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Regularities of many-body systems in the presence of random two-body interactions

Regularities of many-body systems in the presence of random two-body interactions. Yu-Min ZHAO. 1 Department of Physics, Shanghai Jiao Tong University, China; 2 Center of Theoretical Nuclear physics, IMP, CAS, Lanzhou, China; 3 Cyclotron Center, RIKEN, Japan; 4 CCAST, CAS, Beijing, China.

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Regularities of many-body systems in the presence of random two-body interactions

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  1. Regularities of many-body systems in the presence of random two-body interactions Yu-Min ZHAO 1 Department of Physics, Shanghai Jiao Tong University, China; 2 Center of Theoretical Nuclear physics, IMP, CAS, Lanzhou, China; 3 Cyclotron Center, RIKEN, Japan; 4 CCAST, CAS, Beijing, China In collaboration with Prof. ARIMA Akito, YOSHINAGA Naotaka, and others

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

  3. 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) Random matrices and random two-body interactions

  4. One usually chooses 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）

  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

  7. Why this result is interesting？．．．　．．．

  8. 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).

  9. Three interesting results • Phenomenological method by Tokyo group (namely, by us) reasonably applicable to all systems • Geometric method by GANIL group applicable to “simple” systems • Mean field method by Mexico group applicable to sd, sp boson systems

  10. Recent Efforts • By Papenbrock & Weidenmueller by using correlation between Energy radius • By Yoshinaga & Arima & Zhao by using energy centroids and width • Hand waving ideas by a few groups (Zelevinsky, Zuker, Otsuka, and others)

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

  12. Probability of Imax g.s.

  13. A few examples

  14. Collectivity in the IBM under random interactions Taken from PRC62,014303(2000), by R. Bijker and A. Frank

  15. Collectivity in the IBM under random interactions Taken from PRL84,420(2000), by R. Bijker and A. Frank

  16. Collectivity in the SD subspace

  17. Energy centroids with fixed spin

  18. a few examples

  19. 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.

  20. 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

  21. 谢谢! arigedo! Acknowledgements: Akito Arima (Tokyo) Naotaka Yoshinaga (Saitama) Kengo Ogawa (Chiba) Noritake Shimizu(Tokyo) Nobuaki Yoshida (Kansai) Stuart Pittel (Delaware) R. Bijker (Mexico) J. N. Ginocchio (Los Alamos) Olaf Scholten (Groningen) V. K. B. Kota (Ahmedabad)

  22. Empirical method by Tokyo group

  23. d玻色子情形

  24. d玻色子情形

  25. Four fermions in a single-j shell

  26. 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．

  27. Collectivity in the SD subspace

  28. Taken from YMZ,AA and KO,to appear in PRC

  29. Collectivity in the SD subspace

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