# Triplet-odd pairing in finite nuclear systems: Even-even singly closed
nuclei

Nobuo Hinohara<EMAIL_ADDRESS>Center for Computational
Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan  Faculty of
Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571,
Japan  Facility for Rare Isotope Beams, Michigan State University, East
Lansing, Michigan 48824, USA  Tomohiro Oishi<EMAIL_ADDRESS>RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama 351-0198,
Japan  Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto
606-8502, Japan  Kenichi Yoshida<EMAIL_ADDRESS>Research Center
for Nuclear Physics, Osaka University, Ibaraki, Osaka 567-0047, Japan  Center
for Computational Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8577,
Japan  RIKEN Nishina Center for Accelerator-Based Science, Wako, Saitama
351-0198, Japan

###### Abstract

Background

The appearance of the pairing condensate is an essential feature of many-
fermion systems. There are two possible types of pairing: spin-singlet and
spin-triplet. However, an open question remains as to whether the spin-triplet
pairing condensate emerges in finite nuclei.

Purpose

The aim of this work is to examine the coexistence of the spin-singlet and
spin-triplet like-particle pairing condensates in nuclei. We also discuss the
dependence on the type of pairing functional.

Method

The Hartree-Fock-Bogoliubov calculations with a Skyrme $+$ local-pair energy-
density functional (EDF) are performed to investigate the pairing condensate
in the spherical ground states of Ca and Sn isotopes.

Results

The spin-singlet pair EDF induces not only the spin-singlet but also the spin-
triplet pairing condensates due to a strong spin-orbit splitting. By
discarding the spin-orbit EDF, only the spin-singlet pairing condensate
appears. The spin-triplet pair EDF, however, induces the spin-orbit splitting
and accordingly the spin-singlet pairing condensate.

Conclusions

The spin-orbit splitting plays an essential role in the coexistence of the
spin-singlet and spin-triplet pairing condensates in nuclei.

##  I introduction

The pairing is universal in many-fermion systems [1, 2]. A mean-field model
was first introduced by Bardeen, Cooper, and Schrieffer (BCS) for describing
the electronic superconductivity [3]. Within the original BCS theory, assumed
is the condensation of a Cooper pair with a relative angular momentum being
$s$ wave, the total spin zero, and the center-of-mass momentum zero. A variety
of forms of pairing, unconventional pairings, that are different from the BCS
type have been also found or predicted; see the reviews [4, 5, 6, 7] and the
references therein. Especially, in electronic and cold-atomic systems, the
spin-triplet Cooper pair has been actively investigated. The first example is
the superfluidity of helium-3 atoms [8, 9, 10], where the spin-fluctuation
interaction induces the spin-triplet Cooper pairs of fermionic atoms. The
Fulde-Ferrell-Larkin-Ovchinnikov type of superconductivity, where the center-
of-mass momentum is not zero, has been also discussed for decades [11, 12, 13,
14, 15]. In several spices of heavy-fermion metals and ferromagnetic Mott
insulators [16, 17, 18, 19, 20], the spin-triplet type of superconductivity is
expected. It is worthwhile to mention that spin-triplet pairing is a basic
concept of topological superconductivity. For the emergence of spin-triplet
pairing, the spin-orbit interaction often plays an essential role [4, 6, 21,
22, 23].

The pair correlation by a nucleon Cooper pair contributes significantly to
low-energy nuclear physics. The BCS theory was applied to atomic nuclei soon
after the original work [3] by Bohr et al. [24, 25]. The like-particle spin-
singlet pairing has been investigated mostly and is a key to understanding the
low-energy properties of the nuclear structure, for example, the odd-even
staggering (OES) of nuclear masses, the collectivity of the low-lying
$J^{\pi}=2^{+}$ states in even-even nuclei, and the moments of inertia of
deformed nuclei [26, 27]. The unconventional pairings have also been studied
in nuclear systems. Since a deuteron is the only two-nucleon system that is
bound in nature, the spin-triplet proton-neutron pairing has been investigated
for a long time and is under lively discussions [7]. Similarly, due to the
attractive nature of the nuclear force in the ${}^{3}P_{2}$ channel at high
momentum, the emergence of the triplet-odd pairing has been predicted and
studied in neutron-star matter [28, 29, 30].

For the study of nuclear superfluidity in medium-mass and heavy nuclei, a
self-consistent mean-field or energy-density functional (EDF) approach has
been adopted [31]. This choice is advantageous as it naturally provides the
anomalous (pair) density and the pairing gap as an order parameter and the
medium effects, which are known to be strong [1, 32], can be captured in the
EDF. Since the proton-neutron spin-triplet pairing, as well as the isovector
spin-singlet ones, can be characterized by local pair densities, there have
been many studies for these types of pairing [33]. On the other side,
discussions on the triplet-odd pairing in nuclei have been less active. In
Refs. [34, 35], the connection between the magnetic-dipole excitation to the
triplet-odd pairing is suggested. However, experimental evidence of the
triplet-odd pairing has not been observed.

In this work, we study the like-particle spin-triplet superfluidity in an EDF
approach. To this end, we introduce the spin-triplet nonlocal pair density as
an order parameter. We also investigate the connection between spin-orbit
splitting and triplet-odd pairing.

##  II formalism

We describe the spin-singlet and spin-triplet pairing within the local density
approximation of the EDF. Details on this framework are well summarized in
Refs. [36, 31, 37], and in particular, we focus on the pairing part in this
section.

