Workshop ESNT, Paris
T=0 Pairing in Coordinate space
Shufang Ban
Royal Institute of Technology (KTH)
Stockholm, Sweden
Outline
1. 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
1.1 Algorithm for using delta force in HFB calculations:
HFB Equation:
ij 
 h   
 U 
U 

   E  
*
*
V 
   h    V 
1
Vijkl kl withint eraction

2 kl
Vijkl   drdr ' i* ( r ) *j ( r '  ')V ( r , r '  ')[k ( r )l ( r '  ')  l ( r )k ( r '  ')]
 '
V ( r , r '  ')   ( r  r ')
Delta force
Vijkl   dr  i* ( r ) *j ( r ')[k ( r )l ( r ')  l ( r )k ( r ')]
 '
 ij   dr  i* ( r ) *j ( r ') k ( r )l ( r ') kl
 '
kl
  dr   ( r ) ( r ')( r ')
 '
*
i
*
j
Anti-symmetric
 kl   lk
( r ')  k ( r )l ( r ') kl
kl
  , ' 
T=1 pairing (nn, pp)
isantisymmetric forexchangek  l :
 ( r ')   ( r '  )
spinshould beantisymmetric
 ( r ')   ( r )( 1)
atevery positionr
( r ) 
1
2
kl
ij   dr  (1)

1
2
k ( r , )l ( r ,  ) kl
1

2
1

2
  , '
Local Delta
potential
i* (r ,  ) *j ( r ,  )( r )
Local in coordinate space, we can calculate the value at every point r.
All the possible pairing correlations:
T=1 paring:
T  1,  1/ 2;   '  1/ 2, S  Sz  0
 nn T  1,Tz  1;S S z  0|  n,  n, T  1 
 pp T  1,Tz  1;S S z  0|  p,  p, T  1 
 np T  1,Tz  0;S S z  0|  n,  p, T  1 |  n,  p, T  1 
T=0 paring:
   '  1/ 2, T  Tz  0;S  1,   1/ 2
 np S  1,S z  1;T Tz  0|  n,  p, T  0 
 np S  1,S z  1;T Tz  0|  n,  p, T  0 
 np S  1,S z  0;T Tz  0|  n,  p, T  0 |  p,  n, T  0 
Alan L. Goodman, Phys. Rev. C 58(1998)R3051
1.2 using delta force in generalized HFB calculations including np-pairing:
 ( r )with   
Wave function:
Vijkl   drdr '
*
*

(
r

)

 i
j ( r '  ' ')V ( r , r '  ' ')[k ( r )l ( r '  ' ')  l ( r )k ( r '  ' ')]
 ' '
V ( r , r '  ' ')   ( r  r ')
Delta force
Tij 1,S 0   dr  ( 1)
1

2
1 1
  ,  ' |1Tz  i* ( r ,  , ) *j ( r ,  , ')( r )
2 2
1

2
1 1
  ,  ' |1S z  i* ( r ,  , ) *j ( r ,  ',  )( r )
2 2
 '

T 0, S 1
ij
  dr  ( 1)
 '
( r )  k ( r ,  , )l ( r ,  ,  ) kl
kl
ij  Tij 1  Tij 0
Local in
Coordinate space
2. Symmetries of the s.p. wave function
 ( r ,  )  Re  ( r ,  )  i Im  ( r ,  )
Four real components:
Parity:
z-signature:
 k 1 ( r )   Re  ( r , 

 

(
r
)
k
2
   Im  ( r , 
 (r )  
 k 3 ( r )   Re  ( r , 

 

(
r
)
 k4
  Im  ( r , 
 ) 
  ) 
 ) 

 ) 
Pˆk (r , )  k (r , )  pkk (r , )with pk  1
ei ( J z 1/ 2)k (r ,  )  k (  x,  y, z,  )  kk ( r ,  )withk  1
Time-reversal:
i sˆ
Tˆk (r , )  K0e yk (r , )  k (r , )  k* (r ,  )
Global Phase convention:
2. Symmetries of the s.p. wave function
ˆ ˆ i ( Jˆz 1/ 2) ( r ,  ) x   x
PTe
k
i Jˆ y
ˆ
ˆ ˆ (r , )  y   y
Pe pkk ( r ,  )  PT
ˆ
ei ( J z 1/ 2) Pˆk ( r ,  ) z   z
x
y
z
1
+
+
p
2
_
_
p
3
_
+
-p
1/8 space
4
+
_
-p
[1] P. Bonche, H. Flocard, and P. H. Heenen, Nucl. Phys. A 443 (1985) 39
2. Symmetries of the s.p. wave function
Time-reversal symmetry is broken by cranking
P. Bonche, et. al., Nucl. Phys. A 467 (1987) 475
Y. Engel, et. al., Nucl. Phys. A 249 (1975) 215
i s y
  E   J z J z  ( l  s ) z , T  K0e
Axial symmetry is broken by np pairing
Rz ( )C jm Rz1 ( )  eim C jm
A. L. Goodman, Nucl. Phys. A 186 (1972) 475
 C jm C jm 
 C jm C jm 
    1




