T=0 Pairing in Coordinate space

T=0 Pairing in Coordinate space

Workshop ESNT, Paris

Shufang Ban

Royal Institute of Technology (KTH) Stockholm, Sweden

Outline1. Introduction: delta force in HFB

2. Symmetry of the s.p. wave function

3. Delta matrix can be real

4. If real kappa is possible?

5. Summary and further work

1. Introduction

* *

'

1int

2

' ( ) ( ' ') ( , ' ')[ ( ) ( ' ') ( ) ( ' ')]

ij ijkl klkl

ijkl i j k l l k

V with eraction

V drdr r r V r r r r r r

HFB Equation:

* *

h U UE

V Vh

( , ' ') ( ')V r r r r

Delta force

1.1 Algorithm for using delta force in HFB calculations:

* *

'

* *

'

* *

'

( ) ( ')[ ( ) ( ') ( ) ( ')]

( ) ( ') ( ) ( ')

( ) ( ') ( ')

ijkl i j k l l k

ij i j k l klkl

i j

V dr r r r r r r

dr r r r r

dr r r r

Anti-symmetric

kl lk

( ') ( ) ( ')k l klkl

r r r

, ' T=1 pairing (nn, pp)

1* *2( 1) ( , ) ( , ) ( )ij i jdr r r r

Local in coordinate space, we can calculate the value at every point r.

1 1( ) ( , ) ( , )

2 2k l klkl

r r r

:is antisymmetric for exchange k l ( ') ( ' )r r

1

2, '( ') ( )( 1)r r

spin should be antisymmetric

at every position r

Local Delta potential

T=1 paring:

1, 1; 0 | , , 1

1, 1; 0 | , , 1

1, 0; 0 | , , 1 | , , 1

nn z z

pp z z

np z z

T T S S n n T

T T S S p p T

T T S S n p T n p T

1, 1/ 2; ' 1/ 2, 0zT S S

T=0 paring:

1, 1; 0 | , , 0

1, 1; 0 | , , 0

1, 0; 0 | , , 0 | , , 0

np z z

np z z

np z z

S S T T n p T

S S T T n p T

S S T T n p T p n T

' 1/ 2, 0; 1, 1/ 2zT T S

Alan L. Goodman, Phys. Rev. C 58(1998)R3051

All the possible pairing correlations:

* *

' '

' ( ) ( ' ' ') ( , ' ' ')[ ( ) ( ' ' ') ( ) ( ' ' ')]ijkl i j k l l kV drdr r r V r r r r r r

( , ' ' ') ( ')V r r r r

Delta force

Wave function: ( )r with

1.2 using delta force in generalized HFB calculations including np-pairing:

( ) ( , , ) ( , , )k l klkl

r r r

11, 0 * *2

'

10, 1 * *2

'

1 1( 1) , ' |1 ( , , ) ( , , ') ( )

2 2

1 1( 1) , ' |1 ( , , ) ( , ', ) ( )

2 2

T Sij z i j

T Sij z i j

dr T r r r

dr S r r r

1 0T Tij ij ij

Local in Coordinate space

2. Symmetries of the s.p. wave function

( , ) Re ( , ) Im ( , )r r i r

Parity: ˆ ( , ) ( , ) ( , ) 1k k k k kP r r p r with p

z-signature:( 1/ 2) ( , ) ( , , , ) ( , ) 1zi J

k k k k ke r x y z r with

Time-reversal:ˆ *

0ˆ ( , ) ( , ) ( , ) ( , )yi s

k k kkT r K e r r r

Global Phase convention:

Four real components:

1

2

3

4

( ) Re ( , )

( ) Im ( , )( )

( ) Re ( , )

Im ( , )( )

k

k

k

k

r r

r rr

r r

rr


x y z

1 + + p

2 _ _ p

3 _ + -p

4 + _ -p

ˆ( 1/ 2)

ˆ

ˆ( 1/ 2)

ˆ ˆ ( , )

ˆ ˆ ˆ( , ) ( , )

