OutlineOutline Simple comments on regularities of many-body sySimple comments on regularities of many-body sy

stems under random interactionsstems under random interactions Number of spin Number of spin II states for single- states for single-jj configuration configuration JJ-pairing interaction-pairing interaction Sum rules of angular momentum recoupling coefficientSum rules of angular momentum recoupling coefficient

ss Number of states with given spin and isospinNumber of states with given spin and isospin Nucleon approximation of the shell modelNucleon approximation of the shell model Prospect and summary Prospect and summary

Simple comments on regularities of many-body sySimple comments on regularities of many-body systems under random interactionsstems under random interactions

Number of spin Number of spin II states for single- states for single-jj configuration configuration JJ-pairing interaction-pairing interaction Sum rules of angular momentum recoupling coefficientSum rules of angular momentum recoupling coefficient

ss Number of states with given spin and isospinNumber of states with given spin and isospin Nucleon approximation of the shell modelNucleon approximation of the shell model Prospect and summary Prospect and summary

Part I A brief introduction to nuclei under random interactions

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

Two-body random ensemble

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 found spin zero ground state dominance can be obtained by random two-body interactions (Phys. Rev. Lett. 80, 2749) ． 　　Ref. C. W. Johnson et al., PRL80, 2749 (1998); R.Bijker et al., PRL84, 420 (2000); L. Kaplan et al., PRB65, 235120 (2002).

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

Intrinsic collectivity based on the sd IBM

Energy centroids of spin I states

,

22 2

2 2

1 ,

1( 1) |} ( ) ( ) .

2

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

approximately (multiplicity of )

( 1),

2

J J JI I J I I

J I

J n nI

K

n n

J JI I

J JI I

J J

I

E G

n n j I j K j J

j I j K j J

d K

n n

Note that

( 1),

2

JJ JI

I IJI

d n nd

A short summary

Spin 0 ground state dominance for even-even nuclei, regularities for energy centroids with given quantum numbers, collectivity, etc.

Open questions: spin distribution in the ground states ； energy centroids ； requirement for nuclear collectivity ； etc.

For a review, See YMZ, AA, NY, Physics Reports, Volume 400, Page 1 (2004).

Themes and Challenges of Modern Science Complex systems arising out of basic constituents How the world, with all its apparent complexity and diversity, can be

constructed out of a few elementary building blocks and their interactions

Simplicity out of complexity How the world of complex systems can display such remarkable regularity

and, often, simplicity

Understanding the nature of the physical universe

Manipulating matter for the benefit of mankind

Part II Number of states for identical particles in a single-j shell

Why we study this number (Ginocchio)?

The number of states is known for

n=1,2, and unknown when n n 2(e. g. , =3, 4, 5).

Can we have formulas of number of states

when 3?n

A simple method in the text-book

1 2

1 2

1 2

1. Find combinatorial number of ,

,

( ).

2. Find combinatorial number of 1,

n

n

n

M m m m I

j m m m j

P I

M m m m I

wi th the

requi rement that We denote

thi s number by

agai n

wi th the requi rement 1 2 ,

( 1).

3. ( ) ( ) ( 1).

Reference:

R.D. Lawson, Theory of Nuclear Shell Model (Clarendon, Oxford, 1980);

A de-Shalit and I. Talmi, Nuclear Shell Model Theor

n

nI

j m m m j

P I

D j P I P I

we cal l thi s

number by

y (Academic,

New York, 1963).

Empirical formulas

YMZ and AA, PRC68, 044310 (2003).

empirical formulas for n=3,4. For example,

3

3I

I

2 3( ) for ;

6

3 3( ) + for ,

6

0 if (3 3 ) mod 6 equals 0 and 1 otherwise.

I

I

ID j I j

j ID j I j

j I

For n=4, results are more complicated (omitted).

A new method

max

0 0 1 2

1 2

1 2

max

( ) be the number of partitions

0 2 1

0 2 .

( 1), we define (0) ( ) 1.