Rz ( )  C jm C jm   Rz ( )   C jm C jm  
   
 2im   
 C jm C jm  
 e C jm C jm  
Signature symmetry is broken by np pairing
ˆ
ˆ
ei J z C jm ei J z  iC jm
 C jm C jm 
 C jm C jm 
    i Jˆ 

i Jˆ z


z
e  C jm C jm  e
  C jm C jm  
   




C
C
(

1)
C
C
jm jm  
 jm jm 

2. Symmetries of the s.p. wave function
 ( r ,  , )  Re  ( r ,  , )  i Im  ( r ,  , )
Static
Cranking
Triaxial-de
np paring
Cranking+
np paring
Yes
Yes
Yes
Yes
Yes
No
Yes
No
Yes
Yes
No
No
Phase
convention
Yes
Yes
Yes
Yes
Calculated
Coor-space
1/8 [1]
1/8 [2]
1/4
1/4
Parity
Timereversal
Signature
[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
3. Pairing matrix can be real
i sˆ y q
i Jˆ y q
q
ˆ
T k ( r )  K 0e k ( r )  pk e k ( r )
Phase convention:
1 1
1
1
kq ( r )  kq ( r ,  ) |  | q  [kq ( r , ) |  kq ( r ,  ) |  ] | q 
2 2
2
2

Re kq ( x,  y, z,  )  Re kq ( x, y, z,  )
ˆ ˆ q (r )
PT
k
Im  ( x,  y, z,  )   Im  ( x, y, z,  )
q
k
q
k
(1)
(iq )( jq ')  T(iq1)( jq ')  T(iq)(0 jq ')
T(iq )( jq ')   d 3r i* (r , q,  ) *j ( r , q ',  ')T (r ; ,  '; q, q ')
 '
1
 T 1 ( r ; ,  '; q, q ')   vT 1 ( r ) [k ( r , q,  )l ( r , q ',  ')  k ( r , q,  ')l ( r , q ',  )] ( kq )( lq ') 
2
kl
1
 T 0 ( r ; ,  '; q, q ')   vT 0 ( r ) [k ( r , q,  )l ( r , q ',  ')  k ( r , q,  ')l ( r , q ',  )] ( kq )( lq ')
2
kl
vT 1,0  GT 1,0 (1   ( r ) /  c )  ( r )   ( x, y, z )   ( x,  y, z )
Assume real  kl
and using the wave function symmetry (1)
Re T 0,1 ( x,  y, z; ,  ')  Re T 0,1 ( x, y, z;  ,  ')
Im T 0,1 ( x,  y, z; ,  ')   Im T 0,1 ( x, y, z;  ,  ')
(2)
Im  ( iq )( j q )  Im  d 3r  iq* ( r ,  ) j q* ( r ,  ')[ T 1 ( r ,  ,  ')  ( 1)1/ 2q T 0 ( r ,  ,  ')]
 '
  d 3r  {[Re iq ( r ,  ) Re  j q ( r ,  ') Im iq ( r ,  ) Im  j q ( r ,  ')]
 '
 Im[ T 1 ( r ,  ,  ')  ( 1)1/ 2q  T 0 ( r ,  ,  ')]
 [Re iq ( r ,  ) Im  j q ( r ,  ')  Im iq ( r ,  ) Re  j q ( r ,  ')]
 Re[ T 1 ( r ,  ,  ')  ( 1)1/ 2q T 0 ( r ,  ,  ')]}
The integrand {…} is anti-symmetric under inversion
y— -y, there for we have
Paring matrix
can be real
4. If real  kl is possible?
1
 T 1 ( r ; ,  '; q, q ')   vT 1 ( r ) [k ( r , q,  )l ( r , q ',  ')  k ( r , q,  ')l ( r , q ',  )] ( kq )( lq ') 
2
kl
1
 T 0 ( r ; ,  '; q, q ')   vT 0 ( r ) [k ( r , q,  )l ( r , q ',  ')  k ( r , q,  ')l ( r , q ',  )] ( kq )( lq ')
2
kl
4. If real  kl is possible?
 kl is real.
4. If real  kl is possible?
Re  T 1,S  S z 0 ( r, , , q, q ')  0; Im  T 1,S  S z 0 ( r, , , q, q ')  0
Re  T 0,S z 1 ( r, , , n, p )  Re  T 0,S z 1 ( r, , , n, p );
Im  T 0,S z 1 ( r, , , n, p )   Im  T 0,Sz 1 ( r, , , n, p )
Re  T 0,S z 0 ( r, , , n, p )  0; Im  T 0,S z 0 ( r, , , n, p )  0
Re
Complex
Im
4. If real  kl is possible?
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?
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.
2.
3.
Make sure if the kappa can be real?
Construct the pairing matrix
Construct the calculation space by the symmetries
……
Aim: develop the code cr8 with np pairing included.
Thank you !
 ( iq )( jq ')
1
( kq1 )( lq2 )
( kq2 )( lq1 )
  [V( iq )( jq ')  ( kq1 )( lq2 ) V( iq )( jq ')  ( kq2 )( lq1 ) 
2 klq1q2
  V((iqkq)(1 )(jqlq')2 ) ( kq1 )( lq2 )
klq1q2
1
1
V ( r, r ')  GT 1 (1  P )(1   ( r ) /  c ) ( r  r ') (1  P )
2
2
1
1
 GT 0 (1  P )(1   ( r ) /  c ) ( r  r ') (1  P )
2
2
 G (1   ( r ) /  c ) ( r  r ')if GT 1  GT 0  G
(iq )( jq ')  T(iq1)( jq ')  T(iq)(0 jq ')
(iq )( jq ')  T(iq1)( jq ')  T(iq)(0 jq ')
2. Symmetries of the s.p. wave function
 ( r ,  )  Re  ( r ,  )  i Im  ( r ,  )
Static:
k  1
 k 1 ( r )   Re  ( r , 