###  II.1 Nonlocal pair density

The building block of the pairing in the Hartree-Fock-Bogoliubov (HFB) theory
is the pair density matrix. We define it with the standard phase for the case
without the proton-neutron pairing as

$\displaystyle\hat{\tilde{\rho}}(\bm{r}_{1}s_{1},\bm{r}_{2}s_{2};t)$
$\displaystyle=-2s_{2}\langle\Psi|\hat{c}_{\bm{r}_{2}-s_{2}t}\hat{c}_{\bm{r}_{1}s_{1}t}|\Psi\rangle,$
(1)

where $\hat{c}_{\bm{r}st}$ represents the nucleon annihilation operator at
position $\bm{r}$, spin $s$, and isospin $t$, and $|\Psi\rangle$ is the HFB
state.

The spin-singlet and spin-triplet nonlocal pair densities are defined by

$\displaystyle\tilde{\rho}_{t}(\bm{r}_{1},\bm{r}_{2})$
$\displaystyle=\sum_{s}\hat{\tilde{\rho}}(\bm{r}_{1}s,\bm{r}_{2}s;t),$ (2)
$\displaystyle\tilde{\bm{s}}_{t}(\bm{r}_{1},\bm{r}_{2})$
$\displaystyle=\sum_{s_{1}s_{2}}\hat{\tilde{\rho}}(\bm{r}_{1}s_{1},\bm{r}_{2}s_{2};t)\hat{\bm{\sigma}}_{s_{2}s_{1}}.$
(3)

One can express the pair density matrix as

$\displaystyle\hat{\tilde{\rho}}(\bm{r}_{1}s_{1},\bm{r}_{2}s_{2};t)=\frac{1}{2}\tilde{\rho}_{t}(\bm{r}_{1},\bm{r}_{2})\delta_{s_{1}s_{2}}+\frac{1}{2}\tilde{\bm{s}}_{t}(\bm{r}_{1},\bm{r}_{2})\cdot\hat{\bm{\sigma}}_{s_{1}s_{2}},$
(4)

where $\hat{\bm{\sigma}}$ is the spin Pauli matrix. We note that the nonlocal
pair densities show the spatial property of the nucleon pair; the spin-singlet
pair is symmetric and the spin-triplet pair is antisymmetric for the exchange
of the coordinate variables,

$\displaystyle\tilde{\rho}_{t}(\bm{r}_{1},\bm{r}_{2})$
$\displaystyle=\tilde{\rho}_{t}(\bm{r}_{2},\bm{r}_{1}),$ (5)
$\displaystyle\tilde{\bm{s}}_{t}(\bm{r}_{1},\bm{r}_{2})$
$\displaystyle=-\tilde{\bm{s}}_{t}(\bm{r}_{2},\bm{r}_{1}).$ (6)

The spin-triplet nonlocal pair density vanishes at $\bm{r}_{1}=\bm{r}_{2}$,
indicating that a simple local density approximation does not work for the
spin-triplet pair, and the nonlocality plays a major role here.

###  II.2 Density matrix expansion

The nuclear interaction energy derived from a local two-body interaction can
be expressed with the local densities by a density matrix expansion technique.
This has been discussed in Ref. [31] for the particle-hole part of the
interaction. Here we apply the density matrix expansion for the pair density
matrix (nonlocal pair density).

We introduce the following coordinates of the pair:

$\displaystyle\bm{r}_{1}=\bm{r}+\frac{\bm{r}_{\rm
rel}}{2},\quad\bm{r}_{2}=\bm{r}-\frac{\bm{r}_{\rm rel}}{2},$ (7)

assuming that the pair density matrix vanishes quickly with increasing
$\bm{r}_{\rm rel}$. This allows us to expand the nonlocal pair densities in
terms of the relative coordinate $\bm{r}_{\rm rel}$:

$\displaystyle\tilde{\rho}_{t}(\bm{r}_{1},\bm{r}_{2})$
$\displaystyle=\tilde{\rho}_{t}\left(\bm{r}+\frac{\bm{r}_{\rm
rel}}{2},\bm{r}-\frac{\bm{r}_{\rm rel}}{2}\right)$
$\displaystyle=\tilde{\rho}_{t}(\bm{r},\bm{r})$
$\displaystyle\quad+\frac{\partial}{\partial\bm{r}_{\rm
rel}}\tilde{\rho}_{t}\left(\bm{r}+\frac{\bm{r}_{\rm
rel}}{2},\bm{r}-\frac{\bm{r}_{\rm rel}}{2}\right)\Big{|}_{\bm{r}_{\rm
rel}=\bm{0}}\cdot\bm{r}_{\rm rel}$
$\displaystyle\quad+\frac{1}{2}\frac{\partial^{2}}{\partial\bm{r}_{\rm
rel}^{2}}\tilde{\rho}_{t}\left(\bm{r}+\frac{\bm{r}_{\rm
rel}}{2},\bm{r}-\frac{\bm{r}_{\rm rel}}{2}\right)\Big{|}_{\bm{r}_{\rm
rel}=\bm{0}}\bm{r}_{\rm rel}^{2}$ $\displaystyle\quad+{\cal O}(|\bm{r}_{\rm
rel}|^{3})$
$\displaystyle=\tilde{\rho}_{t}(\bm{r})+\frac{1}{2}(\bm{\nabla}_{1}-\bm{\nabla}_{2})\tilde{\rho}_{t}(\bm{r}_{1},\bm{r}_{2})\Big{|}_{\bm{r}_{1}=\bm{r}_{2}=\bm{r}}\cdot\bm{r}_{\rm
rel}$
$\displaystyle\quad+\frac{1}{8}(\bm{\nabla}_{1}-\bm{\nabla}_{2})^{2}\tilde{\rho}_{t}(\bm{r}_{1},\bm{r}_{2})\Big{|}_{\bm{r}_{1}=\bm{r}_{2}=\bm{r}}\bm{r}_{\rm
rel}^{2}$ $\displaystyle\quad+{\cal O}(|\bm{r}_{\rm rel}|^{3})$
$\displaystyle=\tilde{\rho}_{t}(\bm{r})+\frac{1}{8}\left[\Delta\tilde{\rho}_{t}(\bm{r})-4\tilde{\tau}_{t}(\bm{r})\right]\bm{r}_{\rm
rel}^{2}+{\cal O}(|\bm{r}_{\rm rel}|^{3}).$ (8)

