Recent Progress on the 0 + Dominance

Transcription

1 Recent Progress on the 0 + Dominance Naotaka Yoshinaga Saitama University, APAN N. Yoshinaga-2006-CNS 1

2 Outline of my talk ntroduction Formulation in single j-shells Estimation of the ground state energy Application to the P() Summary n collaboration with A. Arima and Y.M. Zhao N. Yoshinaga-2006-CNS 2

3 ntroduction ohnson, Bertsch and Dean (1998) 0 + predominance N. Yoshinaga-2006-CNS 3

4 C. W. ohnson, G. F. Bertsch, D.. Dean and. Talmi Second paper Calculation on O, Ca, Mg RQE : random quasiparticle ensemble TBRE: two-body random ensemble RQE-NP: random quasiparticle ensemble-no pairing RQE-SPE: random quasiparticle ensemble with single-particle energies N. Yoshinaga-2006-CNS 4

5 References Y.M. Zhao, A. Arima, N.Yoshinaga, Phys. Rep. 400 (2004) 1. C.W. ohnson, G.F. Bertsch, D.. Dean, Phys. Rev. Lett. 80 (1998) C.W. ohnson, G.F. Bertsch, D.. Dean,. Talmi, Phys. Rev. C 61 (1999) Y.M. Zhao, A. Arima, Phys. Rev. C 64 (2001) Y.M. Zhao, A. Arima, N.Yoshinaga, Phys. Rev. C 66 (2002) A. Arima, N.Yoshinaga,Y.M. Zhao, Eur. Phys.. A 13 (2002) 105. Y.M. Zhao, A. Arima, N.Yoshinaga, Phys. Rev. C 66 (2002) N.Yoshinaga, A. Arima,Y.M. Zhao,. Phys. A 35 (2002) Y.M. Zhao, A. Arima, N.Yoshinaga, Phys. Rev. C 68 (2003) A. Arima, N.Yoshinaga,Y.M. Zhao, Nucl. Phys. A 722 (2003) 234c. Y.M. Zhao, A. Arima,.N. Ginocchio, N.Yoshinaga, Phys. Rev. C 68(2003) Y. M. Zhao, A. Arima, N. Shimizu, K. Ogawa, N. Yoshinaga, and O. Scholten, Phys. Rev. C 70 (2004) V. Zelevinsky, A. Volya, Phys. Rep. 391 (2004) 311. N. Yoshinaga-2006-CNS 5

6 Formulation in single-j shells n order to simplify argument, we take j n -configuration. Hamiltonian 2 j-1 ( ) ( ) (0) = å + é ë ùû = 0 Hˆ 2 1 G A A 1 1 A = a a A = - a a 2 ë û 2 ë û ( ) ( ) ( ) ( ) é j j ù, é j j ù Two-body random ensemble (TBRE) : The ensemble average : 1 é G r( G ) = exp ê - 2p ë 2 G = 0, G G = d 2 K K ù ú û N. Yoshinaga-2006-CNS 6

7 Matrix elements of with dimensiton Ĥ 2 j-1 n ˆ n bg =< b g >= å a bg = 0 H j H j G n( n -1) a = < j Kd, j } j b >< j Kd, j } j g > å n-2 2 n n-2 2 n bg 2 Kd Definition of the matrix α ( α ) º a bg ( b, g = 1,, d ) bg d for spin- states c.f.p. with dimensiton c.f.p. N. Yoshinaga-2006-CNS 7 d

8 0 + dominance examples Probabilities of Spin= ground states : P( ) All probabilities are obtained by 1000 runs of the TBRE Hamiltonian in One sees clearly the Spin = 0 dominance in this figure. N. Yoshinaga-2006-CNS 8 j 4

9 Empirical approach to predict P() We set one of the two-body matrix elements G = -1 and all others 0. We find which angular momentum gives the lowest eigenvalue among all the eigenvalues of the shell model diagonalization. N. Yoshinaga-2006-CNS 9

10 How many times does a certain angular momentum gives the lowest eigenvalues among all the possible eigenvalues? We predict that the probability of where g.s. is given by P( ) = N / N N G ( = (2 j + 1) / 2) is the number of N. Yoshinaga-2006-CNS 10 N

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12 Two main problems What is the origin of spin=0 dominance? What quantities characterize the ground state energy for each angular momentum? -- How to estimate the ground state energy in many-body problems? N. Yoshinaga-2006-CNS 12

13 Estimation of the ground state energy We assume the ground state energy of spin- states as follows The distribution of enegies are assumed to be Gaussian (min) E = E - F( d ) s { G } Average Width N. Yoshinaga-2006-CNS 13

