outline a short history of spin zero ground state dominance present status of this study @...
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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
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
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
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
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
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
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)
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