1. 上海交通大学物理系 赵玉民

2. 提纲 随机相互作用原子核低激发态主要结果 随机相互作用原子核低激发态主要结果 最近其他研究组几个工作 最近其他研究组几个工作 我们最近的工作 我们最近的工作 展望 展望

3. Part I 随机相互作用下原子核的 随机相互作用下原子核的 规则结构的主要结果 规则结构的主要结果

4. 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) Two-body Random ensemble (TBRE)

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

6. 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. Two-body random ensemble （ＴＢＲＥ）

7. A Simple example

9. Where this result is interesting?

10. Available Results Empircal method Zhao & Arima & Yoshinaga (2002) Mean-field method Bijker-Frank (2003) Geometrid method Chau et al. (2003) ------------------------------------------------------------ Time reversal invariance (TRI) Zuker et al. (2002); Time reversal invariance? Bijker&Frank&Pittel (1999); Width ? Bijker&Frank (2000); off-diagonal matrix elements for I=0 states Drozdz et al. (2001) Highest symmetry &Time Reveral Otsuka&Shimizu(2004-2007) Spectral Radius Papenbrock & Weidenmueller (2004-2007) Semi-empirical formula Yoshinaga, Arima and Zhao(2006-2007)

11. References after Johnson, Bertsch and Dean 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); P.H-T.Chau, A. Frank, N.A.Smirnova, and P.V.Isacker, Phys. Rev. C66, 061301 (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); T. Papenbrock and H. A. Weidenmueller, Phys. Rev. Lett. 93, 132503 (2004); T. Papenbrock and H. A. Weidenmueller, Phys. Rev. C 73 014311 (2006); 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); 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); Y.M.Zhao, A. Arima, K. Ogawa, Phys. Rev. C71, 017304 (2005); Y. M. Zhao, A. Arima, N. Yoshida, K. Ogawa, N. Yoshinaga, and V.K.B.Kota, Phys. Rev. C72, 064314 (2005); N. Yoshinaga, A. Arima, and Y. M. Zhao, Phys. Rev. C73, 017303 (2006); Y. M. Zhao, J. L. Ping, A. Arima, Phys. Rev. C76, 054318 (2007); J. J. Shen, Y. M. Zhao, A. Arima, N. Yoshinaga, Physic. Rev. C77, 054312 (2008); J. J. Shen, A. Arima, Y. M. Zhao, N. Yoshinagan, Phys. Rev. C78, in press (2008); etc. 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); P.H-T.Chau, A. Frank, N.A.Smirnova, and P.V.Isacker, Phys. Rev. C66, 061301 (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); T. Papenbrock and H. A. Weidenmueller, Phys. Rev. Lett. 93, 132503 (2004); T. Papenbrock and H. A. Weidenmueller, Phys. Rev. C 73 014311 (2006); 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); 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); Y.M.Zhao, A. Arima, K. Ogawa, Phys. Rev. C71, 017304 (2005); Y. M. Zhao, A. Arima, N. Yoshida, K. Ogawa, N. Yoshinaga, and V.K.B.Kota, Phys. Rev. C72, 064314 (2005); N. Yoshinaga, A. Arima, and Y. M. Zhao, Phys. Rev. C73, 017303 (2006); Y. M. Zhao, J. L. Ping, A. Arima, Phys. Rev. C76, 054318 (2007); J. J. Shen, Y. M. Zhao, A. Arima, N. Yoshinaga, Physic. Rev. C77, 054312 (2008); J. J. Shen, A. Arima, Y. M. Zhao, N. Yoshinagan, Phys. Rev. C78, in press (2008); etc. Review paper ： Review paper ： Y.M. Zhao, A. Arima, and N. Yoshinaga, Physics Reports 400, 1 (2004). Y.M. Zhao, A. Arima, and N. Yoshinaga, Physics Reports 400, 1 (2004).

