OutlineOutline A short history of spin zero ground state domA short history of spin zero ground state dom

inanceinance Present status of this studyPresent status of this study @ Physical mechanism remains unclear @ Physical mechanism remains unclear @ Collectivity of low-lying states by using TBR@ Collectivity of low-lying states by using TBR

EE @ Energy centroids of fixed spin states@ Energy centroids of fixed spin states Perspectives Perspectives @ Some simpler quantities can be studied firs@ Some simpler quantities can be studied firs

tt @Searching for other regularities@Searching for other regularities

Random two-body interactions

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

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).

An example

† (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

9 , 4

2j n

Some recent papersSome 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).

Study of the ori gi n of 0 g. s. domi nance:

* For "Si mpl e" systems

So cal l ed "geometry" method

suggested by GANI L group.

They can expl ai n systems i n whi ch

al l ei genval ues can be wri tten i n

terms of l i near combi nati on of two-body

matri x el ements.

7For example,

2j

d

si ngl e shel l ;

bosons。

Two interesting results

Empirical method by Tokyo group

　 reasonably applicable to all systems Mean field method by Mexico group

　 sd, sp boson systems

Empirical method by Tokyo group( ) ( )

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

d玻色子情形

0 2 2 4 4

max

spin of the lowest and highest state when 1,

and other parameters are zero.

(min) (min) (max) (min) (max)

6 0 0

Lc

n c c c c c

k I

max

max max

max max

0

6 1 2 2 2

6 1 0 2 2

6 1 2 0

I

k I I

k I I

k

max max

max max

max max

0

6 1 0 2 2

6 1 2 2 2

I I

k I I

k I I

dd 玻色子情形玻色子情形

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

Four fermions in a single-j shell

Why Why P(0) staggers P(0) staggers periodically?periodically?

对四个粒子情形，如果对四个粒子情形，如果 GJ=-1GJ=-1 其他两体力为其他两体力为零，Ｉ＝０的态只有一个非零的本征值．零，Ｉ＝０的态只有一个非零的本征值．

Ｉ＝０的态的数量随Ｉ＝０的态的数量随 jj 呈规则涨落呈规则涨落．．

( )0 06J j

I IJ

E D

最大自旋态作基态的几率

A few examples

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

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.

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

(

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

单位: %)

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 44.

B

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

Collectivity in the IBM under random interactions

Collectivity in the IBM under random interactions

Energy centroids with fixed spin

,

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

( being the lowest)IP E

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

Acknowledgements: Akito Arima (Tokyo) Naotaka Yoshinagana (Saitama) Kengo Ogawa (Chiba) Stuart Pittel (Delaware) R. Bijker (Mexico)　 J. N. Ginocchio (Los Alamos) Rick Casten (New Haven) Olaf Scholten (Groningen) V. K. B. Kota (Ahmedabad) Noritake Shimizu(Tokyo) Nobuaki Yoshida (Kansai) Igal Talmi (Weizman)