We use
$\tilde{\rho}_{t}(\bm{r}_{1},\bm{r}_{2})=\tilde{\rho}_{t}(\bm{r}_{2},\bm{r}_{1})$
to remove the first-order term, and the local pair density and kinetic pair
density are defined as

$\displaystyle\tilde{\rho}_{t}(\bm{r})$
$\displaystyle=\tilde{\rho}_{t}(\bm{r},\bm{r}),$ (9)
$\displaystyle\tilde{\tau}_{t}(\bm{r})$
$\displaystyle=\left(\bm{\nabla}_{1}\cdot\bm{\nabla}_{2}\right)\tilde{\rho}_{t}(\bm{r}_{1},\bm{r}_{2})\Big{|}_{\bm{r}_{1}=\bm{r}_{2}=\bm{r}}.$
(10)

The spin-triplet nonlocal pair density is expanded as

$\displaystyle\tilde{\bm{s}}_{t}(\bm{r}_{1},\bm{r}_{2})$
$\displaystyle=\tilde{\bm{s}}_{t}\left(\bm{r}+\frac{\bm{r}_{\rm
rel}}{2},\bm{r}-\frac{\bm{r}_{\rm rel}}{2}\right)$ $\displaystyle=\bm{r}_{\rm
rel}\cdot\left[\frac{\partial}{\partial\bm{r}_{\rm
rel}}\otimes\tilde{\bm{s}}_{t}\left(\bm{r}+\frac{\bm{r}_{\rm
rel}}{2},\bm{r}-\frac{\bm{r}_{\rm rel}}{2}\right)\right]_{\bm{r}_{\rm
rel}=\bm{0}}$ $\displaystyle\quad+{\cal O}(|\bm{r}_{\rm rel}|^{2})$
$\displaystyle=\frac{1}{2}\bm{r}_{\rm
rel}\cdot\left[(\bm{\nabla}_{1}-\bm{\nabla}_{2})\otimes\tilde{\bm{s}}_{t}(\bm{r}_{1},\bm{r}_{2})\right]\Big{|}_{\bm{r}_{1}=\bm{r}_{2}=\bm{r}}$
$\displaystyle\quad+{\cal O}(|\bm{r}_{\rm rel}|^{2})$
$\displaystyle=i\bm{r}_{\rm rel}\cdot\tilde{\sf J}_{t}(\bm{r})+{\cal
O}(|\bm{r}_{\rm rel}|^{2}),$ (11)

where

$\displaystyle\tilde{\sf
J}_{t}(\bm{r})=\frac{1}{2i}(\bm{\nabla}_{1}-\bm{\nabla}_{2})\otimes\tilde{\bm{s}}_{t}(\bm{r}_{1},\bm{r}_{2})\Big{|}_{\bm{r}_{1}=\bm{r}_{2}=\bm{r}}$
(12)

is the tensor (spin-current) pair density, we use
$\tilde{\bm{s}}_{t}(\bm{r}_{1},\bm{r}_{2})=-\tilde{\bm{s}}_{t}(\bm{r}_{2},\bm{r}_{1})$
to remove the zeroth-order term, and
$\bm{v}\cdot(\bm{u}\otimes\bm{w})\equiv(\bm{v}\cdot\bm{u})\bm{w}$. Spin-
triplet anisotropic pairing in condensed-matter physics requires an odd wave-
number $\bm{k}$ dependence, and the tensor pair density corresponds to the
order parameters of the $p$-wave superfluidity that consists of nine
components [4].

The density matrix expansion of the nonlocal pair density provides the local
EDF starting from a local two-body spin-singlet and spin-triplet pairing
interaction. The general form of the pair EDF is given by [38]

$\displaystyle E_{{\rm pair},t}^{S=0}=\int d\bm{r}_{1}d\bm{r}_{2}v^{S=0}_{{\rm
pair},t}(|\bm{r}_{1}-\bm{r}_{2}|)|\tilde{\rho}_{t}(\bm{r}_{1},\bm{r}_{2})|^{2},$
(13) $\displaystyle E_{{\rm pair},t}^{S=1}=\int
d\bm{r}_{1}d\bm{r}_{2}v^{S=1}_{{\rm
pair},t}(|\bm{r}_{1}-\bm{r}_{2}|)|\tilde{\bm{s}}_{t}(\bm{r}_{1},\bm{r}_{2})|^{2},$
(14)

where $v^{S=0}_{{\rm pair},t}$ and $v^{S=1}_{{\rm pair},t}$ are the spin-
singlet and spin-triplet pairing interaction strengths that depend on the
absolute value of the relative coordinate of the two nucleons.