ˆ ( , )

z

y

z

i Jk

i Jk k

i Jk

PTe r x x

Pe p r PT r y y

e P r z z

[1] P. Bonche, H. Flocard, and P. H. Heenen, Nucl. Phys. A 443 (1985) 39

1/8 space


Signature symmetry is broken by np pairing

ˆ ˆ

( 1)

z z

jm jm jm jm

i J i Jjm jm jm jm

jm jm jm jm

C C C C

e C C e C C

C C C C

ˆ ˆz zi J i J

jm jme C e iC

Time-reversal symmetry is broken by cranking

zE J 0( ) , yi sz zJ l s T K e

P. Bonche, et. al., Nucl. Phys. A 467 (1987) 475Y. Engel, et. al., Nucl. Phys. A 249 (1975) 215

Axial symmetry is broken by np pairing

1( ) ( ) imz jm z jmR C R e C

1

2

( ) ( )

jm jm jm jm

z jm jm z jm jm

imjm jm jm jm

C C C C

R C C R C C

C C e C C

A. L. Goodman, Nucl. Phys. A 186 (1972) 475


Static Triaxial-de

Cranking np paringCranking+ np paring

Parity Yes Yes Yes Yes

Time-reversal

Yes No Yes No

Signature Yes Yes No NoPhase convention Yes Yes Yes Yes

Calculated Coor-space 1/8 [1] 1/8 [2] 1/4 1/4

( , , ) Re ( , , ) Im ( , , )r r i r

[1] P. Bonche, H. Flocard, and P. H. Heenen, Nucl. Phys. A 443 (1985) 39 [2] P. Bonche, H. Flocard, and P. H. Heenen, Nucl. Phys. A 467 (1987) 475

1 1 ( )( ')

0 0 ( )( ')

1( ; , '; , ') ( ) [ ( , , ) ( , ', ') ( , , ') ( , ', )]

2

1( ; , '; , ') ( ) [ ( , , ) ( , ', ') ( , , ') ( , ', )]

2

T T k l k l kq lqkl

T T k l k l kq lqkl

r q q v r r q r q r q r q


1 0( )( ') ( )( ') ( )( ')

T Tiq jq iq jq iq jq

3 * *( )( ')

'

( , , ) ( , ', ') ( ; , '; , ')Tiq jq i j Td r r q r q r q q

3. Pairing matrix can be real

Phase convention:ˆˆ

0ˆ ( ) ( ) ( )y yi s i Jq q q

k k k kT r K e r p e r

1 1 1 1( ) ( , ) | | [ ( , ) | ( , ) | ] |

2 2 2 2q q q qk k k kr r q r r q

ˆ ˆ ( )qkPT r Re ( , , , ) Re ( , , , )

Im ( , , , ) Im ( , , , )

q qk k

q qk k

x y z x y z

x y z x y z

(1)

3 * * 1/ 2( )( ) 1 0

'

3

'

Im Im ( , ) ( , ')[ ( , , ') ( 1) ( , , ')]

{[Re ( , ) Re ( , ') Im ( , ) Im ( , ')]

q q qiq j q i j T T

q q q qi j i j

d r r r r r

d r r r r r

1/ 21 0

1

Im[ ( , , ') ( 1) ( , , ')]

[Re ( , ) Im ( , ') Im ( , ) Re ( , ')]

Re[ ( , ,

qT T

q q q qi j i j

T

r r

r r r r

r

1/ 20') ( 1) ( , , ')]}q

T r

( ) ( , , ) ( , , )r x y z x y z

1,0 1,0(1 ( ) / )T T cv G r

0,1 0,1

0,1 0,1

Re ( , , ; , ') Re ( , , ; , ')

Im ( , , ; , ') Im ( , , ; , ')T T

T T

x y z x y z

x y z x y z

(2)

Assume real kl and using the wave function symmetry (1)

The integrand {…} is anti-symmetric under inversion y— -y, there for we have

Paring matrix can be real

1 1 ( )( ')

0 0 ( )( ')

1( ; , '; , ') ( ) [ ( , , ) ( , ', ') ( , , ') ( , ', )]

2

1( ; , '; , ') ( ) [ ( , , ) ( , ', ') ( , , ') ( , ', )]

