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1、 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最近其他研究组几个工作 n我们最近的工作n展望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.
2、Ref: Ann. Math. 6767, 325 (1958)1970s 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. Flore
3、s, Phys. Lett. B34, 261 (1970). Other applications: complicated systems (e.g., quantum chaos)Two-body Random ensemble (TBRE) 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 interac
4、tions.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).One usually choose Gaussian distribution for two-body random i
5、nteractionsThere 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 ResultsAv
6、ailable ResultsEmpircalEmpircal method method Zhao & Zhao & ArimaArima & & YoshinagaYoshinaga (2002) (2002)Mean-field method Mean-field method BijkerBijker-Frank (2003)-Frank (2003)Geometrid method Geometrid method ChauChau et al. (2003) et al. (2003) -Time reversal invariance (TRI)Time reversal inv
7、ariance (TRI) Zuker et al. (2002); Zuker et al. (2002);Time reversal invariance? Time reversal invariance? Bijker&Frank&PittelBijker&Frank&Pittel (1999); (1999);Width ?Width ? Bijker&FrankBijker&Frank (2000); (2000);off-diagonal matrix elements for I=0 states off-diagonal matrix elements for I=0 sta
8、tes DrozdzDrozdz et al. (2001) et al. (2001) Highest symmetry &Time Highest symmetry &Time ReveralReveral Otsuka&Shimizu(2004-2007) Otsuka&Shimizu(2004-2007) Spectral RadiusSpectral Radius PapenbrockPapenbrock & Weidenmueller (2004-2007) & Weidenmueller (2004-2007)Semi-empirical formulaSemi-empirica
9、l formula YoshinagaYoshinaga, Arima and Zhao(2006-2007), Arima and Zhao(2006-2007)References after Johnson, References after Johnson, BertschBertsch and Dean and Dean R. R. BijkerBijker, A. Frank, and S. Pittel, Phys. Rev. C60, 021302(1999); D. , A. Frank, and S. Pittel, Phys. Rev. C60, 021302(1999)
10、; D. MulhallMulhall, A. , A. VolyaVolya, and V. , and V. ZelevinskyZelevinsky, Phys. Rev. Lett.85, 4016(2000); , Phys. Rev. Lett.85, 4016(2000); NuclNucl. Phys. A682, 229c(2001); V. . Phys. A682, 229c(2001); V. ZelevinskyZelevinsky, D. , D. MulhallMulhall, and A. , and A. VolyaVolya, , YadYad. . Fiz
11、Fiz. 64, 579(2001); D. . 64, 579(2001); D. KusnezovKusnezov, Phys. Rev. , Phys. Rev. LettLett. 85, 3773(2000); ibid. . 85, 3773(2000); ibid. 87, 029202 (2001); L. Kaplan and T. 87, 029202 (2001); L. Kaplan and T. PapenbrockPapenbrock, Phys. Rev. , Phys. Rev. LettLett. 84, 4553(2000); . 84, 4553(2000
12、); R.BijkerR.Bijker and and A.FrankA.Frank, Phys. Rev. Lett.87, 029201(2001); S. , Phys. Rev. Lett.87, 029201(2001); S. DrozdzDrozdz and M. and M. WojcikWojcik, , PhysicaPhysica A301, 291(2001); L. A301, 291(2001); L. Kaplan, T. Kaplan, T. PapenbrockPapenbrock, and C. W. Johnson, Phys. Rev. C63, 014
13、307(2001); R. , and C. W. Johnson, Phys. Rev. C63, 014307(2001); R. BijkerBijker and A. Frank, and A. Frank, Phys. Rev. C64, (R)061303(2001); R. Phys. Rev. C64, (R)061303(2001); R. BijkerBijker and A. Frank, Phys. Rev. C65, 044316(2002); P.H- and A. Frank, Phys. Rev. C65, 044316(2002); P.H-T.ChauT.C
