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Outline Simple comments on regularities of many-body systems under random interactions Number of spin I states for single-j configuration J-pairing interaction Sum rules of angular momentum recoupling coefficients Number of states with given spin and isospin Nucleon approximation of the shell model Prospect and summary Simple comments on regularities of many-body systems under random interactions Number of spin I states for single-j configuration J-pairing interaction Sum rules of angular momentum recoupling coefficients Number of states with given spin and isospin Nucleon approximation of the shell model Prospect and summary
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Part I A brief introduction to nuclei under random interactions Two-body random ensemble
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In 1998, Johnson, Bertsch, and Dean found spin zero ground state dominance can be obtained by random two-body interactions (Phys. Rev. Lett. 80, 2749) . Ref. C. W. Johnson et al., PRL80, 2749 (1998); R.Bijker et al., PRL84, 420 (2000); L. Kaplan et al., PRB65, 235120 (2002).
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Intrinsic collectivity based on the sd IBM
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Energy centroids of spin I states
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A short summary Spin 0 ground state dominance for even-even nuclei, regularities for energy centroids with given quantum numbers, collectivity, etc. Open questions: spin distribution in the ground states ; energy centroids ; requirement for nuclear collectivity ; etc. For a review, See YMZ, AA, NY, Physics Reports, Volume 400, Page 1 (2004).
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Themes and Challenges of Modern Science Complex systems arising out of basic constituents How the world, with all its apparent complexity and diversity, can be constructed out of a few elementary building blocks and their interactions Simplicity out of complexity How the world of complex systems can display such remarkable regularity and, often, simplicity Understanding the nature of the physical universe Manipulating matter for the benefit of mankind
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Part II Number of states for identical particles in a single-j shell Why we study this number (Ginocchio)?
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A simple method in the text-book
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Empirical formulas YMZ and AA, PRC68, 044310 (2003). empirical formulas for n=3,4. For example, For n=4, results are more complicated (omitted).
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A new method
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Conjugates of
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An Example
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An example: n=4 Here we should study bosons with SU(5)symmetry, i.e., d bosons. The number of states for d bosons has been studied in the interacting boson model. By using results of the IBM, we were able to obtain dimension for d bosons. Then we quickly get the number of states for n=4 of fermions and bosons. What about odd number of particles? YMZ and AA, PRC71, 047304 (2005)
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Other works Dimension for n>3: J. N. Ginocchio and W. C. Haxton, “Symmetries in Science VI”, Edited by B. Gruber and M. Ramek, (Plenum, New York, 1993). Number of states for Zamick et al. Physical Review C71, 054308 (2005).
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For n=3, we should study SU(4) reduction rule. However, Talmi proved our results for n=3(Physical Review C72,037302(2005)). He obtained some recursion formulas and proved the formulas by reduction method. This method can be used to prove any formulas (in principle) but it can not be used to find new formulas.
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The guideline of Talmi’s efforts First, he assumes the formula is correct for j-1 shell, then it suffices to show that it is also correct to j shell. Next he enumerates the effect by changing m1= j-1 to j. Summing this effect, he obtains the dimension of j shell.
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Part III J-pairing interaction Fermions in a single-j shell:
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J –pair truncation of the shell model Empirically we find that J-pair truncation is good for J- pairing interaction.
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PartIV Sum rules of angular momentum recoupling coefficients For n=3, J-pair truncation gives exact solution.
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An example
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Similar things can be done for n=4 Ref.: YMZ & AA (Physical Review C72, 054307 (2005) Here we obtain sum rules of 9-j symbols. The difference is that here situation is more complicated. Generally speaking, the number of nonzero eigenvalues is not always one for J pairing interaction and each of eigenvalue is unknown. However, the trace of eigenvalues for each I states with only J-pairing interactions is always a constant with respect to orthogonal transformation:
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Proof
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Sum rules Summing over J, we obtain sum rules of 9-j symbols
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Sum rules with odd J and odd K
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Sum rules with both even and odd J,K
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Two examples
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Part V Number of states for nucleons in a single-j shell [An application of these sum rules] References: L.Zamick et al., Physical Review C72, 044317 (2005); Y.M.Z. and A.A., Physical Review C72, 064333 (2005).
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JT-pairing interaction Nucleons in a single-j shell:
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Similarly,
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The case of T=0
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Part VI Our next effort on pair approximation IBM SD collective pairs diagonalization of the shell model Hamiltonian in nucleon pair subspace. How far one can go? How reliable is “collective pair” approximation? What can one calculate by using pair appro. ?
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We have done following work Validity of SD pair truncation for special cases Application to A=130 even-even nuclei (rather successful calculation) We proposed an efficient algorithm to describe even systems and odd-A (also doubly odd) systems on the same footing.
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The nuclei we shall try in the near future Mass number A around 140, neutron rich side (both even and odd A, both positive and negative parity) Validity of SD truncation, realistic cases. [How good or not good is the pair truncation?] Extension of pairs [S1, S2, D1, D2, F pairs, G pairs, etc. ]
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Part VII Summary & prospect Nuclear structure under random interactions Number of states with given spin (&isospin), sum rule of 6j and 9j symbols Our next project: Nucleon pair approximation of the shell model: odd and doubly odd nuclei. Applications to realistic systems
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Acknowledgements: Akito Arima (Tokyo) Naotaka Yoshinaga (Saitama) Kengo Ogawa (Chiba) Nobuaki Yoshida (Kansai)
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International Conference on Nuclear Structure Physics, Shanghai, June 12-17th, 2006.
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