2

T T k l k l kq lqkl

T T k l k l kq lqkl



4. If real is possible? kl


is real.kl


0

1, 0 1, 0

0, 1 0, 1

0, 1 0, 1

0, 0, 0

Re ( , , , , ') 0; Im ( , , , , ') 0

Re ( , , , , ) Re ( , , , , );

Im ( , , , , ) Im ( , , , , )

Re ( , , , , ) 0; Im ( , , , , ) 0

z z

z z

z z

z z

T S S T S S

T S T S

T S T S

T S T S

r q q r q q

r n p r n p

r n p r n p

r n p r n p

Complex

Im

Re

Chose complex wave function and assume real kl


the np pairing can be described in general.

Remained question:

1. If complex wave function, real kappa are equivalent to real wave function, complex kappa? Is there any transformation between them?

2. How we can construct the input wave functions of general case from the wave function of T=1 case?


1. Using delta force, we can get the local pairing matrix, for both with or without np pairing cases.

2. The np pairing breaks axial and signature symmetries, we must calculate it in ¼ space when parity and phase convention are required.

3. Chose complex wave function, assume real kappa, the pairing matrix can be real.

4. Using complex wave function and real kappa, the np pairing can be described without lose generality. There are still remained questions.

Further work: 1. Make sure if the kappa can be real?2. Construct the pairing matrix3. Construct the calculation space by the symmetries ……Aim: develop the code cr8 with np pairing included.

Thank you !Thank you !

1 2 2 1

1 2 2 1

1 2

1 2

1 2

1 2

( )( ) ( )( )( )( ') ( )( ') ( )( ) ( )( ') ( )( )

( )( )( )( ') ( )( )

1[

2kq lq kq lq

iq jq iq jq kq lq iq jq kq lqklq q

kq lqiq jq kq lq

klq q

V V

V

1 0( )( ') ( )( ') ( )( ')


1

0

1 0

1 1( , ') (1 )(1 ( ) / ) ( ') (1 )

2 21 1

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

(1 ( ) / ) ( ')

T c

T c

c T T

V r r G P r r r P

G P r r r P

G r r r if G G G

1 0( )( ') ( )( ') ( )( ')



( , ) Re ( , ) Im ( , )r r i r

x y z

1 + + p

2 - - p

3 - + -p

4 + - -p

1

2

3

4

( ) Re ( , )

( ) Im ( , )( )

( ) Re ( , )

Im ( , )( )

k

k

k

k

r r

r rr

r r

rr

1k

Static:

1

2

3

4

( ) Re ( , )

( ) Im ( , )

( ) Re ( , )

Im ( , )( )

k

k

k

k

r r

r r

r r

rr

1k

1

2

3

4

( ) Re ( , )

( ) Im ( , )

( ) Re ( , )

Im ( , )( )

k

k

k

k

r r

r r

r r

rr

1k

Cranking:

* *

' '

' ( ) ( ' ' ') ( , ' ' ')[ ( ) ( ' ' ') ( ) ( ' ' ')]ijkl i j k l l kV drdr r r V r r r r r r

( , ' ' ') ( ')V r r r r

Delta force

* *

' '

( ) ( ' ') ( ' ')ij i jdr r r r

1/ 2, ' '

1/ 2, ' , '

( ' ') ( ) ( ' ')

( )( 1)

( )( 1)

k l klkl

r r r

r

or r

Wave function: ( )r with

1.2 using delta force in generalized HFB calculations including np-pairing:

antisymmetric ask

spin antisymmetric and isospin symmetric

or isospin antisymmetric and spin symmetric

at every position r

( ) ( , , ) ( , , )k l klkl

r r r


Cranking triaxial-deformed wave function as input:

Construct density matrix and T=1 pairing matrix in ¼ space

Get the H for HFB equation, solve it and get first U, V

Assure starting values for the delta potential, T=0 paring

Calculate new density matrix and pairing matrix

Calculate new density matrix and diagonalizes in canonical basis

Mixing neutron proton in quasi-particle basis

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T=0 Pairing in Coordinate space