Inserting the density matrix expansion in the nonlocal pair densities, we have

$\displaystyle E^{S=0}_{{\rm pair},t}$ $\displaystyle=\int
d\bm{r}\left\\{\tilde{C}^{\rho}_{t}|\tilde{\rho}_{t}(\bm{r})|^{2}+\tilde{C}^{\Delta\rho}_{t}{\rm
Re}\left[\tilde{\rho}^{\ast}_{t}(\bm{r})\Delta\tilde{\rho}_{t}(\bm{r})\right]\right.$
$\displaystyle\quad\left.+\tilde{C}^{\tau}_{t}{\rm
Re}\left[\tilde{\rho}^{\ast}_{t}(\bm{r})\tilde{\tau}_{t}(\bm{r})\right]\right\\},$
(15) $\displaystyle E^{S=1}_{{\rm pair},t}$ $\displaystyle=\int
d\bm{r}\tilde{C}^{J}_{t}|\tilde{\sf J}_{t}(\bm{r})|^{2}.$ (16)

The coupling constants are related to the local potential as

$\displaystyle\tilde{C}^{\rho}_{t}$ $\displaystyle=\int d\bm{r}_{\rm
rel}v^{S=0}_{{\rm pair},t}(|\bm{r}_{\rm rel}|),$ (17)
$\displaystyle\tilde{C}^{\Delta\rho}_{t}$
$\displaystyle=-\frac{1}{4}\tilde{C}^{\tau}_{t}=\frac{1}{4}\int d\bm{r}_{\rm
rel}\bm{r}_{\rm rel}^{2}v^{S=0}_{{\rm pair},t}(|\bm{r}_{\rm rel}|),$ (18)
$\displaystyle\tilde{C}_{t}^{J}$ $\displaystyle=\int d\bm{r}_{\rm
rel}\bm{r}_{\rm rel}^{2}v^{S=1}_{{\rm pair},t}(|\bm{r}_{\rm rel}|).$ (19)

These are the coupling constants for the spin-singlet pairing
$\tilde{C}^{\rho}_{t}$ and its next-order terms $\tilde{C}^{\Delta\rho}_{t}$
and $\tilde{C}^{\tau}_{t}$, and the spin-triplet coupling constant
$\tilde{C}^{J}_{t}$.

By using the G3RS-1E-1 potential introduced by Tamagaki [28], for instance,
$\tilde{C}^{\rho}_{t}=-697.087$ MeV fm3 and
$\tilde{C}^{\Delta\rho}_{t}=1363.253$ MeV fm5 are obtained for the spin-
singlet coupling constants. For the spin-triplet coupling, on the other hand,
by using the G3RS-3O-1 potential for $v^{S=1}_{{\rm pair},t}$, we obtain
$\tilde{C}^{J}_{t}=6794.724$ MeV fm5. Note that we assumed the ${}^{3}P_{1}$
channel to obtain this value and that this is repulsive in this channel.

In the lowest order in terms of the nonlocality, the local pair density
$\tilde{\rho}_{t}(\bm{r})$ and the pair EDF that is proportional to
$|\tilde{\rho}_{t}(\bm{r})|^{2}$ represent the spin-singlet pair condensation
and EDF, while the spin-current pair density $\tilde{\sf J}_{t}(\bm{r})$ and
the term proportional to $|\tilde{\sf J}_{t}(\bm{r})|^{2}$ represent the spin-
triplet pair condensation and EDF.

Zero-range Skyrme interactions also produce the terms related to the spin-
singlet and spin-triplet pair condensation. Only the SkP interaction [39]
includes the spin-singlet and spin-triplet terms, and other standard Skyrme
EDFs do not consider pairing terms other than those proportional to
$|\tilde{\rho}_{t}(\bm{r})|^{2}$ due to unrealistic pairing properties [31].

We note that the spin-triplet pair density and thus the EDF can be decomposed
into trace (pseudoscalar), antisymmetric (vector), and symmetric
(pseudotensor) parts [37]:

$\displaystyle\tilde{J}_{t}(\bm{r})$ $\displaystyle=\sum_{a=x,y,z}\tilde{\sf
J}_{taa}(\bm{r}),$ (20) $\displaystyle\tilde{\bm{J}}_{ta}(\bm{r})$
$\displaystyle=\sum_{b,c=x,y,z}\epsilon_{abc}\tilde{\sf J}_{tbc}(\bm{r}),$
(21) $\displaystyle\tilde{\underline{\sf J}}_{tab}(\bm{r})$
$\displaystyle=\frac{1}{2}\tilde{\sf J}_{tab}(\bm{r})+\frac{1}{2}\tilde{\sf
J}_{tba}(\bm{r})-\frac{1}{3}\tilde{J}_{t}(\bm{r})\delta_{ab}.$ (22)

This decomposition of the spin-current quantity is also applied in the
discussion of ${}^{3}P_{2}$ superfluidity [40, 41], and these three components
are relevant to ${}^{3}P_{0}$, ${}^{3}P_{1}$, and ${}^{3}P_{2}$ superfluidity,
respectively.

For a general pair EDF that is not based on an effective interaction, each
coupling constant in the spin-singlet pair EDF (II.2) can be taken
independently, except that the relation between $\tilde{C}_{t}^{\Delta\rho}$
and $\tilde{C}_{t}^{\tau}$ in Eq. (18) is a requirement from the local gauge
invariance [37]. The spin-triplet pair EDF has a structure similar to that of
the tensor functional in the particle-hole EDF and can have a more general
form [42]:

$\displaystyle E^{S=1}_{{\rm pait},t}=\int
d\bm{r}\tilde{C}_{t}^{J0}|\tilde{J}_{t}(\bm{r})|^{2}+\tilde{C}_{t}^{J1}|\tilde{\bm{J}}_{t}(\bm{r})|^{2}+\tilde{C}_{t}^{J2}|\tilde{\underline{\sf
J}}_{t}(\bm{r})|^{2}.$ (23)

Unlike the particle-hole part, these three coupling constants are not
constrained by the local gauge invariance and can be taken independently. When
the pair EDF is derived from a nonlocal effective interaction of the form
(14), three coupling constants are related by
$\tilde{C}^{J}_{t}=3\tilde{C}_{t}^{J0}=2\tilde{C}_{t}^{J1}=\tilde{C}_{t}^{J2}$.
However, the tensor and spin-orbit interactions, which are not in the form of
Eq. (14), allow independent contributions to the three coupling constants.

