2. Outline  A short history of spin zero ground state dominance  Present status of this study @ Physical mechanism remains unclear @ Physical mechanism remains unclear @ Collectivity of low-lying states by using TBRE @ Collectivity of low-lying states by using TBRE @ Energy centroids of fixed spin states @ Energy centroids of fixed spin states  Perspectives @ Some simpler quantities can be studied first @ Some simpler quantities can be studied first @Searching for other regularities @Searching for other regularities

3. Random two-body interactions 1958 Wigner introduced Gaussian orthogonal ensemble of random matrices (GOE) in understanding the spacings of energy levels observed in resonances of slow neutron scattering on heavy nuclei. Ref: Ann. Math. 67, 325 (1958) 1970’s French, Wong, Bohigas, Flores introduced two-body random ensemble (TBRE) Ref: Rev. Mod. Phys. 53, 385 (1981); Phys. Rep. 299, (1998); Phys. Rep. 347, 223 (2001). Original References: J. B. French and S.S.M.Wong, Phys. Lett. B33, 449(1970); O. Bohigas and J. Flores, Phys. Lett. B34, 261 (1970). Other applications: complicated systems (e.g., quantum chaos)

4. Two-body random ensemble （ＴＢＲＥ） One usually choose Gaussian distribution for two-body random interactions There are some people who use other distributions, for example, A uniform distribution between -1 and 1. For our study, it is found that these different distribution present similar statistics.

5. In 1998, Johnson, Bertsch, and Dean discovered that spin parity =0+ ground state dominance can be obtained by using random two-body interactions (Phys. Rev. Lett. 80, 2749) ． This result is called 0 g.s. dominance. Similar phenomenon was found in other systems, say, sd-boson systems. Ref. C. W. Johnson et al., PRL80, 2749 (1998); R.Bijker et al., PRL84, 420 (2000); L. Kaplan et al., PRB65, 235120 (2002).

6. An example

9. Some recent papers R. Bijker, A. Frank, and S. Pittel, Phys. Rev. C60, 021302(1999); D. Mulhall, A. Volya, and V. Zelevinsky, Phys. Rev. Lett.85, 4016(2000); Nucl. Phys. A682, 229c(2001); V. Zelevinsky, D. Mulhall, and A. Volya, Yad. Fiz. 64, 579(2001); D. Kusnezov, Phys. Rev. Lett. 85, 3773(2000); ibid. 87, 029202 (2001); L. Kaplan and T. Papenbrock, Phys. Rev. Lett. 84, 4553(2000); R.Bijker and A.Frank, Phys. Rev. Lett.87, 029201(2001); S. Drozdz and M. Wojcik, Physica A301, 291(2001); L. Kaplan, T. Papenbrock, and C. W. Johnson, Phys. Rev. C63, 014307(2001); R. Bijker and A. Frank, Phys. Rev. C64, (R)061303(2001); R. Bijker and A. Frank, Phys. Rev. C65, 044316(2002); L. Kaplan, T.Papenbrock, and G.F. Bertsch, Phys. Rev. B65, 235120(2002); L. F. Santos, D. Kusnezov, and P. Jacquod, Phys. Lett. B537, 62(2002); Y.M. Zhao and A. Arima, Phys. Rev.C64, (R)041301(2001); A. Arima, N. Yoshinaga, and Y.M. Zhao, Eur.J.Phys. A13, 105(2002); N. Yoshinaga, A. Arima, and Y.M. Zhao, J. Phys. A35, 8575(2002); Y. M. Zhao, A. Arima, and N. Yoshinaga, Phys. Rev.C66, 034302(2002); Y. M. Zhao, A. Arima, and N. Yoshinaga, Phys. Rev. C66, 064322(2002); P.H-T.Chau, A. Frank, N.A.Smirnova, and P.V.Isacker, Phys. Rev. C66, 061301 (2002); Y.M.Zhao, A. Arima, N. Yoshinaga, Phys.Rev.C66, 064323 (2002); Y. M. Zhao, S. Pittel, R. Bijker, A. Frank, and A. Arima, Phys. Rev. C66, R41301 (2002); Y. M. Zhao, A. Arima, G. J. Ginocchio, and N. Yoshinaga, Phys. Rev. C66,034320(2003); Y. M. Zhao, A. Arima, N. Yoshinga, Phys. Rev. C68, 14322 (2003); Y. M. Zhao, A. Arima, N. Shimizu, K. Ogawa, N. Yoshinaga, O. Scholten, Phys. Rev. C70, 054322 (2004); T. Papenbrock and H. A. Weidenmueller, Phys. Rev. Lett. 93, 132503 (2004); Y.M.Zhao, A. Arima, K. Ogawa, Phys. Rev. C (in press) Review papers ： Y.M.Zhao, A. Arima, and N. Yoshinaga, Phys. Rep. 400, 1(2004); V. Zelevinsky and A. Volya, Phys. Rep. 391, 311 (2004).

11. Two interesting results  Empirical method by Tokyo group reasonably applicable to all systems  Mean field method by Mexico group sd, sp boson systems

12. Empirical method by Tokyo group

13. d 玻色子情形

15. Phenomenological method Let find the lowest eigenvalue; Repeat this process for all.

16. Four fermions in a single-j shell

18. Why P(0) staggers periodically?  对四个粒子情形，如果 GJ=-1 其他两体力为 零，Ｉ＝０的态只有一个非零的本征值．  Ｉ＝０的态的数量随 j 呈规则涨落．

19. 最大自旋态作基态的几率

20. A few examples

21. Parity distribution in the ground states  (A) Both protons and neutrons are in the shell which corresponds to nuclei with both proton number Z and neutron number N ~40;  (B) Protons in the shell and neutrons in the shell which correspond to nuclei with Z~40 and N~50;  (C) Both protons and neutrons are in the shell which correspond to nuclei with Z and N~82;  (D) Protons in the shell and neutrons in the shell which correspond to nuclei with Z~50 and N~82.

23. Collectivity in the IBM under random interactions

28. Energy centroids with fixed spin

34. Conclusion and prospect  Regularities of many-body systems under random interactions, including spin zero ground state dominance, energy centroids with various quantum numbers, collectivity, etc.  Suggestion: Try any physical quantities by random interactions  Questions: parity distribution, energy centroids, constraints of collectivity, and spin 0 g.s. dominance

35. Acknowledgements: Akito Arima (Tokyo) Naotaka Yoshinagana (Saitama) Kengo Ogawa (Chiba) Stuart Pittel (Delaware) R. Bijker (Mexico) J. N. Ginocchio (Los Alamos) Rick Casten (New Haven) Olaf Scholten (Groningen) V. K. B. Kota (Ahmedabad) Noritake Shimizu(Tokyo) Nobuaki Yoshida (Kansai) Igal Talmi (Weizman)