Spin-triplet pairing in nuclear DFT

Nobuo HinoharaCenter for Computational Sciences, University of Tsukuba

Nobuo Hinohara, Tomohiro Oishi, and Kenichi Yoshida,accepted for publication in Phys. Rev. C, arXiv:2308.02617.
Feb. 21, 2024

DFT2024

(singlet-even)

Nuclear
force
space
spin
isospinTwo-nucleon
exchangeobservables

1

symmetric(s-wave etc)
anti-symmetric(singlet)symmetric(triplet)anti-symmetricstandard pairing
mass, level density

anti-symmetric(p-wave etc)
symmetric(triplet)symmetric(triplet)anti-symmetric
2 superfluidity
in neutron matter

symmetric(s-wave etc)
symmetric(triplet)anti-symmetric(singlet)
np-pair
anti-symmetricisoscalar pairing
β-decay half-life
ββ-decay half-life

anti-symmetric(p-wave etc)
anti-symmetric(singlet)anti-symmetric(singlet)
np-pair
anti-symmetric(triplet-even)
3

E

1O

(singlet-odd)??

standard nuclear pairing: spin-singlet pair condensation
this talk focuses on: spin-triplet pair condensation
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3P<latexitO

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E<latexit

Pairing in nuclei

p Nucleons has spin and isospin degrees of freedom.
p Four channels in the nucleon pair
Local density
In nuclear DFT⇢˜k , ⌧˜kJ̃ks̃0 , T̃0 , F̃0j̃0sha1_base64="5/hyN0OF561OxyXmXzE9x3UDwBM=">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</latexit><latexit

BW(Balian-Werthamer) (1963)
Aµj = A µj
ˆ =AA(k̂) = A(k̂x + ik̂y )ẑsha1_base64="vg8sLoGCrCmkUJtlY2zD2tI5KdM=">AAACsnicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQvoxrik5pQkxld7+dbGxSQWl2hkayrYKmjoGmrG+SrAJRV0fWuBMvECygZ6BmCggMkwhDKUGaAgIF9gO0MMQwpDPkMyQylDLkMqQx5DCZCdw5DIUAyE0QyGDAYMBUCxWIZqoFgRkJUJlk9lqGXgAuotBapKBapIBIpmA8l0IC8aKpoH5IPMLAbrTgbakgPERUCdCgyqBlcNVhp8NjhhsNrgpcEfnGZVg80AuaUSSCdB9KYWxPN3SQR/J6grF0iXMGQgdOHRkQRUjd9PJQxpDBZgv2QC/VYAFgH5MhliflnV9M/BVkGq1WoGiwxeA/230OCmwWGgD/PKviQvDUwNms3ABYwgQ/TowGSEGekZmumZBpooOzhBo4qDQZpBiUEDGB/mDA4MHgwBDKFAeycz7GQ4xHCYyYQpiimRKRmilIkRqkeYAQUw5QAAKYyfkA==</latexit>

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m= 1<latexit

・neutron 3P2 superfluidity in 2/3⇢0  ⇢  3⇢0⇤M・five order parameters in 3P2 superfluidity JM (k) = ( 1)J M (k)・not many discussions in finite nuclei (Oishi EPJA57,180(2021))
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Spin-triplet pairing in electron and nuclear systems

Electron system (3He superfluidity)

・A and B phases (ABM and BW)・p-wave superfluidity
・order parameter
1
X
X
(3)
(
k̂)
=
A
Y
(
k̂)
=
Aµj k̂jÂµµm 1m= A(k) · i 2
恒藤敏彦、現代物理学叢書 超伝導・超流動(岩波, 2001)
北孝文、統計力学から理解する超伝導理論 (サイエンス社, 2013)
玉垣良三、物理学最前線15 (1986)

Aμj: 3P order parameter (3✕3)

j

ABM(Anderson-Brinkman-Morel)(1972)0✓

Nuclear system (3P2 superfluidity in neutron star)0@Aµj = A 01k̂x + ik̂y
k̂z

0
0
i

k̂z
k̂x + ik̂y
◆

1
0
0A
0

⇢˜t (r, r ) = ⇢˜t (r , r)

