structures and dynamics self- consistently

4. Characteristics of weakly nuclei ： Diffused surfaceFermi surface near 0 Halo-core decouplingContinuum coupling Weakly bound effects ： deformed halo ， soft modes, Borromean, BEC-BCS pairing ， tensor force ， shell evolutions ， clustering

5. Problems ： HO basis has been widely used in nuclear physics, is efficient and has excellent mathematical properties, but insufficient for weakly bound systems Halo-core decouplingContinuum coupling Deformed coordinate-space Hartree-Fock-Bogoliubov To solve the self-consistency problem, simultaneously describe the deformations, pairing, halo, continuum together Diffused surfaceFermi surface near 0

6. 2D-Coordinate-space HFB Firstly we developed the 2D coordinate-space HFB: 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

7. 3D-Coordinate-space HFB Toward the 3D coordinate-space HFB ： a big challenge Conventional methods such as finite-difference, B-splines, are too expensive So we collaborate with applied mathematicians, 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

8. 3D-Coordinate-space HFB Finally we realize the 3D coordinates-space HFB: MADNESS-HFB 入选为 PRC Editors’ Suggestion ( 仅 3% 文章入选 ) Multi-wavelets firstly applied to nuclear physics MADNESS-HFB potential applications to complex superfluid systems includes: weakly-bound nuclei, fission, cold atoms, neutron stars Benchmark Trapped Cold Fermi gases HO basis compare

9. 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  HFB too costly ？ We explored 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

10. Coordinate-space HFB development  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 （ Pei et al., 2015 ）

11. Coordinate-space HFB development Densse quasiparticle spectrum ∝ L 3  Based on stabilization method ， we obtain the widths of quasiparticle resonances Stabilization has been widely applied in atomic phyics 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)

12. Coordinate-space HFB describe weakly bound nuclei Deformed halo ： core-halo deformation decoupling ， in principle various structures exist Expt. deformed halo ： 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 We predicted an“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. Coordinate-space HFB describe weakly bound nuclei Stabilization method definitely see that non-resonant continuum are responsible Large coordinate space HFB calculations are essential for continuum discretization Qusiparticle E Pei et al., PRC 87, 051302(2013)(R). Zhang et al., PRC 88, 054305(2013).

15. Coordinate-space HFB describe weakly bound nuclei  Large coordinate space HFB is essential  Pairing density are more spatially extended Advantages ： simultaneously treat the continuum, deformation, pairing, halo Zhang, Pei*, Xu, PRC 88, 054305(2013). Pairing density Particle density

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

17. 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 aim ： microscopic description of survival probabilities in stead of statistical models 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

18. 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

19. Exotic excitations in weakly bound nuclei 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 P. Avogadro and T. Nakatsukasa, PRC 2011

20. 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 halos impressibility Khan, et al, PRL 2012 Vandebrouck, et al, PRL 2014

21. Benchmarks FAM-QRPA test of 40Mg Be careful of spurious states Large coordinate space is essential Zr100 (mix pairing, surface pairing)

22. Exotic excitations in weakly bound nuclei Surface pairing density vibration （ non-resonant continuum ） Not so collective Emergent soft monopole modes Provide a suitable tool to study novel excitation of weakly bound nuclei

23. Three-body Force in Skyrme Forces and New Parameterization Xueyu Xiong and Junchen Pei

24. Motivation- 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

25. Motivation- Skyrme force  low-density Fermi gas with hard core  The corresponding energy density ε: T. D. Lee and C. N. Yang, Phys. Rev. 105, 1119 (1956). C. DeDominicis and P. C. Martin Phys. Rev. 105, 1417 –1957 A. Gezerlis and G. F. Bertsch, Phys. Rev. Lett. 105, 212501(2010). Compared 3-body force and the Lee-Yang (NLO)  Effective 3-body force  High-order density-dependent terms due to finite-size effect of nucleons a=r/d

26. Motivation- Skyrme force  Three-body force is induced due to the cutoff in model space and is more important in soft interactions With different momentum cutoff Λ K. Moghrabi, M. Grasso, G. Colò, and N. Van Giai, PRL 105, 262501 (2010) With a lower cutoff: Second order contribution divergence with zero-range force The 2-body term decreases and density dependency increase

27. Motivation- Skyrme force  Other versions of Skyrme forces UNEDF forces: global fitting the Skyrme force and the limits has been reached Brussels forces: Extended Skyrme forces with generalizations of t1, t2 M. Kortelainen et al., Phys. Rev. C 82, 024313 (2010). HFB-19,20,21…26 rms:0.58MeV S. Goriely, N. Chamel, and J. M. Pearson Phys. Rev. C 82, 035804 (2010) UNEDF1: fission barriers UNEDF2: tensor forces, shells

28. Motivation- Skyrme force  Other versions of DFT/MF Ab initio EFT-DFT: M.Stoitsov, D. Furnstahl, S. Bogner, Density matrix expansion (DME) : N2LO EFT to local DFT (in particular pion interactions) M. Stoitsov, et al., PRC 82, 054307 (2010) N3LO DFT: J. Dobaczewski B. G. Carlsson, et al.,PRC 78, 044326 (2008) DFT as Tensor products Skyrme Pseudopotential

29. Present work - high-order terms Arguments:  A single density dependence might be too simplistic; test Lee-Yang in nuclei  Consider the higher order density-dependent terms and correlations  S-wave should be studied well before we study tensor terms  Impact on high density region and isovector properties

30. 30 Present work- Nuclear Matter property  Energy per nucleon symmetric matter at the saturation point  The pressure at the saturation point Fitting inputs

31. Present work- Nuclear Matter property  The incompressibility at the saturation point  The symmetry energy coefficient at the saturation point  The equation of state(EOS) of nuclear matter The results of UV14+UV2 ： R. B.Wiringa and V. Fiks ， Phys. Rev. C 38, 2 (1988).

32. Present work- Fitting approach  Simulated Annealing Method: It is a randomized optimization algorithm based on the Monte- Carlo iterative solution method, which accept a worse solution than the current one with a certain probability. Therefore, the search may jump out of the local optimal solution and seek out the global optimal solution.  The Cooling principle : The probability of cooling with the energy difference dE is P (dE) at temperature T, where P(dE) = exp( -dE/(kT). The move to a worse solution is considered as the temperature transition process and accept the move with the probability P (dE) in Simulated Annealing.

33. Present work- Fitting approach  parameter setting.  Minimize the quantity  The charge radii  Fitting 6 parameters (4 parameters are free with nuclear matter properties)  Other parameters are the same as SLy4

34. Simulated annealing approach Present work- results of fitting

35.  Fitted new Skyrme parameters Overall improvement compared to SLy4: radii and energies

36. Present work- results of fitting  The radii of fitting four nuclei comparing to SLy4 ExperimentNew par.SLy4 40 Ca 3.483.4903.493 48 Ca 3.483.5083.511 56 Ni 3.753.7693.772 208 Pb 5.505.4945.498 χ=1.81 ×10 -2 χ=2.00×10 -2

37. Present work- calculation results  The calculation binding energy using new parameter Drip Line nuclei

38. Present work- calculation results  The calculation of symmetry energy

39. Present work- calculation results  The global calculations of 600 e-e nuclei

40. Three-body Force in Skyrme Interaction and New Parameters Summary  The higher-order density dependent terms has been studied  The high-order term is essential (15-30%) and should be included in new parameter fittings. Outlook:  Fitting more nuclei and study more physics  Fitting terms corresponding tensor force and spin-orbit coupling Thank you for your attention ！