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Regularities of many-body systems in the presence of random two-body interactions. Yu-Min ZHAO. 1 Department of Physics, Shanghai Jiao Tong University, China; 2 Center of Theoretical Nuclear physics, IMP, CAS, Lanzhou, China; 3 Cyclotron Center, RIKEN, Japan; 4 CCAST, CAS, Beijing, China. - PowerPoint PPT Presentation
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A short history of spin zero ground state dominance
Present status of this study @ Physical mechanism remains unclear @ Collectivity of low-lying states by using TBRE @ Energy centroids of fixed spin states Perspectives @ Some simpler quantities can be studied first @Searching for other regularities
Random matrices and 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 chooses 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).
Three interesting results
Phenomenological method by Tokyo group (namely, by us)
reasonably applicable to all systems Geometric method by GANIL group
applicable to “simple” systems Mean field method by Mexico group
applicable to sd, sp boson systems
Recent Efforts
By Papenbrock & Weidenmueller by using correlation between Energy radius
By Yoshinaga & Arima & Zhao by using energy centroids and width
Hand waving ideas by a few groups
(Zelevinsky, Zuker, Otsuka, and others)
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
Probability of Imax g.s.
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
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
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
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 Yoshinaga (Saitama) Kengo Ogawa (Chiba) Noritake Shimizu(Tokyo) Nobuaki Yoshida (Kansai) Stuart Pittel (Delaware) R. Bijker (Mexico) J. N. Ginocchio (Los Alamos) Olaf Scholten (Groningen) V. K. B. Kota (Ahmedabad)
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
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