12. Phenomenological method by our group (Zhao, Arima and Yoshinaga): reasonably applicable to all systems Phenomenological method by our group (Zhao, Arima and Yoshinaga): reasonably applicable to all systems Mean field method by Bijker and Frank group: sd, sp boson systems (Kusnezov also considered sp bosons in a similar way) Mean field method by Bijker and Frank group: sd, sp boson systems (Kusnezov also considered sp bosons in a similar way) Geometric method suggested by Chau, Frank, Smirnova, and Isacker goes along the same line of our method (provided a foundation of our method for simple systems in which eigenvalues are in linear combinations of two- body interactions). Geometric method suggested by Chau, Frank, Smirnova, and Isacker goes along the same line of our method (provided a foundation of our method for simple systems in which eigenvalues are in linear combinations of two- body interactions).

13. Applications of our method to realistic systems

14. Spin Imax Ground state probabilities

15. By using our phenomenological method, one can trace back what interactions, not only monopole pairing interaction but also some other terms with specific features, are responsible for 0 g.s. dominance. We understand that the Imax g.s. probability comes from the Jmax pairing interaction for single-j shell (also for bosons). The phenomenology also predicts spin I g.s. probabilities well. On the other hand, the reason of success of this method is not fully understood at a deep level, i.e., starting from a fundamental symmetry. By using our phenomenological method, one can trace back what interactions, not only monopole pairing interaction but also some other terms with specific features, are responsible for 0 g.s. dominance. We understand that the Imax g.s. probability comes from the Jmax pairing interaction for single-j shell (also for bosons). The phenomenology also predicts spin I g.s. probabilities well. On the other hand, the reason of success of this method is not fully understood at a deep level, i.e., starting from a fundamental symmetry. Bijker-Frank mean field applies very well to sp bosons and reasonably well to sd bosons. Bijker-Frank mean field applies very well to sp bosons and reasonably well to sd bosons. Geometry method Chau, Frank, Sminova and Isacker is applicable to simple systems. Geometry method Chau, Frank, Sminova and Isacker is applicable to simple systems. Summary of understanding of the 0 g.s. dominance

16. Time reversal invariance Zuker et al. (2002); Time reversal invariance Zuker et al. (2002); Time reversal invariance? Bijker&Frank&Pittel (1999); Time reversal invariance? Bijker&Frank&Pittel (1999); Width ? Bijker&Frank (2000); off-diagonal matrix elements for I=0 states Drozdz et al. (2001), Highest symmetry hypothesis Otsuka&Shimizu(~2004), Spectral Radius by Papenbrock & Weidenmueller (2004-2006) Semi-empirical formula by Yoshinaga, Arima and Zhao(2006). Other works

17. 2. Energy centroids of spin I states under random interactions

21. Other works on energy centroids Mulhall, Volya, and Zelevinsky, PRL(2000) Mulhall, Volya, and Zelevinsky, PRL(2000) Kota, PRC(2005) Kota, PRC(2005) YMZ, AA, Yoshida, Ogawa, Yoshinaga, and Kota, PRC(2005) YMZ, AA, Yoshida, Ogawa, Yoshinaga, and Kota, PRC(2005) YMZ, AA, and Ogawa PRC(2005) YMZ, AA, and Ogawa PRC(2005)

22. 3. Collective motion in the presence of random interactions

23. Collectivity in the IBM under random interactions

25. Shell model: Horoi, Zelevinsky, Volya, PRC, PRL; Velazquez, Zuker, Frank, PRC; Dean et al., PRC; IBM: Kusnezov, Casten, et al., PRL; Geometric model: Zhang, Casten, PRC; Other works

26. Part II. Recent efforts on nuclei under random interactions

27. Recent efforts on 0 g.s. dominance Highest symmetry &Time Reveral Otsuka & Shimizu(2004-2007) Spectral Radius Papenbrock & Weidenmueller (2004-2007) Semi-empirical formula Yoshinaga, Arima and Zhao(2006-2007)

28. YMZ, Pittel, Bijker, Frank, and AA, PRC66, 041301 (2002). (By using usual SD pairs) YMZ, J. L. Ping, and AA, PRC76, 054318 (2007). (By using symmetry dictated pairs--FDSM) Calvin W. Johnson, Hai Ah Nam, PRC75, 047305 (2007). Shell model calculations 集体运动模式

29. (A) Both protons and neutrons are in the shell which corresponds to nuclei with both proton number Z and neutron number N ~40; (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; (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; (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. (D) Protons in the shell and neutrons in the shell which correspond to nuclei with Z~50 and N~82. 随机相互作用下宇称分布规律