14、hau, A. Frank, , A. Frank, N.A.SmirnovaN.A.Smirnova, and , and P.V.IsackerP.V.Isacker , Phys. Rev. C66, 061301 (2002); L. Kaplan, , Phys. Rev. C66, 061301 (2002); L. Kaplan, T.PapenbrockT.Papenbrock, and G.F. , and G.F. BertschBertsch, Phys. Rev. B65, 235120(2002); L. F. Santos, D. , Phys. Rev. B65,
15、 235120(2002); L. F. Santos, D. KusnezovKusnezov, and P. , and P. JacquodJacquod, Phys. , Phys. LettLett. B537, 62(2002); T. . B537, 62(2002); T. PapenbrockPapenbrock and H. A. Weidenmueller, Phys. Rev. and H. A. Weidenmueller, Phys. Rev. LettLett. 93, . 93, 132503 (2004); T. 132503 (2004); T. Papen
16、brockPapenbrock and H. A. Weidenmueller, Phys. Rev. C and H. A. Weidenmueller, Phys. Rev. C 7373 014311 (2006); 014311 (2006); Y.M. Y.M. Zhao and A. Zhao and A. ArimaArima, Phys. Rev.C64, (R)041301(2001); A. , Phys. Rev.C64, (R)041301(2001); A. ArimaArima, N. , N. YoshinagaYoshinaga, and Y.M. Zhao,
17、, and Y.M. Zhao, Eur.J.PhysEur.J.Phys. A13, 105(2002); N. . A13, 105(2002); N. YoshinagaYoshinaga, A. , A. ArimaArima, and Y.M. Zhao, J. Phys. A35, 8575(2002); Y. , and Y.M. Zhao, J. Phys. A35, 8575(2002); Y. M. Zhao, A. M. Zhao, A. ArimaArima, and N. , and N. YoshinagaYoshinaga, Phys. Rev.C66, 0343
18、02(2002); Y. M. Zhao, A. , Phys. Rev.C66, 034302(2002); Y. M. Zhao, A. ArimaArima, and N. , and N. YoshinagaYoshinaga, Phys. Rev. C66, 064322(2002); , Phys. Rev. C66, 064322(2002); Y.M.ZhaoY.M.Zhao, A. , A. ArimaArima, N. , N. YoshinagaYoshinaga, Phys.Rev.C66, , Phys.Rev.C66, 064323 (2002); Y. M. Zh
19、ao, S. Pittel, R. 064323 (2002); Y. M. Zhao, S. Pittel, R. BijkerBijker, A. Frank, and A. , A. Frank, and A. ArimaArima, Phys. Rev. C66, R41301 , Phys. Rev. C66, R41301 (2002); Y. M. Zhao, A. (2002); Y. M. Zhao, A. ArimaArima, G. J. Ginocchio, and N. , G. J. Ginocchio, and N. YoshinagaYoshinaga, Phy
20、s. Rev. C66,034320(2003); , Phys. Rev. C66,034320(2003); Y. M. Zhao, A. Y. M. Zhao, A. ArimaArima, N. , N. YoshingaYoshinga, Phys. Rev. C68, 14322 (2003); Y. M. Zhao, A. , Phys. Rev. C68, 14322 (2003); Y. M. Zhao, A. ArimaArima, N. , N. Shimizu, K. Ogawa, N. Shimizu, K. Ogawa, N. YoshinagaYoshinaga,
21、 O. , O. ScholtenScholten, Phys. Rev. C70, 054322 (2004); , Phys. Rev. C70, 054322 (2004); Y.M.ZhaoY.M.Zhao, A. , A. ArimaArima, K. Ogawa, Phys. Rev. C71, 017304 (2005); Y. M. Zhao, A. , K. Ogawa, Phys. Rev. C71, 017304 (2005); Y. M. Zhao, A. ArimaArima, N. Yoshida, K. Ogawa, N. , N. Yoshida, K. Oga
22、wa, N. YoshinagaYoshinaga, and , and V.K.B.KotaV.K.B.Kota , Phys. Rev. C72, 064314 (2005); N. , Phys. Rev. C72, 064314 (2005); N. YoshinagaYoshinaga, A. , A. ArimaArima, and Y. M. , and Y. M. Zhao, Phys. Rev. C73, 017303 (2006); Y. M. Zhao, J. L. Ping, A. Zhao, Phys. Rev. C73, 017303 (2006); Y. M. Z
23、hao, J. L. Ping, A. ArimaArima, Phys. Rev. C76, 054318 , Phys. Rev. C76, 054318 (2007); (2007); J. J. Shen, Y. M. Zhao, A. J. J. Shen, Y. M. Zhao, A. ArimaArima, N. , N. YoshinagaYoshinaga, Physic. Rev. C77, 054312 (2008); J. J. , Physic. Rev. C77, 054312 (2008); J. J. Shen, A. Shen, A. ArimaArima,
24、Y. M. Zhao, N. , Y. M. Zhao, N. YoshinaganYoshinagan, Phys. Rev. C78, in press (2008);, Phys. Rev. C78, in press (2008); etc. etc. Review paper Review paper： Y.M. Zhao , A. Y.M. Zhao , A. ArimaArima, and N. , and N. YoshinagaYoshinaga, , Physics ReportsPhysics Reports 400, 1 (2004). 400, 1 (2004). n
