Regular structure of atomic nuclei in thepresence of random interactions

Outline• A brief introduction of spin zero ground state (0 g.

s.) dominance• Present status along this line 1. Main results on studies of the 0 g.s. dominance 2. Positive parity ground state dominance 3. Collectivity of low-lying states by using the TBRE 4. Energy centroids of given spin states• Perspectives * Some simpler quantities (e.g., parity positive g.s.

dominance, energy centroids) can be studied first * Searching for other regularities (e.g., collectivity)

A brief introduction to the problem of

the 0 g.s. dominance

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. 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)

Two-body Random ensemble (TBRE)

21 2 3 4

1 2 3 4

( )

2

1( ) exp( )

21,

1/ 2,

JTj j j jGJT

j j j j xGx

x

1 2 3 4if | | ;

otherwise.

j j JT j j JT

One usually choose 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.

1 2 3 4 1 2 3 4| |JTj j j jG j j JT V j j JT

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 interactions (Phys. Rev. Lett. 80, 2749) ．

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).

† (0)

2 2

† † † ( )

2

A single- shell Hamiltonian:

2 1( ) ,

| | , 0,2, ,2 1

1( ) ,

21 1

( ) exp( )22

J JJ

J

J

J Jj j

J J

j

H J A A G

G j J V j J J j

A a a

G G

(1) (1) (1) (1) (1)0 2 4 6 8

(2) (2) (2) (2) (2)0 2 4 6 8

(3) (3) (3)0 2 4

9 For , 0,2,4,6,8.

2There are five independent two-body matrices.

Set 1: , , , , output (1);

Set 2: , , , , output (2);

Set 3: , ,

j J

G G G G G

G G G G G

G G G

(3) (3)

6 8

(1000) (1000) (1000) (1000) (1000)0 2 4 6 8

, , output (3);

Set 1000: , , , , output (1000);

G G

G G G G G

2

2

1

3

1

3

1

2

1

1

1

Spi n Di mensi on

0

2

3

4

5

6

7

8

9

10

12

j=9/2, n=4

References after Johnson, Bertsch and Dean

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); P.H-T.Chau, A. Frank, N.A.Smirnova, and P.V.Isacker, Phys. Rev. C66, 061301 (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); T. Papenbrock and H. A. Weidenmueller, Phys. Rev. Lett. 93, 132503 (2004); T. Papenbrock and H. A. Weidenmueller, Phys. Rev. C 73 014311 (2006); 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); 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); Y.M.Zhao, A. Arima, K. Ogawa, Phys. Rev. C71, 017304 (2005); Y. M. Zhao, A. Arima, N. Yoshida, K. Ogawa, N. Yoshinaga, and V.K.B.Kota , Phys. Rev. C72, 064314 (2005); N. Yoshinaga, A. Arima, and Y. M. Zhao, Phys. Rev. C73, 017303 (2006); Y. M. Zhao, J. L. Ping, A. Arima, Preprint; etc.

Review papers ：　 YMZ, AA, and N. Yoshinaga, Phys. Rep. 400, 1(2004); V. Zelevinsky and A. Volya, Phys. Rep. 391, 311 (2004).

1. Present status of understanding

the 0 g.s. dominance problem

Present status of this subjectPresent status of this subject

• Phenomenological method by our group (Zhao, Arima and Yoshinaga): reasonably applicable to all systems

• Mean field method by Bijker and Frank group: sd, sp boson systems (Kusnezov also considered sp bosons in a similar way)

• Geometric method suggested by Chau, Frank, Smirnova, and Isacker goes along the same line of our method (provided a foundation of our method for simple systems in which eigenvalues are in linear combinations of two-body interactions).

Our phenomenological method

( ) ( )

0(0) 0 2 4 6

7, 4 ( ) by TBRE exact ( ) pred(1) Pred(2)

25 3 13

19.9% 18.19% 14.3% 25% 6 2 6

32

I v I vE j n P I P I g

E G G G G

f or

2(2) 0 2 4 6

2(4) 2 4 6

3.14

1 11 3 13 1.2% 0.89% 0 0 3.25

2 6 2 642 13

31.7% 33.25% 11 11

E G G G G

E G G G

4(2) 0 2 4 6

4(4) 2 4 6

28.6% 25% 4.12

1 5 5 13 0 0 0 0 3.45

2 6 2 67 13

25.3 11

E G G G G

E G G G

5(4) 2 4 6

6(2) 0 2

0% 22.96% 28.6% 25% 3.68

8 192 26 0 0 0 0 3.62

7 77 111 5

2 6

E G G G

E G G

4 6

8(4) 2 4 6

3 19 0 0.02% 0 0 3.64

2 610 129 127

22.2% 24.15% 28.6% 25% 21 77 33

G G

E G G G

4.22

Our phenomenological method

Let find the lowest eigenvalue;

Repeat this process for all .'1, 0J J JG G

JG

( ) = g.s. probability

Number of time that

=

1 for a single-j shell)

2

I

P I I

N I

N

N j

appears i n the above process

Number of two-body matri x el ements

(

empirical ( ) /IP I N N

Four fermions in a single-j shell

2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

7 0 4 2 8

j J

9 0 4 0 0 12

11 0 4 0 4 8 16

13 0 4 0 2 2 12 20

15 0 4 0 2 0 0 16 24

17 0 4 6 0 4 2 0 20 28

19 0 4 8 0 2 8 2 16 24 32

21 0 4 8 0 2 0 0 0 20 28 36

23 0 4 8 0 2 0 10 2 0 24 32 40

25 0 4 8 0 2 4 8 10 6 0 28 36 44

27 0 4 8 0 2 4 2 0 0 4 20 32 40 48

29 0 4 8 0 0 2 6 8 12 8 0 24 36 44 52

31 0 4 8 0 0 2 0 8 14 16 6 0 32 40 48 56

0 2 4 6 8 10 12

0

20

40

60

80

0 5 10 15 20 25

0

20

40

60

0 2 4 6 8 10 12 140

20

40

60

80

0 5 10 15 20

0

20

40

60

80

a)

