上海交通大学物理系赵玉民提纲提纲n n随机相互作用原子核低激发态主要结果n n最近其他研究组几个工作 n n我们最近的工作n n瞻望Part I 随机相互作用下原子核的 规则结构的主要结果1958 Wigner introduced Gaussian orthogonal ensemble of random 1959 matrices (GOE) in understanding the spacings of energy levels 1960 observed in resonances of slow neutron scattering on heavy nuclei. 1961 Ref: Ann. Math. 67, 325 (1958)19621970s French, Wong, Bohigas, Flores introduced two-body random 1963 ensemble (TBRE)1964 Ref: Rev. Mod. Phys. 53, 385 (1981);1965 Phys. Rep. 299, (2019);1966 Phys. Rep. 347, 223 (2019).1967Original References: 1968 J. B. French and S.S.M.Wong, Phys. Lett. B33, 449(1970);1969 O. Bohigas and J. Flores, Phys. Lett. B34, 261 (1970). 1970Other applications: complicated systems (e.g., quantum chaos)Two-body Random ensemble (TBRE) 1.What does 0 g.s. dominance mean ?2. In 2019, Johnson, Bertsch, and Dean discovered that spin parity =0+ ground state dominance can be obtained by using random two-body interactions.3.This result is called the 0 g.s. dominance. 4.Similar phenomenon was found in other systems, say, sd-boson systems. 5. C. W. Johnson et al., PRL80, 2749 (2019);6. R. Bijker et al., PRL84, 420 (2000);7. L. Kaplan et al., PRB65, 235120 (2019).One usually choose Gaussian distribution for two-body random interactionsThere 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） A Simple exampleWhere this result is interesting?Available ResultsAvailable ResultsEmpircal method Zhao & Arima & Yoshinaga (2019)Empircal method Zhao & Arima & Yoshinaga (2019)Mean-field method Bijker-Frank (2019)Mean-field method Bijker-Frank (2019)Geometrid method Chau et al. (2019) Geometrid method Chau et al. (2019) -Time reversal invariance (TRI) Zuker et al. (2019);Time reversal invariance (TRI) Zuker et al. (2019);Time reversal invariance? Bijker&Frank&Pittel (2019);Time reversal invariance? Bijker&Frank&Pittel (2019);Width ? Bijker&Frank (2000);Width ? Bijker&Frank (2000);off-diagonal matrix elements for I=0 states Drozdz et al. (2019) off-diagonal matrix elements for I=0 states Drozdz et al. (2019) Highest symmetry &Time Reveral Otsuka&Shimizu(2019-2019) Highest symmetry &Time Reveral Otsuka&Shimizu(2019-2019) Spectral Radius Papenbrock & Weidenmueller (2019-2019)Spectral Radius Papenbrock & Weidenmueller (2019-2019)Semi-empirical formula Yoshinaga, Arima and Zhao(2019-2019)Semi-empirical formula Yoshinaga, Arima and Zhao(2019-2019)References after Johnson, Bertsch and DeanReferences after Johnson, Bertsch and Dean R. Bijker, A. Frank, and S. Pittel, Phys. Rev. C60, R. Bijker, A. Frank, and S. Pittel, Phys. Rev. C60, 021302(2019); D. Mulhall, A. Volya, and V. Zelevinsky, 021302(2019); D. Mulhall, A. Volya, and V. Zelevinsky, Phys. Rev. Lett.85, 4016(2000); Nucl. Phys. A682, Phys. Rev. Lett.85, 4016(2000); Nucl. Phys. A682, 229c(2019); V. Zelevinsky, D. Mulhall, and A. Volya, Yad. Fiz. 229c(2019); V. Zelevinsky, D. Mulhall, and A. Volya, Yad. Fiz. 64, 579(2019); D. Kusnezov, Phys. Rev. Lett. 85, 64, 579(2019); D. Kusnezov, Phys. Rev. Lett. 85, 3773(2000); ibid. 87, 029202 (2019); L. Kaplan and T. 3773(2000); ibid. 87, 029202 (2019); L. Kaplan and T. Papenbrock, Phys. Rev. Lett. 84, 4553(2000); R.Bijker and Papenbrock, Phys. Rev. Lett. 84, 4553(2000); R.Bijker and A.Frank, Phys. Rev. Lett.87, 029201(2019); S. Drozdz and A.Frank, Phys. Rev. Lett.87, 029201(2019); S. Drozdz and