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spin-current (tensor) pair density J̃t (r) = 1 [(rsha1_base64="zwN7x72gSfOFVZPqEClOhEm4zX0=">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</latexit>

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Local densities<latexit0⇢˜t (r) = ⇢˜t (r, r)

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pair density matrix<latexit

sha1_base64="HN7E7QZeIb/9et5ENuANb8oycw0=">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</latexit><latexit

Spin-singlet and triplet pairing in nuclear DFT
2s0 h |âr0 s0 t0 , ârst | inon-local pair density (spin-singlet condensation)s̃t (r, r ) =2i0(spin-singlet: s=sʼ)⇢˜t (r, r 0 ) =snon-local spin pair density (spin-triplet condensation) s̃t (r, r 0 ) =s̃t (r , r)

0

X

X

⇢ˆ
˜(rst, r 0 st)⇢˜(rst, r 0 s0 t) ˆ s0 s

ss0order parameter for spin-singlet pairing⇣
rrel ⌘rrel,Rs̃t (r, r 0 ) = s̃t R +2
 2⇣rrel ⌘@
rrel,R= s̃t (R, R) + rrel ·⌦ st R ++ O(|rrel |2 )
@rrel
2
2
rrel =0
1+ O(|rrel |2 )= rrel · (r r0 ) ⌦ s̃t (r, r 0 )02
r=r =R2= irrel · J̃t (R) + O(|rrel | )r0 ) ⌦ s̃t (r, r 0 )]r=r0 ← nine order parameters
Richardson, Phys. Rev. D 5, 1883 (1972).
Masaki et al Phys. Rev. Res. 2, 013193 (2020)

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J˜t (r) = J˜tr (r)er<latexit

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1 pseudoscalar (3P0)X
J̃tii (r)J˜t (r) =J˜t (r) = J˜tr (r)er + J˜tz (r)ez
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J˜tk (r) =<latexitiij

spin-current pair density has the same spatial symmetry as the particle-hole spin-current (tensor) density has.

X
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1[(rJ̃t (r) =2i<latexit

spin-current pair density<latexit

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J˜t (r) = 0<latexit

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Symmetry property of spin-current pair density0r ) ⌦ s̃t (r, r )]r=r0 (3 spatial x 3 spin)

3 vectors (3P1)0Rohozinski et al. Phys. Rev. C 81, 014313 (2010)

5 pseudotensors (3P2)"ijk J̃tij (r)J̃tij (r) =J̃tr (r),1
1J̃tij (r) + J̃tji (r)22

In the case of axial symmetry (with z-simplex symmetry) (r,φ, z)J̃tz (r) 6= 0

p 3P1 channel is active in spherical nuclei.
p Is there spin-triplet pair condensation in finite nuclei ?
p What is the property of the spin-current pair density?

1˜
Jt (r) ij
3

We consider spherical finite nuclei (with parity symmetry)J̃t (r) = 0

radial component only: spin and relative momentum are orthogonal to radial direction

Sρn or SJn (fm-3)

spin-singlet and spin-triplet condensation relevantZquantities (proportional to pair EDFs)

S⇢ n =

0.25

15
20
25

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Z

0.3

Ca
Sρn
S Jn

0.2

0.15

0.1

0.05

30
35

N
40

Sρn or SJn (fm-3)

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Spin-triplet pair condensate

dr|˜⇢n (r)|2

S Jn = R 2

0
45
50
0.4

50

dr|J˜n (r)|2

0.5

Sn

60
70
80
90

R2=10fm2

Ca and Sn isotopes: only neutron is super
Sρn
SJn

0.3

0.2

0.1
0
100 110 120 130

N

p pair EDF: spin-singlet only (volume-type pairing)p spin-singlet pair EDF induces spin-triplet pair condensate
p spin-triplet pair EDF has different orbital dependence (large at high-j intruder)sha1_base64="/ju9xhIFtz/S7NUZcbGd7rfUzb4=">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</latexit><latexit⇢˜(r) =

nlj

singlet-pair EDF

20
25
30

p Spin-orbit EDF induces the spin-triplet pair condensation fromthe spin-singlet pair EDF.
p Coupling constant of spin-triplet pair EDF is adjusted to provide the same pairing energy in 44Ca.
p Spin-triplet pair EDF alone (not realistic) shows similar pairing property.
1 X(2j + 1)u1 (nlj, r)u2 (nlj, r)
4⇡r2