TBRE, pred.j=9/2 shell with 4 fermions

I g

.s. p

rob

ab

ilitie

s (

%)

TBRE, pred.j=9/2 shell with 5 fermions

c)

b)

TBRE, pred.7 fermions in the j1=7/2,j

2=5/2 orbits

angular momenta I

d)

TBRE, pred.10 sd bosons system

Applications of our method to realistic systems

Spin Imax Ground state probabilities

• By using our phenomenological method, one can trace back what interactions, not only monopole pairing interaction but also some other terms with specific features, are responsible for 0 g.s. dominance. We understand that the Imax g.s. probability comes from the Jmax pairing interaction for single-j shell (also for bosons). The phenomenology also predicts spin I g.s. probabilities well. On the other hand, the reason of success of this method is not fully understood at a deep level, i.e., starting from a fundamental symmetry.

• Bijker-Frank mean field applies very well to sp bosons and reasonably well to sd bosons.

• Geometry method Chau, Frank, Sminova and Isacker is applicable to simple systems.

Summary of understandingof the 0 g.s. dominance

Time reversal invariance Zuker et al. (2002);

Time reversal invariance? Bijker&Frank&Pittel (1999);

Width ? Bijker&Frank (2000);

off-diagonal matrix elements for I=0 states Drozdz et al. (2001),

Highest symmetry hypothesis Otsuka&Shimizu(~2004),

Spectral Radius by Papenbrock & Weidenmueller (2004-2006)

Semi-empirical formula by Yoshinaga, Arima and Zhao(2006).

Other works

2. Parity distribution in the ground

States under random interactions

Present status of this subjectPresent status of this subject

• (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.

7 / 2 5/ 2g d

5/ 2 1/ 2 9 / 2f p g

11/ 2 1/ 2 3/ 2h s d

5/ 2 1/ 2 9 / 2f p g

11/ 2 1/ 2 3/ 2h s d

7 / 2 5/ 2g d

(unit

Basis ( )

(0,4) (0,6) (2,2) (2,4) (2,6)

86.6 86.2 93.1 81.8 88.8

(2,3) (1,4) (0,5) (6,1) (2,1) (1,3) (1,5)

42.8 38.6 45.0

A

: %)

38.4 31.2 77.1 69.8

Basis ( )

(2,2) (2,4) (4, 2)

72.7 80.5 81.0

(3,4) (2,3) (3,2) (4,1) (1,4) (5,0) (3,3) (5,1)

42.5 72.4 39.1 75.1 26.4 4

B

4.1 79.4 42.9

Basis ( )

(2,2) (2,4) (4,0) (6,0)

92.2 81.1 80.9 82.4

(2,3) (5,0) (4,1) (1,5) (1,3)

52.0 42.6 56.5 64.4 73.0

Basis ( )

(2,2) (4,2

C

D

) (2, 4) (0,6)

67.2 76.1 74.6 83.0

(3,2) (2,3) (0,5) (3,3)

54.2 54.0 45.9 54.5

3. Collective motion in the presence

of random interactions

Present status of this subjectPresent status of this subject

Collectivity in the IBM under random interactions

Shell model: Horoi, Zelevinsky, Volya, PRC, PRL; Velazquez, Zuker, Frank, PRC; Dean et al., PRC; IBM:Kusnezov, Casten, et al., PRL; Geometric model: Zhang, Casten, PRC;

Other works

YMZ, Pittel, Bijker, Frank, and AA, PRC66, 041301 (2002). (By using usual SD pairs)

YMZ, J. L. Ping, and AA, preprint.(By using symmetry dictated pairs)

Our works by using SD pairs

4. Energy centroids of spin I states

under random interactions

Present status of this subjectPresent status of this subject

,

22 2

2 2

proportional to

1 ,

1( 1) |} ( ) ( ) .

2

Suppose that |} ( ) ( ) 's are random.

multiplicity number of ( , )

J J JI I J I I

J I

J n nI

K

n n

J JI I

JI

J

E G

n n j I j K j J

j I j K j J

d K

Note that ( 1)

,2

( 1),

2

JI

J

JJ JII I I

JI

n n

d n nd

Other works on energy centroids

• Mulhall, Volya, and Zelevinsky, PRL(2000)

• Kota, PRC(2005)

• YMZ, AA, Yoshida, Ogawa, Yoshinaga, and Kota, PRC(2005)

• YMZ, AA, and Ogawa PRC(2005)

Conclusion and perspective Conclusion and perspective

• Regularities of many-body systems under random interactions, including spin zero ground state dominance, parity distribution,

collectivity, energy centroids with various quantum numbers.

• Suggestions and Questions: Study of simpler quantities: parity distribution in ground states, energy centroids, constraints of Hamiltonian in order to obtain correct collectivity, and spin 0 g.s. dominance

Acknowledgements: Akito Arima (Tokyo) Naotaka Yoshinagana (Saitama) Stuart Pittel (Delaware) Kengo Ogawa (Chiba) Nobuaki Yoshida (Kansai) R. Bijker (Mexico)　 Olaf Scholten (Groningen) V. K. B. Kota (Ahmedabad) Noritake Shimizu(Tokyo)

Thank you for your attention!