2

Then

n

n

n

nI

P I I i i i

i i i j n

i i i l

n nI nj P D j

D

Let

wi th the requi rement that

for fermi ons and for bosons

Here

max 0 0 0( ) ( ) ( 1).

For example,

j, j-1, j-2, j-3 0,0,0,0

j, j-1, j-2, j-4 1,0,0,0

j, j-1, j-3, j-4 1,1,0,0 j, j-1, j-2, j-5 2,0,0,0

nI I I j P I P I

Conjugates of 0( )P I

max

bosons with spin / 2.

bosons 2 , and for fermions 2 1- .

n L n

n l n j n

I n

Transform every Young di agrams to i ts correspondi ng conj ugate, then

we swi tch to the Young di agrams for For

Fi rst, the of

0 1 2

bosons with spin / 2 bosons with spin

and that of fermions in a single-j shell.

( ) " " (0 2 )

ALWAY

n

L n n

l n

P I n i i i L n

equal s that of

Second, of bosons

0 1 2

0 1 2

S EQUALS

( ) bosons with spin (0 2 )

or ( ) fermions in a single-j shell (0 2 1 )n

n

P I n l i i i l

P I n i i i j n

of

of .

0

The reason is as follows:

( ) bosons with spin corresponds to Young diagram with

at most rows and 2 columns or with rows and 2 1 columns.

The conjugate of these Young diagrams hav

P I n L

n l n j n of

0

e at most

2 rows or 2 1 rows and columns which correspond to

( ) fermions in a shell with spin .

This means that ( ) ( ) if 2 2 1- . This can be

also readily confi

n nI I

l j n n

P I n j n l

D j D l l j n

,

of or

0

0

rmed by computer.

The ( ) bosons with spin is relatively easier to obtain,

because here we study the reduction rule U( +1) SO(3).

When we study ( ) of fermions in a single- shell or b

P I n L

n

P I n j

of

S

osons with

spin , one has to study the symmetry of SU(2 +1) or SU(2 +1)

which has or dependence.

l l j

j l

An Example

1 2 2 1 1 1

7 / 2 with 4 :

0 ,2 ,4 ,5 ,6 ,8

bosons ( 3 / 2 2)

with 4 has the same dimension.

Similarly, fermions with 11/ 2,

6 has the same dimension as

bosons with -5 / 2 3 and 6, etc.

j n

I

d l j

n

j

n

l j n

An example: n=4

Here we should study bosons with SU(5)symmetry, i.e., d bosons. The number of states for d bosons has been studied in the interacting boson model.

By using results of the IBM, we were able to obtain dimension for d bosons. Then we quickly get the number of states for n=4 of fermions and bosons.

What about odd number of particles?

YMZ and AA, PRC71, 047304 (2005)

Other works

Dimension for n>3: J. N. Ginocchio and W. C. Haxton, “Symmetries in Science VI”, Edited by B. Grube

r and M. Ramek, (Plenum, New York, 1993). Number of states for Zamick et al. Physical Review C71, 054308 (2005).

40 ( )ID j

4 30 ( ) ( ).I I jD j D j

For n=3, we should study SU(4) reduction rule.

However, Talmi proved our results for n=3(Physical Review C72,037302(2005)) . He obtained some recursion formulas and proved the formulas by reduction method.

This method can be used to prove any formulas (in principle) but it can not be used to find new formulas.

The guideline of Talmi’s efforts

First, he assumes the formula is correct for j-1 shell, then it suffices to show that it is also

correct to j shell. Next he enumerates the effect by changing

m1= j-1 to j. Summing this effect, he obtains the dimension

of j shell.

For the case with ,I j

Part III J-pairing interaction

Fermions in a single-j shell:

† (0)

† † † ( )

2 1( ) ,

10,2, , 2 1; ( ) .

2

J JJ

J Jj j

H J A A

J j A a a

J –pair truncation of the shell model Empirically we find that J-pair truncation is good for J-

pairing interaction.

max

max

max

When , ( 3,4,5), we find following facts but

our understanding is not so deep :

When total spin of the system, , is not very close to

eigenvalues are extremely close to integers ( -pai

J J n

I I

J

,

r picture )

or asympototically some special fractions ( " " picture ).