(a)

triplet-pair EDF

(c)

35
40
20

˜ =
J(r)

1 X
(2j + 1) j(j + 1)
4⇡r3

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singlet-pair EDF

25

N

nlj

30

(b)

No LS
0.5 LS
1.0 LS

triplet-pair EDF

(d)

35

2hl · siSJn (fm-3)

0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
3.5
3
2.5
2
1.5
1
0.5
0
SJn (fm-3)

singlet condensation<latexit

0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0

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Ca isotopes<latexit

spin-triplet
pair EDF
Sρn (fm-3)

spin-singlet
pair EDF

Sρn (fm-3)

Spin-singlet/triplet condensations and EDFs
triplet condensation

40Nl(l + 1)3
u1 (nlj, r)u2 (nlj, r)
4

pairing gap

=
1
Nt
dr
ss0 µ

Z
X

h
i
dr Ũt (r)⇢t (r) + M̃t (r)⌧t (r) + B̃t (r) · Jt (r) ,
Z
(t)⇤
(t)
0 (t)
2 (µ, rs )h̃s0 s (r) 2 (µ, rs)

dr

X

sµ

pairing energy|Δn| (MeV)
2.5| 2 (µ, rs)|2
(t)

2

0
0
-3
-6
-9
-12
-15

singlet-pair EDF

20
25
30

N

(a)

1.5
1

singlet-pair EDF

35
40

triplet-pair EDF
No LS
0.5 LS
1.0 LS

20
25
30

(b)

0.5
triplet-pair EDF

(c)
(d)

35
0.4

0.3

0.2

0.1

0
0
-3
-6
-9
-12
-15

|Δn| (MeV)

spin-singlet pair EDF

Epair,nS=1 (MeV)t =Z

Epair,nS=0 (MeV)

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Pairing gap and pairing energy
spin-triplet pair EDF
0.5

40

N

p Pairing gap: an extended definition of density-averaged gap
p Pairing energy is similar (coupling constant is fitted at 44Ca) in singlet-pair and triplet pair EDFs.
p Pairing gap does not correspond to OES when spin-triplet pair EDF is used

Observables
What (ground-state) observable is relevant to spin-triplet pair condensation?
two-neutron separation energy (S2n)and pairing rotational MOI

87

singlettriplet

1f7/2
1f5/2

6
Nnlj

5
4

S2n (MeV)

Occupation probabilities

Ca

2p3/2

2

2p1/2

1
0
20

24

28

32
N

36

40

Jn (MeV-1)

3

22
20
18
16
14
12
10
8
6
4
2
25

(a)

singlet-pair EDF
triplet-pair EDF
Exp.

20

(b)

15
10
5
0
20

24

28

32

36

40

p Spin-triplet pair EDF does not affect s.p. occupation nor S2n. N
p Some difference in pairing rotational MOI: ΔS2n=S2n(N,Z) - S2n(N+2,Z) = 4/Jn.

Summaryp Spin-triplet like-particle (triplet-odd) pairing in finite nuclei
p Spin-current pair density is an order parameter for spin-triplet pairing
within the LDA.
p Spin-triplet pair condensation is induced by the spin-singlet pair EDF
with spin-orbit EDF.
p Spin-singlet and spin-triplet pair has different orbital dependence.
p Spin-triplet pair EDF induces spin-orbit splitting and spin-singlet pair condensation.
p Related observables
p NH, Tomohiro Oishi, and Kenichi Yoshida, arXiv:2308.02617.