25、 nPhenomenological methodPhenomenological method by our group (Zhao, by our group (Zhao, ArimaArima and and YoshinagaYoshinaga): reasonably applicable to all systems ): reasonably applicable to all systems n nMean field method by Mean field method by BijkerBijker and Frank group: and Frank group: sd
26、sd, sp , sp boson systems (boson systems (KusnezovKusnezov also considered sp bosons in also considered sp bosons in a similar way)a similar way)n nGeometric method suggested by Geometric method suggested by ChauChau, Frank, , Frank, SmirnovaSmirnova, and Isacker goes along the same line of our , an
27、d Isacker goes along the same line of our method (provided a foundation of our method for method (provided a foundation of our method for simple systems in which simple systems in which eigenvalueseigenvalues are in linear are in linear combinations of two-body interactions). combinations of two-bod
28、y interactions). Applications of our method to realistic systemsSpin Imax Ground state probabilities n nBy using our phenomenological method, one can trace back By using our phenomenological method, one can trace back what interactions, not only monopole pairing interaction but what interactions, no
29、t only monopole pairing interaction but also some other terms with specific features, are responsible also some other terms with specific features, are responsible for 0 for 0 g.sg.s. dominance. We understand that the Imax . dominance. We understand that the Imax g.sg.s. . probability comes from the
30、 probability comes from the JmaxJmax pairing interaction for pairing interaction for single-j shell (also for bosons). The phenomenology also single-j shell (also for bosons). The phenomenology also predicts spin I predicts spin I g.sg.s. probabilities well. On the other hand, the . probabilities we
31、ll. On the other hand, the reason of success of this method is not fully understood at a reason of success of this method is not fully understood at a deep level, i.e., starting from a fundamental symmetry. deep level, i.e., starting from a fundamental symmetry. n nBijkerBijker-Frank mean field appl
32、ies very well to sp bosons and -Frank mean field applies very well to sp bosons and reasonably well to reasonably well to sdsd bosons. bosons. n nGeometry method Geometry method ChauChau, Frank, , Frank, SminovaSminova and Isacker is applicable and Isacker is applicable to simple systems. to simple
33、systems. Summary of understandingof the 0 g.s. dominance Time reversal invariance Time reversal invariance Zuker et al. (2002); Zuker et al. (2002); Time reversal invariance? Time reversal invariance? Bijker&Frank&PittelBijker&Frank&Pittel (1999); (1999);Width ?Width ? Bijker&FrankBijker&Frank (2000
34、); (2000);off-diagonal matrix elements for I=0 states off-diagonal matrix elements for I=0 states DrozdzDrozdz et al. (2001), et al. (2001), Highest symmetry hypothesis Highest symmetry hypothesis Otsuka&Shimizu(2004),Otsuka&Shimizu(2004),Spectral RadiusSpectral Radius by Papenbrock & Weidenmueller
