# Coexistence of isospin $I=0$ and $I=1$ pairings in asymmetric nuclear matter

Y. -J. Yan  Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou
730000, China  School of Nuclear Science and Technology, University of Chinese
Academy of Sciences, Beijing 100049, China  CAS Key Laboratory of High
Precision Nuclear Spectroscopy, Institute of Modern Physics, Chinese Academy
of Sciences, Lanzhou 730000, China  X. -L. Shang<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China  School of Nuclear Science and Technology, University of Chinese Academy
of Sciences, Beijing 100049, China  CAS Key Laboratory of High Precision
Nuclear Spectroscopy, Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  J.-M. Dong  Institute of Modern Physics,
Chinese Academy of Sciences, Lanzhou 730000, China  School of Nuclear Science
and Technology, University of Chinese Academy of Sciences, Beijing 100049,
China  CAS Key Laboratory of High Precision Nuclear Spectroscopy, Institute of
Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China  W. Zuo
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China  School of Nuclear Science and Technology, University of Chinese Academy
of Sciences, Beijing 100049, China  CAS Key Laboratory of High Precision
Nuclear Spectroscopy, Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China

###### Abstract

The coexistence of neutron-neutron (n-n), proton-proton (p-p), and neutron-
proton (n-p) pairings is investigated by adopting an effective density-
dependent contact pairing potential. These three types of pairings can coexist
only if the n-p pairing is stronger than the n-n and p-p pairings for isospin
asymmetric nuclear matter. In addition, the existence of n-n and p-p pairs
might enhance n-p pairings in asymmetric nuclear matter when the n-p pairing
strength is significantly stronger than the n-n and p-p ones. Conversely, the
n-p pairing is reduced by the n-n and p-p pairs when the n-p pairing
interaction approaches n-n and p-p pairings.

###### pacs:

21.60.De, 21.45.Ff, 21.65.Cd, 21.30.Fe

##  I INTRODUCTION

The importance of pairing correlation in nuclear systems was realized very
early bohr . In finite nuclei, neutron-neutron (n-n) and proton-proton (p-p)
pairing effects are realized in several nuclear properties such as
deformation, moments of inertia, alignments, and mass systematics nnimp1 ;
nnimp2 ; nnimp3 . In extended systems, nuclear pairing is expected to occur in
the dense matter inside the neutron stars ns1 ; ns2 . This pairing is crucial
for understanding various phenomena in neutron star physics, from the cooling
of new born stars ns3 ; ns4  to the afterburst relaxation in X-ray transients
ns5 , as well as in the understanding of glitches ns6 . However, insufficient
attention is paid to the isospin-singlet pairing, i.e., the neutron-proton
(n-p) paring. Recently, it was suggested that the isospin-singlet pairing is
possibly crucial in understanding of some nuclear structural issues, such as
the Gamow-Teller transition npimp1 ; npimp2 . In addition, considering the
spin and isospin degrees of freedom, the nuclear Cooper pairs contain very
interesting inner structures huang .

It is well-known that pair correlations crucially depend on the pairing near
the Fermi surface. Because neutrons and protons share the same Fermi energy in
symmetric nuclear matter, n-n (p-p) pairs compete intensely with n-p pairs .
Generally, the most energetically favored excludes the others. The
investigations on nuclear pairs almost focus on a single pairing structure,
i.e., either the n-n (p-p) or n-p pair only shen ; nn1 ; nn2 ; sh2 ; sdbaldo ;
np1 ; np2 ; np3 ; np4 ; sh3 . Nevertheless, coexistence may emerge in special
cases such as in the case of isospin asymmetric nuclear matter. In a neutron-
rich system, although the isospin-singlet n-p pairing may be favored, the
excess neutrons can as well form isospin-triplet n-n pairs coexisting with the
other, and they can influence each other. Furthermore, the nuclei far from the
beta-stability line, i.e. the exotic nuclei, can be obtained from heavy-ion
collisions (HIC), which has been addressed as a laboratory for the dynamic
evolution of the superfluid state of nuclear matter lab1 . New aspects of
pairing could appear in these exotic nuclei with regard to isospin
asymmetries, one of which might be the interplay between n-n and n-p pairings
in the nuclei owing to the significant overlap of proton and neutron orbits
huang ; lab2 .

In Ref. huang , the coexistence of isospin $I=1$ and $I=0$ pairing are
considered to study the inner phase structure and phase transition at low
density where the BCS-BEC crossover occurs. The result obtained indicates that
including the $I=1$ channel pairing significantly alters the phase structure
and phase transition properties. In nuclear matter, another concern is the
interplay between the $I=1$ and $I=0$ pairings. Based on this motivation, to
investigate the coexistence of the n-n, p-p, and n-p pairing in asymmetric
nuclear matter with effective contact pairing interaction in this study, we
employ the extend Nambu-Gorkov propagator, which includes the isospin triplet
n-n and p-p pairings and the isospin singlet n-p pairing.