###  II.3 Mean-field approach

The mean-field equations for protons and neutrons, obtained by the functional
derivative of the EDF, are given in Refs. [39, 37]. The pair Hamiltonian has
the following form:

$\displaystyle\tilde{h}^{(t)}_{ss^{\prime}}(\bm{r})$
$\displaystyle=\left[\tilde{U}_{t}(\bm{r})-\bm{\nabla}\tilde{M}_{t}(\bm{r})\cdot\bm{\nabla}\right]\delta_{ss^{\prime}}$
$\displaystyle\quad+\frac{1}{2i}\left\\{\bm{\nabla}\cdot\left[\tilde{\sf
B}_{t}(\bm{r})\cdot\hat{\bm{\sigma}}_{ss^{\prime}}\right]+\left[\tilde{\sf
B}_{t}(\bm{r})\cdot\hat{\bm{\sigma}}_{ss^{\prime}}\right]\cdot\bm{\nabla}\right\\},$
(24)

where the potential energy $\tilde{U}_{t}$, the effective inertia parameter
$\tilde{M}_{t}$, and the spin-orbit form factors $\tilde{\sf B}_{t}$ are given
by

$\displaystyle\tilde{U}_{t}(\bm{r})$
$\displaystyle=2\tilde{C}_{t}^{\rho}\tilde{\rho}_{t}(\bm{r})+2\tilde{C}^{\Delta\rho}_{t}\Delta\tilde{\rho}_{t}(\bm{r})+\tilde{C}_{t}^{\tau}\tilde{\tau}_{t}(\bm{r}),$
(25) $\displaystyle\tilde{M}_{t}(\bm{r})$
$\displaystyle=\tilde{C}_{t}^{\tau}\tilde{\rho}_{t}(\bm{r}),$ (26)
$\displaystyle\tilde{\sf B}_{tab}(\bm{r})$
$\displaystyle=2\tilde{C}_{t}^{J0}\tilde{J}_{t}(\bm{r})\delta_{ab}-2\tilde{C}_{t}^{J1}\sum_{c=x,y,z}\epsilon_{acb}\tilde{\bm{J}}_{tc}(\bm{r})$
$\displaystyle\quad+2\tilde{C}_{t}^{J2}\tilde{\underline{\sf
J}}_{tab}(\bm{r}).$ (27)

As the pair Hamiltonian (24) depends on the spins $s$ and $s^{\prime}$ when
spin-triplet pair EDF is considered, we define the pairing gap by averaging
out the pair Hamiltonian using the spin-dependent lower component of the
quasiparticle wave function $\phi_{2}^{(t)}(\mu,\bm{r}s)$ as

$\displaystyle\Delta_{t}$ $\displaystyle=\frac{\displaystyle\int
d\bm{r}\sum_{ss^{\prime}\mu}\phi^{(t)\ast}_{2}(\mu,\bm{r}s^{\prime})\tilde{h}_{s^{\prime}s}^{(t)}(\bm{r})\phi^{(t)}_{2}(\mu,\bm{r}s)}{\displaystyle\int
d\bm{r}\sum_{s\mu}|\phi^{(t)}_{2}(\mu,\bm{r}s)|^{2}}$
$\displaystyle=\frac{1}{N_{t}}\int
d\bm{r}\left[\tilde{U}_{t}(\bm{r}){\rho}_{t}(\bm{r})+\tilde{M}_{t}(\bm{r}){\tau}_{t}(\bm{r})+\tilde{\sf
B}_{t}(\bm{r})\cdot{\sf J}_{t}(\bm{r})\right],$ (28)

where $N_{t}=N$ or $Z$, and the density $\rho_{t}$, the kinetic density
$\tau_{t}$, and the tensor (spin-current) density ${\sf J}_{t}$ are defined
using the nonlocal particle-hole densities [defined in a similar way as Eqs.
(2) and (3) but for the particle-hole density matrix] as

$\displaystyle\rho_{t}(\bm{r})$ $\displaystyle=\rho_{t}(\bm{r},\bm{r}),$ (29)
$\displaystyle\tau_{t}(\bm{r})$
$\displaystyle=(\bm{\nabla}_{1}\cdot\bm{\nabla}_{2})\rho_{t}(\bm{r}_{1},\bm{r}_{2})\Big{|}_{\bm{r}_{1}=\bm{r}_{2}=\bm{r}},$
(30) $\displaystyle{\sf J}_{t}(\bm{r})$
$\displaystyle=\frac{1}{2i}(\bm{\nabla}_{1}-\bm{\nabla}_{2})\otimes\bm{s}_{t}(\bm{r}_{1},\bm{r}_{2})\Big{|}_{\bm{r}_{1}=\bm{r}_{2}=\bm{r}}.$
(31)

Although Eq. (28) is a natural extension of the average gap for a generalized
pair Hamiltonian [43], the discrepancy of the pairing gap and experimental OES
has been pointed out when the singlet-pair EDF contains the kinetic terms
$\tilde{C}^{\tau}_{t}$ and $\tilde{C}^{\Delta\rho}_{t}$ [44].