We have calculated the overlaps between this picture wavefunction

and that by exact diagonalizations (very close to 1!).

What i

cluster

s the hidden symmetry of Jmax pairing interaction ?

References:

YMZ, AA, JNG, NY, PRC68, 044320 (2003);

YMZ and AA, PRC70, 034306 (2004).

PartIV Sum rules of angular momentum recoupling coefficients

For n=3, J-pair truncation gives exact solution.

3

3

For each with , there is one and only one

non-zero eigenvalue which equals

1 2(2 1) .

Summing over ( is even), we obtain 3 ( ).

If we know ( ), then we can obtain sum rules

J

I

I

I H H

j j JJ

j I J

J J D j

D j

of the above

-j symbol. 6

An example

even

2 3 13

6 22(2 1)

3 3 3 13 3

6 2

(e.g., using Talmi's book), one can

obtain 2(2 1) 0,

JI

J

II I j

j j JJ

j I J j I j II j

j j JJ

j I J

Usi ng some recoupl i ng techni que

Then we al s

odd even

2(2 1) 2(2 1)J J

j j J j j JJ J

j I J j I J

o have

Similar things can be done for n=4 Ref.: YMZ & AA (Physical Review C72, 054307 (2005) Here we obtain sum rules of 9-j symbols. The difference is that here situation is more complicated.

Generally speaking, the number of nonzero eigenvalues is not always one for J pairing interaction and each of eigenvalue is unknown.

However, the trace of eigenvalues for each I states with only J-pairing interactions is always a constant with respect to orthogonal transformation:

† †0 | | 0

1 ( ) 4(2 1)(2 1)

I IJ K J K

M MK

IJK

K

A A A A

j j J

J K j j K

J K I

Proof

4 ( ) ( ) 4 2 2

24 4 4 2 2

( ) ( ) 4 4 ( ) ( )

( ) ( ) ( ) ( )

| 0 6 }( ) , ( ) : ;

| | 6 }( ) , ( ) :

0 | | | 0

0 | | 0

IJ K

M

JK

I IJ K J K

M MK

I IJ K J K

M MK

I j M A A j I j J j K I

j I H j I j I j J j K I

A A I j I j A A

A A A A

Sum rulesSumming over J, we obtain sum rules of 9-j symbols

4

even even

even even

even even

6 ( ) 0 | | 0

1 ( ) 4(2 1)(2 1)

Next, we should evaluate values of 1 ( ) .

This can be done by

I IJ K J KI M M

J K

IJK

J K

IJK

J K

D j A A A A

j j J

J K j j K

J K I

studying the number of non-zero cfps'

(details are omitted here).

Sum rules with odd J and odd K

4

odd odd

odd odd

4

6 ( ) 0 | | 0

1 ( ) 4(2 1)(2 1)

Here we must be careful that we have to use ( ) which

corresponds to dimension for four

I IJ K J KI M M

J K

IJK

J K

I

D j A A A A

j j J

J K j j K

J K I

D j

odd

fictitious fermions (wave

function is symmetric with exchange two identical fermions) ;

This can be obtained easily by studying again bosons.

Next, we should evaluate values of 1 ( ) . IJK

d

odd

J K

Sum rules with both even and odd J,K

1

, ( )

4 ( 1) / 2 , 24(2 1)(2 1)

4 4 (4 ) / 2 , 2

This result can be obtained by using other (old) sum rules and evaluating

( ) . By using this

J K

J

J K IJK

j j JI I j

J K j j Kj I I j

J K I

result, sum rules with even , sum rules

with odd , we can easily find sum rules with even and odd

or odd and even .

The same thing was be done for bosons with spin (which is an integer).

JK

JK J K

J K

l

Two examples

even odd

max

even

2 2 , is even;4

2(2 1)(2 1) 2 2 2 , is odd;4

2 24

2 2 2 , is even;4

2(2 1)(2 1)

24

J K

K

II j I

j j JI

J K j j K I j I

J K II I

I j

Il l J I j I

J K l l KI

J K I

even 2 , is odd.

etc.

