1. Nuclear Low-lying Spectrum and Quantum Phase Transition Zhipan Li School of Physical Science and Technology Southwest University 17th Nuclear Physics Workshop, Kazimierz Dolny, Poland

2. www.swu.edu.cn Outline Introduction 1 Theoretical framework 2 Results and discussion 3 Summary and outlook 4

3. Quantum Phase Transition in finite system  Quantum Phase Transition (QPT) between competing ground-state phases induced by variation of a non- thermal control parameter at zero temperature.  In atomic nuclei:  1 st and 2 nd order QPT: abrupt transition in shapes.  Control Par. Number of nucleons  Two approaches to study QPT  Method of Landau based on potentials (not observables)  Direct computation of order parameters (integer con. par.)  Combine both approaches in a self-consistent microscopic framework Spherical Deformed ECritical β Potential Order par. Potential Order par. F. Iachello, PRL2004 P. Cejnar et al., RMP82, 2155 (2010)

4. Covariant Energy Density Functional (CEDF)  CEDF: nuclear structure over almost the whole nuclide chart  Scalar and vector fields: nuclear saturation properties  Spin-orbit splitting  Origin of the pseudo-spin symmetry  Spin symmetry in anti-nucleon spectrum  ……  Spectrum: beyond the mean-field approximation  Restoration of broken symmetry, e.g. rotational  Mixing of different shape configurations Ring1996, Vretenar2005, Meng2006 PES AMP+GCM: Niksic2006, Yao2010 5D Collective Hamiltonian based on CEDF

5. Brief Review of the model Construct 5D Collective Hamiltonian (vib + rot) E(J π ), BE2 … Cal. Exp. 3D covariant Density Functional ph + pp Coll. Potential Moments of inertia Mass parameters Diagonalize: Nuclear spectroscopy Niksic, Li, Vretenar, Prochniak, Meng & Ring, PRC79, 034303 (09) Libert, Girod & Delaroche, PRC60, 054301 (99) Prochniak & Rohozinski, JPG36, 123101 (09)

6.  Spherical to prolate 1 st order QPT [Z.P. Li, T. Niksic, D. Vretenar, J. Meng, G.A. Lalazissis, P. Ring, PRC79, 054301(2009)]  Analysis of order parameter [Z.P. Li, T. Niksic, D. Vretenar, J. Meng, PRC80, 061301(R) (2009)]  Spherical to γ-unstable 2 nd order QPT [Z.P. Li, T. Niksic, D. Vretenar, J. Meng, PRC81, 034316 (2010)] Microscopic Analysis of nuclear QPT

7.  Potential Energy Surfaces (PESs) Discontinuity First order QPT

8.  Potential Energy Surfaces (PESs) along β along γ First order QPT

9.  Spectrum detailed spectroscopy has been reproduced well !! First order QPT

10.  Spectrum  Characteristic features: Sharp increase of R 42 =E(4 1 )/E(2 1 ) and B(E2; 2 1 →0 1 ) in the yrast band X(5) First order QPT

11.  Single-particle levels First order QPT 150 Nd

12. Microscopic analysis of Order parameters  Finite size effect (nuclei as mesoscopic systems)  Microscopic signatures (order parameter)  In finite systems, the discontinuities of QPT will be smoothed out  1 st order 2 nd order; 2 nd order crossover F. Iachello, PRL2004 based on IBM F. Iachello, PRL2004 based on IBM 1.Isotope shift & isomer shift 2.Sharp peak at N~90 in (a) 3.Abrupt decrease; change sign in (b)

13.  Microscopic signatures (order parameter) Conclusion: even though the control parameter is finite number of nucleons, the phase transition does not appear to be significantly smoothed out by the finiteness of the nuclear system. Microscopic analysis of Order parameters

14. Second order QPT  Are the remarkable results for 1 st order QPT accidental ?  Can the same EDF describe other types of QPT in different mass regions ? R. Casten, PRL2000 F. Iachello, PRL2000

15. Second order QPT  PESs of Ba isotopes

16. Second order QPT  PESs of Xe isotopes

17. Second order QPT  Evolution of shape fluctuation: Δβ / 〈 β 〉, Δγ / 〈 γ 〉

18. Second order QPT  Spectrum of 134 Ba  Microscopic predictions consist with data and E(5) for g.s. band  Sequence of 2 2, 3 1, 4 2 : well structure / ~0.3 MeV higher  The order of two excited 0+ states is reversed

19.  Microscopic analysis of nuclear QPT  PESs display clear shape transitions  The spectrum and characteristic features have been reproduced well for both 1 st & 2 nd order QPT  The microscopic signatures have shown that the phase transition does not appear to be significantly smoothed out by the finiteness of nuclear system.  Further development of the model:  Time-odd part for inertia parameters  Coupling between the pairing & quadruple vibration 22 Summary and outlook

20. J. Meng & JCNP group D. Vretenar & T. Niksic P. Ring L. Prochniak G. A. Lalazissis

21. 6 Collective Hamiltonian

22. 7 Collective Parameter

23. 7

24. 7