###  II.4 Expression within spherical symmetry

We assume the spherical symmetry for the HFB state for simplicity. The
spherical symmetry cancels the two of the spin-current pair densities
$\tilde{J}_{t}(\bm{r})$ and $\tilde{\underline{\sf J}}_{t}(\bm{r})$, and only
the radial component of the vector spin-current pair density can exist,
$\tilde{\bm{J}}_{t}(\bm{r})=\tilde{J}_{tr}(r)\bm{e}_{r}$ [45]. Within the
spherical symmetry, the quasiparticle wave function can be decomposed into the
radial and angular part

$\displaystyle\phi_{i}^{(t)}(E,\bm{r}s)=\frac{u_{i}^{(t)}(nlj,r)}{r}Y_{lm_{l}}(\bm{\hat{r}})\langle
lm_{l}\frac{1}{2}s|jm\rangle\quad(i=1,2),$ (32)

where $i=1$ and 2 correspond to the upper and lower components, respectively.
The local pair density and the spin-current pair density are given by

$\displaystyle\tilde{\rho}_{t}(r)$ $\displaystyle=-\frac{1}{4\pi
r^{2}}\sum_{nlj}(2j+1)u_{1}^{(t)}(nlj,r)u_{2}^{(t)}(nlj,r),$ (33)
$\displaystyle\tilde{J}_{tr}(r)$ $\displaystyle=-\frac{2}{4\pi
r^{3}}\sum_{nlj}(2j+1)\langle\bm{l}\cdot\bm{s}\rangle
u_{1}^{(t)}(nlj,r)u_{2}^{(t)}(nlj,r),$ (34)

where
$\langle\bm{l}\cdot\bm{s}\rangle=\displaystyle\frac{1}{2}\left[j(j+1)-l(l+1)-\displaystyle\frac{3}{4}\right]$.
Notice that these quantities have different dimensions.

Figure 1: Spin-singlet and triplet pairing components of neutrons,
$S_{\rho_{n}}$ and $S_{J_{n}}$, respectively, in the Ca and Sn isotope chains.

##  III Numerical calculations

###  III.1 Spin-singlet pair EDF

We utilize the hfbrad code [46] for spherical Skyrme-HFB calculations in the
following. The SLy4 and spin-singlet volume-type contact pair EDF with the
strength $\tilde{C}_{n}^{\rho}=-46.625$ MeV fm3 (in the standard notation
$V_{0}=4\tilde{C}_{n}^{\rho}=-186.5$ MeV fm3) is employed with the cutoff
parameter of 60 MeV. This strength has been adjusted to reproduce the neutron
pairing gap 1.245 MeV in 120Sn.

We evaluate the spin-singlet and spin-triplet pair condensations with the
following pairing components:

$\displaystyle S_{\rho_{n}}$ $\displaystyle=\int
d\bm{r}|\tilde{\rho}_{t}(\bm{r})|^{2},$ (35) $\displaystyle S_{J_{n}}$
$\displaystyle=R^{2}\int d\bm{r}|\tilde{\bm{J}}_{t}(\bm{r})|^{2}.$ (36)

They have exactly the same local density dependence that appears in the pair
energy. The constant $R^{2}=10$ fm2 is estimated from the ratio
$|\tilde{C}_{t}^{J}/\tilde{C}_{t}^{\rho}|$ of the G3RS potential and is
introduced to make the units of the two quantities identical.

In Fig. 1, the results from neutron pair densities in the Ca and Sn isotopes
are presented. The spin-singlet component $S_{\rho_{n}}$ shows finite values
except in neutron closed-shell nuclei. Even though the attractive pair EDF is
present only in the spin-singlet channel, our results show nonzero values for
the spin-triplet component $S_{J_{n}}$, namely, a coexistence of the spin-
singlet and spin-triplet condensates is suggested in finite nuclei. This is
also expected in Eqs. (33) and (34). A similar feature has been discussed in
the case of the $T=1$ proton-neutron pair [47], in condensed matter [21], and
in ultracold Fermi gas [22, 23, 48] in the presence of the spin-orbit
($\bm{k}\otimes\bm{\sigma}$ type) interaction. Notice that a direct comparison
of $S_{\rho_{n}}$ and $S_{J_{n}}$ does not make sense as their relative value
depends on the introduced constant $R^{2}$. However, the isotopic dependence
indicates that the spin-triplet pairing is more sensitive to the shell orbits
involved than the spin-singlet one is. $S_{\rho_{n}}$ is enhanced in the
midshell region with high degeneracy, such as in the $f_{7/2}$ and $f_{5/2}$
orbits in Ca isotopes and $50<N<82$ and $82<N<126$ in Sn isotopes, while
$S_{J_{n}}$ shows a stronger orbital dependence; we see an enhancement (a
reduction) in $S_{J_{n}}$ in the isotope where the neutron Fermi energy is
around $j_{>}$ ($j_{<}$) orbit in $f_{7/2}$ and $f_{5/2}$ in Ca isotopes and
an enhancement in the intruder region in the middle shell in Sn isotopes.

Figure 2:  The neutron local pair density $\tilde{\rho}_{n}(r)$ and the radial
component of the neutron tensor pair density $\tilde{J}_{nr}(r)$ of 42Ca and
56Ca and the contributions from $1f_{7/2}$ and $1f_{5/2}$ orbits.