M. Wojcik, Physica A301, 291(2019); L. Kaplan, T. M. Wojcik, Physica A301, 291(2019); L. Kaplan, T. Papenbrock, and C. W. Johnson, Phys. Rev. C63, Papenbrock, and C. W. Johnson, Phys. Rev. C63, 014307(2019); R. Bijker and A. Frank, Phys. Rev. C64, 014307(2019); R. Bijker and A. Frank, Phys. Rev. C64, (R)061303(2019); R. Bijker and A. Frank, Phys. Rev. C65, (R)061303(2019); R. Bijker and A. Frank, Phys. Rev. C65, 044316(2019); P.H-T.Chau, A. Frank, N.A.Smirnova, and 044316(2019); P.H-T.Chau, A. Frank, N.A.Smirnova, and P.V.Isacker, Phys. Rev. C66, 061301 (2019); L. Kaplan, P.V.Isacker, Phys. Rev. C66, 061301 (2019); L. Kaplan, T.Papenbrock, and G.F. Bertsch, Phys. Rev. B65, T.Papenbrock, and G.F. Bertsch, Phys. Rev. B65, 235120(2019); L. F. Santos, D. Kusnezov, and P. Jacquod, 235120(2019); L. F. Santos, D. Kusnezov, and P. Jacquod, Phys. Lett. B537, 62(2019); T. Papenbrock and H. A. Phys. Lett. B537, 62(2019); T. Papenbrock and H. A. Weidenmueller, Phys. Rev. Lett. 93, 132503 (2019); T. Weidenmueller, Phys. Rev. Lett. 93, 132503 (2019); T. Papenbrock and H. A. Weidenmueller, Phys. Rev. C 73 Papenbrock and H. A. Weidenmueller, Phys. Rev. C 73 014311 (2019); Y.M. Zhao and A. Arima, Phys. Rev.C64, 014311 (2019); Y.M. Zhao and A. Arima, Phys. Rev.C64, (R)041301(2019); A. Arima, N. Yoshinaga, and Y.M. Zhao, (R)041301(2019); A. Arima, N. Yoshinaga, and Y.M. Zhao, Eur.J.Phys. A13, 105(2019); N. Yoshinaga, A. Arima, and Eur.J.Phys. A13, 105(2019); N. Yoshinaga, A. Arima, and Y.M. Zhao, J. Phys. A35, 8575(2019); Y. M. Zhao, A. Arima, Y.M. Zhao, J. Phys. A35, 8575(2019); Y. M. Zhao, A. Arima, and N. Yoshinaga, Phys. Rev.C66, 034302(2019); Y. M. and N. Yoshinaga, Phys. Rev.C66, 034302(2019); Y. M. Zhao, A. Arima, and N. Yoshinaga, Phys. Rev. C66, Zhao, A. Arima, and N. Yoshinaga, Phys. Rev. C66, 064322(2019); Y.M.Zhao, A. Arima, N. Yoshinaga, 064322(2019); Y.M.Zhao, A. Arima, N. Yoshinaga, Phys.Rev.C66, 064323 (2019); Y. M. Zhao, S. Pittel, R. Phys.Rev.C66, 064323 (2019); Y. M. Zhao, S. Pittel, R. Bijker, A. Frank, and A. Arima, Phys. Rev. C66, R41301 Bijker, A. Frank, and A. Arima, Phys. Rev. C66, R41301 (2019); Y. M. Zhao, A. Arima, G. J. Ginocchio, and N. (2019); Y. M. Zhao, A. Arima, G. J. Ginocchio, and N. Yoshinaga, Phys. Rev. C66,034320(2019); Y. M. Zhao, A. Yoshinaga, Phys. Rev. C66,034320(2019); Y. M. Zhao, A. Arima, N. Yoshinga, Phys. Rev. C68, 14322 (2019); Y. M. Arima, N. Yoshinga, Phys. Rev. C68, 14322 (2019); Y. M. Zhao, A. Arima, N. Shimizu, K. Ogawa, N. Yoshinaga, O. Zhao, A. Arima, N. Shimizu, K. Ogawa, N. Yoshinaga, O. Scholten, Phys. Rev. C70, 054322 (2019); Y.M.Zhao, A. Scholten, Phys. Rev. C70, 054322 (2019); Y.M.Zhao, A. Arima, K. Ogawa, Phys. Rev. C71, 017304 (2019); Y. M. Arima, K. Ogawa, Phys. Rev. C71, 017304 (2019); Y. M. Zhao, A. Arima, N. Yoshida, K. Ogawa, N. Yoshinaga, and Zhao, A. Arima, N. Yoshida, K. Ogawa, N. Yoshinaga, and V.K.B.Kota , Phys. Rev. C72, 064314 (2019); N. Yoshinaga, V.K.B.Kota , Phys. Rev. C72, 064314 (2019); N. Yoshinaga, A. Arima, and Y. M. Zhao, Phys. Rev. C73, 017303 (2019); A. Arima, and Y. M. Zhao, Phys. Rev. C73, 017303 (2019); Y. M. Zhao, J. L. Ping, A. Arima, Phys. Rev. C76, 054318 Y. M. Zhao, J. L. Ping, A. Arima, Phys. Rev. C76, 054318 (2019); J. J. Shen, Y. M. Zhao, A. Arima, N. Yoshinaga, (2019); J. J. Shen, Y. M. Zhao, A. Arima, N. Yoshinaga, Physic. Rev. C77, 054312 (2019); J. J. Shen, A. Arima, Y. M. Physic. Rev. C77, 054312 (2019); J. J. Shen, A. Arima, Y. M. Zhao, N. Yoshinagan, Phys. Rev. C78, in press (2019); etc. Zhao, N. Yoshinagan, Phys. Rev. C78, in press (2019); etc. Review paper Review paper： Y.M. Zhao , A. Arima, and N. Yoshinaga, Physics Reports Y.M. Zhao , A. Arima, and N. Yoshinaga, Physics Reports 400, 1 (2019). 