t=0 pairing in coordinate space workshop esnt, paris shufang ban royal institute of technology (kth)...
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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
2. Symmetries of the s.p. wave function
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
2. Symmetries of the s.p. wave function
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
2. Symmetries of the s.p. wave function
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
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
r q q v r r q r q r q r q
r q q v r r q r q r q r q
4. If real is possible? kl
4. If real is possible? 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
4. If real is possible? 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?
5. Summary and further work
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.
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( )( ') ( )( ') ( )( ')
T Tiq jq iq jq iq jq
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
2. Symmetries of the s.p. wave function
( , ) 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
5. Summary and further work
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