35、(2004-2006) by Papenbrock & Weidenmueller (2004-2006)Semi-empirical formulaSemi-empirical formula by by YoshinagaYoshinaga, Arima and Zhao(2006)., Arima and Zhao(2006).Other works 2. Energy centroids of spin I statesunder random interactionsOther works on energy centroids n nMulhall, Volya, and Zele
36、vinsky, PRL(2000)n nKota, PRC(2005)n nYMZ, AA, Yoshida, Ogawa, Yoshinaga, and Kota, PRC(2005)n nYMZ, AA, and Ogawa PRC(2005) 3. Collective motion in the presence of random interactionsCollectivity in the IBM under random interactions Shell model: Horoi, Zelevinsky, Volya, PRC, PRL; Velazquez, Zuker,
37、 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(2004-2007) Spectral Radius Papenbrock & Weid
38、enmueller (2004-2007)Semi-empirical formula Yoshinaga, Arima and Zhao(2006-2007)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 (200
39、7). Shell model calculations 集体运动模式n n(A) Both protons and neutrons are in the shell (A) Both protons and neutrons are in the shell which corresponds to nuclei with both proton number Z which corresponds to nuclei with both proton number Z and neutron number N 40;and neutron number N 40;n n(B) Proto
40、ns in the shell and neutrons in the (B) Protons in the shell and neutrons in the shell which correspond to nuclei with Z40 and N50;shell which correspond to nuclei with Z40 and N50;n n(C) Both protons and neutrons are in the shell (C) Both protons and neutrons are in the shell which correspond to nu
41、clei with Z and N82;which correspond to nuclei with Z and N82;n n(D) Protons in the shell and neutrons in the (D) Protons in the shell and neutrons in the shell which correspond to nuclei with Z50 and N82.shell which correspond to nuclei with Z50 and N82. 随机相互作用下宇称分布规律 nThe worst case is P(+)=67%，th
42、e 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”(2008). J. J. Shen, Y. M. Zhao, A. Arima, and N. Yoshinaga, Physical Review C77, 054312. “Strong L
43、inear 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
44、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。 我们这个发现的重要意义我们这个发现的重要意
45、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： 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n nFDSM 与 IBM 的不同： 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展望 Acknowledgements:Acknowledgements:Acknowledgements:A
46、cknowledgements: Akito Arima (Tokyo)Akito Arima (Tokyo) NaotakaNaotaka YoshinaganaYoshinagana (Saitama) (Saitama) 贾力源贾力源贾力源贾力源( ( ( (上海交大本科生上海交大本科生上海交大本科生上海交大本科生, , went to MSU last summerwent to MSU last summer) ) ) ) 张丽华张丽华张丽华张丽华( ( ( (上海交通大學上海交通大學上海交通大學上海交通大學物理系硕博联读物理系硕博联读物理系硕博联读物理系硕博联读,from Apri
47、l ,from April ,from April ,from April 06)06)06)06) 沈佳杰沈佳杰沈佳杰沈佳杰( ( ( (上海交通大學上海交通大學上海交通大學上海交通大學物理系直博生物理系直博生物理系直博生物理系直博生,from Sep.07),from Sep.07),from Sep.07),from Sep.07) 雷雷雷雷 扬扬扬扬( ( ( (上海交通大學上海交通大學上海交通大學上海交通大學物理系直博生物理系直博生物理系直博生物理系直博生,from Sep.07),from Sep.07),from Sep.07),from Sep.07) 徐正宇徐正宇徐正宇徐正宇
48、( ( ( (上海交通大學上海交通大學上海交通大學上海交通大學物理系直硕生物理系直硕生物理系直硕生物理系直硕生,from Sep.07),from Sep.07),from Sep.07),from Sep.07) 姜姜姜姜 慧慧慧慧( ( ( (上海交通大學上海交通大學上海交通大學上海交通大學物理系博士生物理系博士生物理系博士生物理系博士生,from Sep.08),from Sep.08),from Sep.08),from Sep.08) 李晨光李晨光李晨光李晨光( ( ( (上海交大本科生上海交大本科生上海交大本科生上海交大本科生, , 预计预计预计预计0909年直硕年直硕年直硕年直硕) ) 谢谢各位谢谢各位谢谢各位谢谢各位! !