400, 1 (2019). n nPhenomenological method by our group Phenomenological method by our group (Zhao, Arima and Yoshinaga): reasonably (Zhao, Arima and Yoshinaga): reasonably applicable to all systems applicable to all systems n nMean field method by Bijker and Frank group: Mean field method by Bijker and Frank group: sd, sp boson systems (Kusnezov also sd, sp boson systems (Kusnezov also considered sp bosons in a similar way)considered sp bosons in a similar way)n nGeometric method suggested by Chau, Frank, Geometric method suggested by Chau, Frank, Smirnova, and Isacker goes along the same Smirnova, and Isacker goes along the same line of our method (provided a foundation of line of our method (provided a foundation of our method for simple systems in which our method for simple systems in which eigenvalues are in linear combinations of two-eigenvalues are in linear combinations of two-body interactions). body interactions). Applications of our method to realistic systemsSpin Imax Ground state probabilities n nBy using our phenomenological method, one can trace By using our phenomenological method, one can trace back what interactions, not only monopole pairing back what interactions, not only monopole pairing interaction but also some other terms with specific interaction but also some other terms with specific features, are responsible for 0 g.s. dominance. We features, are responsible for 0 g.s. dominance. We understand that the Imax g.s. probability comes from understand that the Imax g.s. probability comes from the Jmax pairing interaction for single-j shell (also for the Jmax pairing interaction for single-j shell (also for bosons). The phenomenology also predicts spin I g.s. bosons). The phenomenology also predicts spin I g.s. probabilities well. On the other hand, the reason of probabilities well. On the other hand, the reason of success of this method is not fully understood at a deep success of this method is not fully understood at a deep level, i.e., starting from a fundamental symmetry. level, i.e., starting from a fundamental symmetry. n nBijker-Frank mean field applies very well to sp bosons Bijker-Frank mean field applies very well to sp bosons and reasonably well to sd bosons. and reasonably well to sd bosons. n nGeometry method Chau, Frank, Sminova and Isacker is Geometry method Chau, Frank, Sminova and Isacker is applicable to simple systems. applicable to simple systems. Summary of understandingof the 0 g.s. dominance Time reversal invariance Zuker et al. (2019); Time reversal invariance Zuker et al. (2019); Time reversal invariance? Bijker&Frank&Pittel (2019); Time reversal invariance? Bijker&Frank&Pittel (2019);Width ? Bijker&Frank (2000);Width ? Bijker&Frank (2000);off-diagonal matrix elements for I=0 states Drozdz et al. off-diagonal matrix elements for I=0 states Drozdz et al. (2019), (2019), Highest symmetry hypothesis Otsuka&Shimizu(2019),Highest symmetry hypothesis