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３つのノルムオーバーラップ計算法の比較：大西公式、 Neergård-Wüst 法、そして Pfaffian 公式

３つのノルムオーバーラップ計算法の比較：大西公式、 Neergård-Wüst 法、そして Pfaffian 公式. 大井万紀人、水崎高浩 （専修大学・自然科学研究所）. The Onishi formula: with a square root. Onishi and Yoshida: Nucl.Phys . 80 (1966) Onishi and Horibata : PTP 64 (1980) . Various methods for a determination of sign. Continuity of a norm overlap

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３つのノルムオーバーラップ計算法の比較：大西公式、 Neergård-Wüst 法、そして Pfaffian 公式

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  1. ３つのノルムオーバーラップ計算法の比較：大西公式、Neergård-Wüst法、そしてPfaffian公式３つのノルムオーバーラップ計算法の比較：大西公式、Neergård-Wüst法、そしてPfaffian公式 大井万紀人、水崎高浩 （専修大学・自然科学研究所）

  2. The Onishi formula: with a square root Onishi and Yoshida: Nucl.Phys. 80 (1966) Onishi and Horibata: PTP 64 (1980)

  3. Various methods for a determination of sign • Continuity of a norm overlap • with respect to the Euler angles • The Neergård-Wüst method • The Pfaffian method

  4. Phase continuity: Nodal lines K.Hara, A.Hayashi, P. Ring, Nucl. Phys. A606 (1980) M. Oi and N. Tajima, Phys. Lett. B 606 (2005) M. Oi, et al., in preparation (2012) --- Limbo-dance method

  5. Robledo’sPfaffian formula Pfaffian: a polynomial M: anti-symmetric L.Robledo, Phys.Rev. C 79 (2009)

  6. Anti-symmetric matrix Bipartite expression

  7. Following works with Pfaffian - G. Bertsch and L.M. Robledo, Phys. Rev. Lett. 108 (2012) 042505 - B. Avez and M. Bender, Phys. Rev. C 85 (2012) 034325 - M. Oiand T. Mizusaki, Phys. Lett. B 707 (2012) 305-310 - T. Mizusaki and M. Oi, Phys. Lett. B 715 (2012) 219-224

  8. Mizusaki-Oi (Wick’s theorem) - T. Mizusaki and M. Oi, Phys. Lett. B 715 (2012) 219-224

  9. Anti-symmetric matrix: inverse

  10. Neergård-Wüst method - K. Neergård and E. Wüst, Nucl. Phys. A402 (1983) 311-321. : A polynomial in x

  11. The condition to be a polynomial Due to the Onishi formula: : diagonalisation for λk A necessity to be double-root structure: (…..)2, or Pair-wise eigenvalues: (λ1, λ1), (λ2, λ2), (λ3, λ3),….

  12. The Neergård-Wüstformula : a general complex matrix ! LINPACK for eigenvalues of a general complex matrix: zgeev.f (based on QR method)

  13. Reputations of the Neergård-Wüstformula Avez-Bender (PRC85, 2012): “the practical application of the NW technique becomes cumbersome in realistic cases, and has been rarely used in practice.” Schmidt (PPNP52, 2004): VAMPIR “This problem has been first solved by NW, who designed a method to determine the sign of the square root in a unique way. This method is also used in all our numerical applications.”

  14. Reputations of the Neergård-Wüstformula Robledo(PRC79, 2009): “Handling the eigenvalues of non-Hermitian matrices is a difficult task, that increases its complexity if the pairwise degenerate eigenvalues have to be obtained numerically without any symmetry enforcing degeneracy.” (Boson)

  15. Numerical result Eignevalues at (0,0,9) Norm values 0.311472372 464998 -7.549971869928256E-008 with the NW method 0.311472372 340810 0.000000000000000E+000 with the Pfaffian

  16. Dimension doubling(bipartite expression)

  17. Dim-doubling: result Eigenvalues 65 1.0000000000000000 -0.1063064338732795 66 0.9999999999999988 -0.1063064338735858 67 1.0000000000000204 -0.1854641225420897 68 0.9999999999999850 -0.1854641225420917 69 1.0000000000000018 -3.7008888331375616 70 1.0000000000000009 -3.7008888331380834 71 1.0000000000000080 -5.6771668708735454 72 0.9999999999999998 -5.6771668708927345 73 0.9999999999999989 -16.2926360844221456 74 0.9999999999999999 -16.2926360844227140 0.3114723724649982.174913856448744 E-007 with the NW method Norm overlap 0.311472372340810 0.000000000000000 E+000 with the Pfaffian 0.3114723723403805.187449527856361 E-010 with the DD NW method

  18. Angular momentum projection 170Dy : cranked HFB, P+QQ

  19. Probability distribution: J=20

  20. Probability distribution: J=2

  21. Probability distribution: odd I

  22. Sum rule: numerical result For J=20 (cranking). Imax = 60 ℏ 1.000000189189: NW(original) 0.999997950656: NW(Dim-double) 0.999870838580: Pfaffian

  23. Computational performace Original NW : 1.634 sec (x1) Pfaffian : 3.199 sec (x2) Dim-double NW : 5.099 sec (x3) with i7-875K (OC) for 180 Euler points • For the safety of numerical accuracy, -no-prec-div in ifort cannot be switched on, costing computational performance at ~16%. • The version of ifort must be 12.1 or higher!

  24. Conclusion ○The Neergård-Wüst method was revisited. ○The pair structure tends to be slightly lost in the original form, but by means of the “dimension-doubling” formula, the accuracy is slightly improved. ○These errors do not cause serious problems in angular momentum projection. This is because the errors scatter randomly in the Euler space, unlike the continuity method. ○ Balance between accuracy and comp. performance.

  25. Final form of the formula M. Oiand T. Mizusaki, Phys. Lett. B 707 (2012) 305-310

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