To analyze the contributions from the $j_{>}$ and $j_{<}$ orbits, we take 42Ca
and 56Ca as representative cases, where two particles are supposed to occupy
the $f_{7/2}$ and $f_{5/2}$ orbits mainly. The pair density distributions
$\tilde{\rho}_{n}(r)$ and $\tilde{J}_{nr}(r)$ together with the contributions
from $f_{7/2}$ and $f_{5/2}$ orbits are plotted in Fig. 2. Both the spin-
singlet and spin-triplet neutron pair densities have finite values in the 42Ca
and 56Ca nuclei. There, the $f_{7/2}$ and $f_{5/2}$ neutrons have dominant
contributions as expected. For the spin-singlet density, they have a coherent
contribution, and the total pair densities are composed of the coherent
addition from the other orbits as well (not shown in the figure), whereas the
neutrons in the $f_{7/2}$ and $f_{5/2}$ orbits contribute in a destructive way
to the spin-triplet density. The dominant contribution for $\tilde{J}_{nr}(r)$
in 42Ca is from the $f_{7/2}$ orbit, while $\tilde{J}_{nr}(r)$ in 56Ca is
composed of the multiple orbits including those not shown in the figure. This
indicates the magicity at $N=34$ is weaker than that at $N=28$.

###  III.2 Spin-triplet pair EDF

In place of the spin-singlet pair EDF, we employ the spin-triplet pair EDF.
The coupling constant of the spin-triplet pair EDF is adjusted to reproduce
the pairing energy of 44Ca obtained in the spin-singlet pair EDF
($\tilde{C}_{n}^{J1}=-46.125$ MeV fm5). The pairing energy is $-5.08$ and
$-5.25$ MeV in the singlet-pair EDF and triplet-pair EDF, as shown in Figs.
4(c) and 4(d).

Figure 3 shows the spin-singlet and spin-triplet pairing components calculated
with either the spin-singlet or the spin-triplet pair EDF for the Ca isotopes.
The spin-singlet pairing component calculated with the spin-singlet pair EDF
and the spin-triplet pairing component calculated with the spin-triplet pair
EDF have very similar properties: one sees the collapse of the pairing at the
magic numbers $N=20,28,32$, and $40$ (there is a tiny difference in $N=34$)
and the neutron-number dependence of the relative size of the pairing
component. We also note that there is little difference in the pairing energy,
the chemical potential, and other observables of the particle-hole type. The
coupling constant of the spin-triplet pair EDF $\tilde{C}^{J1}_{t}$ can be
related to the Skyrme parameters as
$\tilde{C}_{t}^{J1}=[t_{2}(1+x_{2})+5t_{o}+2W_{0})]/8$ and are repulsive for
many Skyrme interactions such as SIII (18.125 MeV fm5) [49], SLy4 (30.75 MeV
fm5), SLy5 (31.5 MeV fm5) [50], and SkP (5.486 MeV fm5), but can be attractive
for the Skyrme interactions that include the tensor interaction such as SLy5 +
T ($-53.5$ MeV fm5) [51], and 14 Skyrme parameters out of 36 T$IJ$ parameter
sets in Ref. [52]. Although the coupling constants of the EDF can be taken
arbitrarily in the framework of the nuclear density functional theory, the
tensor interaction will have a large impact on the property of the spin-
triplet pairing coupling constant.

###  III.3 Roles of the spin-orbit EDF

The spin-orbit splitting is expected to play an important role in the spin-
triplet pairing as anticipated from the expression of the tensor pair density
(34). In Fig. 3, we also present the pairing components $S_{\rho_{n}}$ and
$S_{J_{n}}$ obtained by changing the coupling constant of the spin-orbit EDF
while keeping the pairing coupling constants to the original values. Three
spin-orbit EDFs are considered: the original spin-orbit EDF (1.0 LS), the
reduced one multiplied by a factor of 0.5 (0.5 LS), and no spin-orbit EDF (No
LS). First, suppose that the spin-orbit EDF is neglected. Then, the spin-
singlet pair EDF promotes only the spin-singlet pair condensate, and the spin-
triplet pairing component $S_{J_{n}}$ is zero [Figs. 3(a) and 3(b)]. One is
tempted to the opposite conclusion when the spin-triplet pair EDF is
considered. However, the appearance of the spin-triplet pairing component
inevitably induces the spin-orbit splitting and the spin-singlet pairing
component. This results in a nonzero spin-singlet pairing component, although
the induced amount is tiny. As a result, commonly with the spin-singlet and
spin-triplet EDFs, the corresponding pairing component takes the maximum at
around $N=28$, which is around the half-filled situation of the 14-fold
degenerated $f$ orbit [Figs. 3(a) and 3(d)]. The pairing component becomes
zero at LS-closed shells $N=20$ and $N=40$.

The behavior of the pairing components $S_{\rho_{n}}$ and $S_{J_{n}}$ shows a
drastic change with the value of the coupling constant of the spin-orbit EDF,
although we do not change the pair EDF itself. The spin-orbit EDF decreases
the pairing component due to the lower degeneracy of the single-particle
levels [Figs. 3(a) and 3(d)], but it enhances the coexistence of
$S_{\rho_{n}}$ and $S_{J_{n}}$ [Figs. 3(b) and 3(c)]. 111The mixture of the
spin-singlet (even-parity) and spin-triplet (odd-parity) components due to the
spin-orbit field was discussed in the proton-neutron channel within the three-
body model for 16O$+p+n$ [47].  By comparing the $0.5$ LS and $1.0$ LS cases,
one can see an enhancement of the mixing in the region $20<N<28$ and the
suppression in the neutron-rich side with $N>30$. The suppression is common
for the main pairing component and the induced pairing component due to the
lesser degeneracy of the single-particle levels with increasing the coupling
constant of the spin-orbit EDF.

Figure 3: Spin-singlet and spin-triplet pairing components, $S_{\rho_{n}}$ and
$S_{J_{n}}$, calculated in the Ca isotopes by changing the coupling constant
of the spin-orbit EDF. The labels 1.0 LS, 0.5 LS, and No LS represent the
calculations with the original spin-orbit EDF, the reduced one multiplied by a
factor of 0.5, and the calculation without the spin-orbit EDF, respectively.