31. The worst case is P(+)=67% ， the best case is 99.9% 。 On average P(+)~86% 。 The worst case is P(+)=67% ， the best case is 99.9% 。 On average P(+)~86% 。 No counter example has been found so far ！ No counter example has been found so far ！

32. Physical Review C, in press

33. Part III. 我们最近的工作

34. 我们最近的工作 (1) ：矩阵的本征值问题 ( 最低本征值和所有本征值 ) “Lowest Eigenvalues of Random Hamiltonians”(2008). J. J. Shen, Y. M. Zhao, A. Arima, and N. Yoshinaga, Physical Review C77, 054312. “Strong Linear Correlation Between Eigenvalues and Diagonal Matrix elements”, J. J. Shen, A. Arima, Y. M. Zhao, and N. Yoshinaga, Physical Review C(2008). N. Yoshinaga, A. Arima, J. J. Shen, and Y. M. Zhao, “Functional Dependence of eignevalues and diagonal matrix elements”, submitted to PRC. J. J. Shen and Y. M. Zhao, in preparation. A. Arima, Inter. J. Mod. Phys. E, in press. 这些工作属于无心插柳的性质。当时 (2006 年 ) 沈佳杰大学三年级时要做科研, 当时量 子力学 还没有学过, 所以只能用计算机玩玩。 2006 年吉永教授 (N. Yoshinaga) 、有马教 授 (Akito Arima) 和我得到了一个最低本征值的、非常简单的半经验公式（平均能量、 分布宽度和维数），我希望能够更加精确一些，比如能否引入高级距修正。但是没有 特别的结果。沈佳杰通过有趣的尝试和大量的努力，终于得到 很多结果。

35. 我们这个发现的重要意义 对角化大矩阵是很困难的 对角化大矩阵是很困难的 我们意外发现本征值与对角元之间存在简 单函数关系，壳模型情形呈线性关系。 我们意外发现本征值与对角元之间存在简 单函数关系，壳模型情形呈线性关系。 参考沈佳杰的报告 参考沈佳杰的报告

36. 我们最近的工作 (2) ： FDSM 内的集体运动 FDSM 与 IBM 的相似： FDSM 与 IBM 的相似： 类似的群结构 / SU(3), SU(5) group chains 类似的群结构 / SU(3), SU(5) group chains 费米子 / 玻色子自由度；对力 + 四极力； SD 配 对 费米子 / 玻色子自由度；对力 + 四极力； SD 配 对 FDSM 与 IBM 的不同： FDSM 与 IBM 的不同： SO(8) 没有转动极限 SO(8) 没有转动极限

37. SO(8) 极限 SP(6) 极限

39. 总结 随机相互作用原子核的主要结果 随机相互作用原子核的主要结果 最近的主要进展 最近的主要进展 我们的两个工作 我们的两个工作 展望 展望

40. Acknowledgements: Acknowledgements: Akito Arima (Tokyo) Akito Arima (Tokyo) Naotaka Yoshinagana (Saitama) Naotaka Yoshinagana (Saitama) 贾力源 ( 上海交大本科生, went to MSU last summer ) 贾力源 ( 上海交大本科生, went to MSU last summer ) 张丽华 ( 上海交通大學物理系硕博联读,from April 06) 张丽华 ( 上海交通大學物理系硕博联读,from April 06) 沈佳杰 ( 上海交通大學物理系直博生,from Sep.07) 沈佳杰 ( 上海交通大學物理系直博生,from Sep.07) 雷 扬 ( 上海交通大學物理系直博生,from Sep.07) 雷 扬 ( 上海交通大學物理系直博生,from Sep.07) 徐正宇 ( 上海交通大學物理系直硕生,from Sep.07) 徐正宇 ( 上海交通大學物理系直硕生,from Sep.07) 姜 慧 ( 上海交通大學物理系博士生,from Sep.08) 姜 慧 ( 上海交通大學物理系博士生,from Sep.08) 李晨光 ( 上海交大本科生, 预计 09 年直硕 ) 李晨光 ( 上海交大本科生, 预计 09 年直硕 )

41. 谢谢各位 !