2. Outline  Simple comments on regularities of many-body systems under random interactions  Number of spin I states for single-j configuration  J-pairing interaction  Sum rules of angular momentum recoupling coefficients  Number of states with given spin and isospin  Nucleon approximation of the shell model  Prospect and summary  Simple comments on regularities of many-body systems under random interactions  Number of spin I states for single-j configuration  J-pairing interaction  Sum rules of angular momentum recoupling coefficients  Number of states with given spin and isospin  Nucleon approximation of the shell model  Prospect and summary

3. Part I A brief introduction to nuclei under random interactions Two-body random ensemble

4. In 1998, Johnson, Bertsch, and Dean found spin zero ground state dominance can be obtained by random two-body interactions (Phys. Rev. Lett. 80, 2749) ． Ref. C. W. Johnson et al., PRL80, 2749 (1998); R.Bijker et al., PRL84, 420 (2000); L. Kaplan et al., PRB65, 235120 (2002).

6. Intrinsic collectivity based on the sd IBM

7. Energy centroids of spin I states

8. A short summary  Spin 0 ground state dominance for even-even nuclei, regularities for energy centroids with given quantum numbers, collectivity, etc.  Open questions: spin distribution in the ground states ； energy centroids ； requirement for nuclear collectivity ； etc. For a review, See YMZ, AA, NY, Physics Reports, Volume 400, Page 1 (2004).

9. Themes and Challenges of Modern Science  Complex systems arising out of basic constituents How the world, with all its apparent complexity and diversity, can be constructed out of a few elementary building blocks and their interactions  Simplicity out of complexity How the world of complex systems can display such remarkable regularity and, often, simplicity  Understanding the nature of the physical universe  Manipulating matter for the benefit of mankind

10. Part II Number of states for identical particles in a single-j shell  Why we study this number (Ginocchio)? 

11. A simple method in the text-book

12. Empirical formulas  YMZ and AA, PRC68, 044310 (2003). empirical formulas for n=3,4. For example, For n=4, results are more complicated (omitted).

13. A new method

14.  Conjugates of

16. An Example

17. An example: n=4  Here we should study bosons with SU(5)symmetry, i.e., d bosons. The number of states for d bosons has been studied in the interacting boson model.  By using results of the IBM, we were able to obtain dimension for d bosons. Then we quickly get the number of states for n=4 of fermions and bosons.  What about odd number of particles? YMZ and AA, PRC71, 047304 (2005)

18. Other works  Dimension for n>3: J. N. Ginocchio and W. C. Haxton, “Symmetries in Science VI”, Edited by B. Gruber and M. Ramek, (Plenum, New York, 1993).  Number of states for Zamick et al. Physical Review C71, 054308 (2005).

19.  For n=3, we should study SU(4) reduction rule.  However, Talmi proved our results for n=3(Physical Review C72,037302(2005)). He obtained some recursion formulas and proved the formulas by reduction method.  This method can be used to prove any formulas (in principle) but it can not be used to find new formulas.

20. The guideline of Talmi’s efforts  First, he assumes the formula is correct for j-1 shell, then it suffices to show that it is also correct to j shell.  Next he enumerates the effect by changing m1= j-1 to j.  Summing this effect, he obtains the dimension of j shell.

21. Part III J-pairing interaction  Fermions in a single-j shell:

22. J –pair truncation of the shell model  Empirically we find that J-pair truncation is good for J- pairing interaction. 

23. PartIV Sum rules of angular momentum recoupling coefficients  For n=3, J-pair truncation gives exact solution.

24. An example

25. Similar things can be done for n=4 Ref.: YMZ & AA (Physical Review C72, 054307 (2005)  Here we obtain sum rules of 9-j symbols.  The difference is that here situation is more complicated. Generally speaking, the number of nonzero eigenvalues is not always one for J pairing interaction and each of eigenvalue is unknown.  However, the trace of eigenvalues for each I states with only J-pairing interactions is always a constant with respect to orthogonal transformation:

26. Proof

27. Sum rules Summing over J, we obtain sum rules of 9-j symbols

28. Sum rules with odd J and odd K

29. Sum rules with both even and odd J,K

30. Two examples

31. Part V Number of states for nucleons in a single-j shell [An application of these sum rules]  References: L.Zamick et al., Physical Review C72, 044317 (2005); Y.M.Z. and A.A., Physical Review C72, 064333 (2005).

32. JT-pairing interaction  Nucleons in a single-j shell:

33.  Similarly,

35. The case of T=0

37. Part VI Our next effort on pair approximation  IBM  SD collective pairs  diagonalization of the shell model Hamiltonian in nucleon pair subspace.  How far one can go?  How reliable is “collective pair” approximation?  What can one calculate by using pair appro. ?

38. We have done following work  Validity of SD pair truncation for special cases  Application to A=130 even-even nuclei (rather successful calculation)  We proposed an efficient algorithm to describe even systems and odd-A (also doubly odd) systems on the same footing.

39. The nuclei we shall try in the near future  Mass number A around 140, neutron rich side (both even and odd A, both positive and negative parity)  Validity of SD truncation, realistic cases. [How good or not good is the pair truncation?]  Extension of pairs [S1, S2, D1, D2, F pairs, G pairs, etc. ]

40. Part VII Summary & prospect  Nuclear structure under random interactions  Number of states with given spin (&isospin), sum rule of 6j and 9j symbols  Our next project: Nucleon pair approximation of the shell model: odd and doubly odd nuclei. Applications to realistic systems

41. Acknowledgements: Akito Arima (Tokyo) Naotaka Yoshinaga (Saitama) Kengo Ogawa (Chiba) Nobuaki Yoshida (Kansai)

42. International Conference on Nuclear Structure Physics, Shanghai, June 12-17th, 2006.