14 Let us take the following guess to the lowest eigen-value of the Hamiltonian Eb ( b = 1,2, d ) (min) E Ĥ by assuming that eigen-energies follow a gaussian distribution d é r( E ) = exp ê- 2ps ê ë 2 ( E - E ) 2 ( s ) To estimate, we need to solve the following equation ; E E ò (min) d r ( E ) de = -1 2 F d Estimation of the factor ( ) (min) E 2 N. Yoshinaga-2006-CNS 14 ù ú ú û

15 This is converted to the following equation by the change of variable where the error function is defined as We cannot solve this equation analytically, but for large we get by using the asymptotic expansion of the error function for its large argument. Thus, we have M Erfc( t ) = p d x º é-t ù dt Erfc( ) exp ë M t» - ln d - ln(4p ln d ) (min) E = E - 2ln d -ln(4p ln d ) s ò x t (min) M = 2s N. Yoshinaga-2006-CNS 15 E d û - E F( d )

16 Accordingly we have the estimate of the minimum energy for Here { } G F ( d ) = ln d - ln(4p ln d ) 1 2 s and the width Note that this guess is only valid for d >> 1 { } E = E - ( ) s (min) F d G Average 1 d { G } = Tr é( Hˆ - E ) 1 = å Tr é - - d ë α α ë, K 2 ù û Width K K ( a )( a ) (min) E ù û G G K N. Yoshinaga-2006-CNS 16

17 Numerical Check of our formula Let us check our formula numerically. We calculate E (min) - E as a function of s { G } { } (min) E = E -F( d ) s G N. Yoshinaga-2006-CNS 17

18 j = 31/2 n = 4 = 0 E - E (min) { G } N. Yoshinaga-2006-CNS 18 s F

19 Factors 2 F F ( ) 2 d for j=31/2, n=4 Data points Fit curve Assymptotic factor ln(d ) N. Yoshinaga-2006-CNS 19

20 Application of our new formula (min) E = E - F( d ) s { G } F ( d ) = a ln( d ) + b Surprisingly factors a, b a = 0.99, b = 0.36 are insensitive to the shell-size particle number as far as single-j shells are concerned. We calculate P() using this formula Width-prediction method N. Yoshinaga-2006-CNS 20

21 Application to single- j shell 11 j = shell 2 P( ) g.s. probabilities (%) 40 TBRE Emp pred Width pred ( h) N. Yoshinaga-2006-CNS 21

22 Fermion systems : Single- j j = 7/2 to j = 31/2 n = 4 N. Yoshinaga-2006-CNS 22

23 P( = 0 ) P( max ) P(=0) (%) P( max ) (%) (a) (b) TBRE Emp pred Width pred j ( h) N. Yoshinaga-2006-CNS 23

24 Boson systems : Single- l l = 1 to l = 15 n = 4 N. Yoshinaga-2006-CNS 24

25 P( = 0 ) P( max ) P(=0) (%) P( max ) (%) (a) (b) TBRE Emp pred Width pred l ( h) N. Yoshinaga-2006-CNS 25

26 Papenbrock and Weidenmuller Phys. Rev. Lett. 93(2004) E (min) = - r s P { G } s P 1 { } Tr ( ˆ ) 2 G = é H ù d ë û N. Yoshinaga-2006-CNS 26

27 Their P( ) for j = 19/ 2 and n = 6. N. Yoshinaga-2006-CNS 27

28 Microscopic origin of the 0+ dominance a 2 j-1 n ˆ n bg =< b g >= å a bg = 0 H j H j G n( n -1) a = å< j Kd, j } j b >< j Kd, j } j g > n-2 2 n n-2 2 n bg 2 Kd d 1 1 Tr ( a α ) º å bb = d b d ( ) Tr ( α a ) 1 1 s = - = a a - a d d ( ) å bg gb ( ) c.f.p. b, g c.f.p. N. Yoshinaga-2006-CNS 28

29 Random phase approximation 1 d E å Min Tr é» ( α )( α ) K K K ë û K K, K, K E å - F( d ) = a G - å -a - a ùg G» s s = a G - F( d ) å å s s G G å s å G G G Approx-A Approx-B N. Yoshinaga-2006-CNS 29

30 P(=0) (%) TBRE Approx A Approx B j ( h) N. Yoshinaga-2006-CNS 30

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32 Large fluctuation of alpha and sigma a 2 j-1 n ˆ n bg =< b g >= å a bg = 0 H j H j G n( n -1) a = å< j Kd, j } j b >< j Kd, j } j g > n-2 2 n n-2 2 n bg 2 Kd å d 1 1 Tr ( a α ) º å bb = d b a n( n -1) = 2 d c.f.p. c.f.p. N. Yoshinaga-2006-CNS 32

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34 Conclusion A new formula is proposed to estimate the ground state of spin- states of a TBRE hamiltonian. The probability P( ) using our new formula gives a good agreement with P( ) of TBRE for both fermions and bosons in single orbitals The microscopic origin of the spin-0 dominance is much easier to access by using our new formula. N. Yoshinaga-2006-CNS 34