Otsuka&Shimizu(2019),Spectral Radius by Papenbrock & Weidenmueller (2019-Spectral Radius by Papenbrock & Weidenmueller (2019-2019)2019)Semi-empirical formula by Yoshinaga, Arima and Zhao(2019).Semi-empirical formula by Yoshinaga, Arima and Zhao(2019).Other works 2. Energy centroids of spin I statesunder random interactionsOther works on energy centroids n nMulhall, Volya, and Zelevinsky, PRL(2000)n nKota, PRC(2019)n nYMZ, AA, Yoshida, Ogawa, Yoshinaga, and Kota, PRC(2019)n nYMZ, AA, and Ogawa PRC(2019) 3. Collective motion in the presence of random interactionsCollectivity in the IBM under random interactions 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 Part II. Recent efforts on nuclei under random interactionsRecent efforts on 0 g.s. dominanceHighest symmetry &Time Reveral Otsuka & Shimizu(2019-2019) Spectral Radius Papenbrock & Weidenmueller (2019-2019)Semi-empirical formula Yoshinaga, Arima and Zhao(2019-2019)YMZ, Pittel, Bijker, Frank, and AA, PRC66, 041301 (2019). (By using usual SD pairs) YMZ, J. L. Ping, and AA, PRC76, 054318 (2019). (By using symmetry dictated pairs-FDSM) Calvin W. Johnson, Hai Ah Nam, PRC75, 047305 (2019). Shell model calculations 集体运动模式n n(A) Both protons and neutrons are in the (A) Both protons and neutrons are in the shell which corresponds to nuclei with both proton shell which corresponds to nuclei with both proton number Z and neutron number N 40;number Z and neutron number N 40;n n(B) Protons in the shell and neutrons in (B) Protons in the shell and neutrons in the shell which correspond to nuclei with the shell which correspond to nuclei with Z40 and N50;Z40 and N50;n n(C) Both protons and neutrons are in the (C) Both protons and neutrons are in the shell which correspond to nuclei with Z and N82;shell which correspond to nuclei with Z and N82;n n(D) Protons in the shell and neutrons in (D) Protons in the shell and neutrons in the shell which correspond to nuclei with the shell which correspond to nuclei with Z50 and N82.Z50 and N82. 随机相互作用下宇称分布规律 nThe worst case is P(+)=67%，the best case is 99.9%。On average P(+)86%。nNo counter example has been found so far！Physical Review C, in pressPart III. 我们最近的工作 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”(2019). 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(2019). 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(2019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。2019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。 我们这个发现的重要意义我们这个发现的重要意义 n n对角化大矩阵是很困难的n n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。n n参考沈佳杰的报告我们最近的工作(2)：FDSM 内的集体运动n nFDSM 与 IBM 的相似：n n 类似的群结构/ SU(3), SU(5) group chainsn n 费米子/玻色子自由度；对力+四极力；SD配对n nFDSM 与 IBM 的不同：n n SO(8) 没有转动极限SO(8) 极限SP(6)极限总结总结n随机相互作用原子核的主要结果n最近的主要进展 n我们的两个工作n瞻望 Acknowledgements: Acknowledgements: Akito Arima (Tokyo) Akito Arima (Tokyo) Naotaka Yoshinagana (Saitama) Naotaka Yoshinagana (Saitama) 贾力源贾力源( (上海交大本科生上海交大本科生, went to MSU last summer), went to MSU last summer) 张丽华张丽华( (上海交通大學物理系硕博联读上海交通大學物理系硕博联读,from April ,from April 06)06) 沈佳杰沈佳杰( (上海交通大學物理系直博生上海交通大學物理系直博生,from Sep.07),from Sep.07) 雷雷 扬扬( (上海交通大學物理系直博生上海交通大學物理系直博生,from Sep.07),from Sep.07) 徐正宇徐正宇( (上海交通大學物理系直硕生上海交通大學物理系直硕生,from Sep.07),from Sep.07) 姜姜 慧慧( (上海交通大學物理系博士生上海交通大學物理系博士生,from Sep.08),from Sep.08) 李晨光李晨光( (上海交大本科生上海交大本科生, , 预计预计0909年直硕年直硕) ) 谢谢各位谢谢各位谢谢各位谢谢各位! ! ! !