 

(
r
)
Im  ( r , 
 (r )   k 2   
 k 3 ( r )   Re  ( r , 

 

(
r
)
 k4
  Im  ( r , 
Cranking:
k  1
 k 1 ( r )   Re  ( r , 

 

(
r
)
k
2

   Im  ( r , 
 k 3 ( r )   Re  ( r , 

 

(
r
)
 k4
  Im  ( r , 
 ) 
  ) 
 ) 

 ) 
 ) 
  ) 
 ) 

 ) 
x
y
z
1
+
+
p
2
-
-
p
3
-
+
-p
4
+
-
-p
k  1
 k 1 ( r )   Re  ( r , 

 

(
r
)
k
2

   Im  ( r , 
 k 3 ( r )   Re  ( r , 

 

(
r
)
 k4
  Im  ( r , 
 ) 
 ) 
 ) 

 ) 
1.2 using delta force in generalized HFB calculations including np-pairing:
 ( r )with   
Wave function:
Vijkl   drdr '
*
*

(
r

)

 i
j ( r '  ' ')V ( r , r '  ' ')[k ( r )l ( r '  ' ')  l ( r )k ( r '  ' ')]
 ' '
Delta force
ij   dr
V ( r , r '  ' ')   ( r  r ')

 
'
i* (r ) *j (r ' ')( r ' ')
'
( r ' ')  k ( r )l ( r ' ') kl antisymmetricask
kl
1/ 2 
 ( r )( 1)
 , ' '
or ( r )( 1)1/ 2  , ' , '
( r )  k ( r ,  , )l ( r ,  ,  ) kl
kl
spinantisymmetricand isospinsymmetric
orisospinantisymmetricand spinsymmetric
atevery positionr
5. Summary and further work
Cranking triaxial-deformed wave function as input:
Construct density matrix and T=1 pairing matrix in ¼ space
Assure starting values for the delta potential, T=0 paring
Get the H for HFB equation, solve it and get first U, V
Mixing neutron proton in quasi-particle basis
Calculate new density matrix and pairing matrix
Calculate new density matrix and diagonalizes in canonical basis