J I j I

Part V Number of states for nucleons in a single-j shell [An application of these sum rules]

References: L.Zamick et al., Physical Review C72, 044317 (2005);

Y.M.Z. and A.A., Physical Review C72, 064333 (2005).

space with both protons and neutronsnj

JT-pairing interaction

Nucleons in a single-j shell:

†

† † †

† †

,

1( ) ,

21

( ) ( ) .2

T T

T T

T

T T

JJT JT

JT MM MMM J

JT JTMM jt jt MM

M MJT JTMM jt jt M M

H A A

A a a

A a a

Similarly,

2

2

2 2 2 2

2 2

4 4

' '

'

| |

0 | | 0

6 .

T T

JTJT

IT ITJT KT JT KT

MM MMJK T T

IT

j IT H j IT

A A A A

D

2 2 2 2' '

2

2 2 2

2 2

0 | | 0

1 ( ) 4(2 1)(2 1)

1 1

2 21 1

(2 1)(2 ' 1) '2 2

'

T T

IT ITJT KT JT KT

MM MM

I TJK

A A A A

J K

Tj j J

T T j j K T

J K I T T T

The case of T=0'

2 2

'2 2

( 0)

( , ) (1,1),( , ) (even,even);

( , ) (0,0), ( , ) (odd,odd).

1 11

2 21 1

6 (1 ( ) 36(2 1)(2 1) 12 21 1 0

(1 ( ) 4(2 1)(2 1)

II T

even J even K

I

even J even K

T T J K

T T J K

j j J

D J K j j K

J K I

j j J

J K j j

even even odd odd

even even odd odd

1 10

2 21 1

02 20 0 0

(1+(-) ) (1+(-) )

2 (2 1)(2 1) 2 (2 1)(2 1) .

I IJK JK

J K J K

J K J K

K

J K I

j j J j j J

J K j j K J K j j K

J K I J K I

6 ( 0)

6 3( 0)

6 2( 0)

6 5( 0)

(6 3) / 2 2 ( 6 , [ ],

3( (6 3) / 2) mod 3) :

3(2 6 ) (2 1) ( 3) 1;

23

(2 6 ) (2 1) ( 1);21

(4 6 ) (2 1) ( 1)(3 4);2

(4 6 ) (2

I k T

I k T

I k T

I k T

j kFor I j I k L

m j k

D k L k m k k

D k L k m k k

D k L k m k k

D k L k

6 1( 0)

6 4( 0)

11) (3 1);

21

2 (9 1);2

12( 1) ( 1)(9 4).

2The reason why we need six formulas is because of the

reduction rule of d bosons.

I k T

I k T

m k k

D kj k k

D k j k k

Part VI Our next effort on pair approximation

IBM SD collective pairs

diagonalization of the shell model Hamiltonian

in nucleon pair subspace. How far one can go? How reliable is “collective pair” approximation? What can one calculate by using pair appro. ?

We have done following work

Validity of SD pair truncation for special cases Application to A=130 even-even nuclei (rather successful calculation)

We proposed an efficient algorithm to describe even systems and odd-A (also doubly odd) systems on the same footing.

The nuclei we shall try in the near future

Mass number A around 140, neutron rich side

(both even and odd A, both positive and negative parity) Validity of SD truncation, realistic cases.

[How good or not good is the pair truncation?] Extension of pairs [S1, S2, D1, D2, F pairs, G pairs, etc. ]

Part VII Summary & prospect

Nuclear structure under random interactions Number of states with given spin (&isospin),

sum rule of 6j and 9j symbols Our next project:

Nucleon pair approximation of the shell model: odd and doubly odd nuclei. Applications to realistic systems

Acknowledgements: Akito Arima (Tokyo) Naotaka Yoshinaga (Saitama) Kengo Ogawa (Chiba) Nobuaki Yoshida (Kansai)

International Conference on Nuclear Structure Physics, Shanghai, June 12-17th, 2006.