We also plot the pairing gap and the pairing energy in Fig. 4. Figures 3(a)
and 4(a) show that the singlet-pairing component $S_{\rho_{n}}$ and the
pairing gap $\Delta_{n}$ behave in a similar way in the case of singlet-pair
EDF, while the triplet-pairing component $S_{J_{n}}$ [Fig. 3(d)] and the
pairing gap [Fig. 4(b)] in the case of the triplet-pair EDF behave in a
different way. A strong reduction of the averaged gap for the triplet pairing
in the “No LS” case [Fig. 4(b)] is due to a low tensor density ${\sf J}$. The
pairing energies for the singlet-pair EDF [Fig. 4(c)] and the triplet-pair EDF
[Fig. 4(d)] take similar values as a function of the neutron number, although
the coupling constants of the two pair EDFs are adjusted only at $N=24$. The
agreement of the pairing energy shows that the triplet-pair EDF can include a
contribution similar to that of the singlet-pair EDF.

Figure 4:  Pairing gap $|\Delta_{n}|$ and pairing energy $E^{S=0,1}_{{\rm
pair},n}$ calculated for the singlet- and triplet-pair EDF in the Ca isotopes
by changing the coupling constant of the spin-orbit EDF.

###  III.4 Relevant observables

The small values of the pairing gap defined by Eq. (28) may not correspond to
the experimental OES for the triplet-pair EDF [Fig. 4(b)], similar to when the
singlet-pair Hamiltonian contains derivative terms [44]. We note that another
average gap in which $\phi_{2}^{(t)\ast}$ is replaced by $\phi_{1}^{(t)\ast}$
in Eq. (28) behaves even worse in the case of the triplet-pair EDF, because of
the singlet-pair amplitude in the denominator that is very small as expected
from Fig. 3(c).

To analyze the influence of the type of pairing in the occupation, we plot the
occupation of each neutron orbit in Fig. 5. The occupation number increases in
the order of the single-particle energies with the neutron number increases,
showing that the dominant pair correlation is within a single orbit near the
Fermi energy. There is no significant difference between the occupation
numbers calculated with the singlet- and triplet-pair EDFs. The difference is
barely visible around $34\leq N\leq 38$.

Figure 5:  Occupation numbers of $pf$-shell orbits in Ca isotopes calculated
with the singlet- and triplet-pair EDFs.

The pairing rotational moment of inertia can be a relevant observable of the
pairing condensation [53] even in the presence of the triplet-pair EDF. Figure
6 shows the two-neutron separation energies $S_{2n}(N)=E(N-2)-E(N)$ and the
neutron pairing rotational moments of inertia calculated from the three HFB
energy differences ${\cal J}_{n}(N)=4/\Delta
S_{2n}(N)=4/[E(N+2)-2E(N)+E(N-2)]$. Unlike the relation between the pairing
gap and the OES, the pairing rotational moment of inertia holds a good
correspondence with the experimental values of the double binding-energy
differences. The moment of inertia at $N=30$ becomes negative ($-17.86$ MeV-1)
for the triplet-pair EDF, while the singlet-pair EDF reproduces the
experimental value. The negative value originates from the staggering behavior
of $S_{2n}$ at $N=30$ seen in Fig. 6 (a). Two of the HFB states ($N=28$ and
32) used to computed the inertia at $N=30$ are in the normal states
corresponding to the $f_{7/2}$ and $p_{3/2}$ shell closures, and the inertia
at $N=30$ does not correspond to the pairing indicator, as the expression of
the inertia based on the double binding-energy differences assumes that the
three nearby isotopes have similar pairing structures, and this assumption
does not hold in this case. We can see a difference in the moment of inertia
in the neutron-rich region at $N=36$ and 38 for the singlet-pair and triplet-
pair EDF. The binding energies in the neutron-rich isotopes may determine the
coupling constant of the triplet-pair EDF.

Figure 6:  Two-neutron separation energies and moments of inertia of the
neutron pairing rotation in the Ca isotopes calculated for the singlet- and
triplet-pair EDFs. The inertia calculated with the triplet-pair EDF at $N=30$
($-17.86$ MeV-1) is not shown in the figure.

##  IV Summary

We analyzed the spin-triplet pair condensation of like particles in singly
closed nuclei. The relevant quantity to the spin-triplet pair condensation is
the nine-component spin-current pair density. One component can be finite in
the HFB calculation within the spherical symmetry. We have demonstrated that
the spin-singlet and spin-triplet pairing condensates coexist in open-shell
nuclei, and one component of the pair EDF can induce the other component. The
spin-orbit splitting is shown to play an essential role in the coexistence of
the two types of pair condensates, because the spin is no longer a good
quantum number.

The inclusion of both the spin-singlet and spin-triplet pair EDFs into the
nuclear EDF will enable us more detailed description of the nuclear pairing
condensation, including the isotope and isotone dependence, and deepen the
understanding of the origin of the spin-orbit splitting and the role of the
tensor force in the pairing channel. In describing open-shell nuclei with
deformation, the pseudoscalar and pseudotensor components should be
considered. In subsequent works, we will present such developments.

## Acknowledgments

The authors express their sincere gratitude to T. Naito, W. Nazarewicz, H.
Tajima, and Y. Yanase for their conscientious review of the manuscript and
invaluable insights. Additionally, the authors acknowledge the invaluable
contributions of members of the PHANES Collaboration (M. Dozono, M. Matsuo, S.
Ota, and S. Shimoura) for their insightful discussions. This work was
supported by the JSPS KAKENHI (Grants No. JP19K03824, No. JP19K03872, No.
JP19KK0343, and No. JP20K03964).

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