The paper is organized as follows: In Sec. II, we briefly derive the gap
equation and thermodynamics, as well as introduce the adopted effective
pairing interaction. The numerical results and discussion are presented in
Sec. III, where the results of the coexistence of three types of pairings are
compared with the single pairing at certain density. Finally, a summary and a
conclusion are given in Sec. IV.

##  II Formalism

The Nambu-Gorkov propagator at finite temperatures, including the n-n, n-p,
and p-p pairings huang , is expressed as:

$\displaystyle
G=\left(\begin{array}[]{llll}i\omega_{\upsilon}-\varepsilon_{n}&\ \ \ \ 0&\ \
\Delta_{np}&\ \ \Delta_{nn}\\\ \\\ \ \ \ \
0&i\omega_{\upsilon}-\varepsilon_{p}&\ \ \Delta_{pp}&-\Delta_{np}\\\ \\\ \ \
\Delta_{np}&\ \ \Delta_{pp}&i\omega_{\upsilon}+\varepsilon_{p}&\ \ \ \ 0\\\
\\\ \ \ \Delta_{nn}&-\Delta_{np}&\ \ \ \
0&i\omega_{\upsilon}+\varepsilon_{n}\end{array}\right)^{-1},$ (8)

where $\omega_{\upsilon}=(2\upsilon+1)\pi k_{B}T$ with $\upsilon\in\mathbb{Z}$
represents the Matsubara frequencies.
$\varepsilon_{n/p}=\textbf{p}^{2}/(2m)-\mu_{n/p}$ is the single particle
(s.p.) energy with chemical potential $\mu_{n/p}$. In addition, $\Delta_{nn}$,
$\Delta_{pp}$, and $\Delta_{np}$ are the isospin-triplet n-n, isospin-triplet
p-p, and isospin-singlet n-p pairing gaps, respectively.

###  II.1 Gap equations

The neutron-proton anomalous propagator, which corresponds to $G_{13}$, reads

$\displaystyle F_{np}^{\dagger}(\omega_{\upsilon},\textbf{p})$
$\displaystyle=$
$\displaystyle\frac{-\Delta_{np}[(i\omega_{\upsilon})^{2}+i\omega_{\upsilon}(\varepsilon_{n}-\varepsilon_{p})-\varepsilon_{n}\varepsilon_{p}-\Delta_{np}^{2}-\Delta_{nn}\Delta_{pp}]}{\big{[}(i\omega_{\nu})^{2}-E_{+}^{2}\big{]}\big{[}(i\omega_{\nu})^{2}-E_{-}^{2}\big{]}}$
$\displaystyle=$
$\displaystyle\frac{-\Delta_{np}\big{\\{}[(i\omega_{\upsilon})^{2}-\varepsilon_{+}^{2}]-i\omega_{\upsilon}(2\delta\mu)+2\delta\mu^{2}+\frac{(\Delta_{nn}-\Delta_{pp})^{2}}{2}\big{\\}}}{\big{[}(i\omega_{\nu})^{2}-E_{+}^{2}\big{]}\big{[}(i\omega_{\nu})^{2}-E_{-}^{2}\big{]}},$

where
$E_{\pm}=\sqrt{\varepsilon_{+}^{2}\pm\sqrt{\varepsilon_{-}^{4}+\varepsilon_{\Delta}^{4}}}$
is the quasi-particle energy in the condensate with the definition
$\varepsilon_{\Delta}^{4}=\Delta_{np}^{2}[(\varepsilon_{n}-\varepsilon_{p})^{2}+(\Delta_{nn}-\Delta_{pp})^{2}]$
and
$2\varepsilon_{\pm}^{2}=\varepsilon_{n}^{2}+\Delta_{nn}^{2}+\Delta_{np}^{2}\pm(\varepsilon_{p}^{2}+\Delta_{pp}^{2}+\Delta_{np}^{2})$.
$\delta\mu=(\varepsilon_{p}-\varepsilon_{n})/2=(\mu_{n}-\mu_{p})/2$ represents
the Fermi surface mismatch. The summation over the Matsubara frequencies
provides the density matrix of particles in the condensate, i.e, the n-p
pairing probabilities,

$\displaystyle\nu_{np}(\textbf{p})$ $\displaystyle=$
$\displaystyle-\frac{\Delta_{np}}{2}\Big{\\{}\big{[}\frac{1-2f(E_{+})}{2E_{+}}+\frac{1-2f(E_{-})}{2E_{-}}\big{]}$
$\displaystyle+$
$\displaystyle\frac{2\delta\mu^{2}+\frac{(\Delta_{nn}-\Delta_{pp})^{2}}{2}}{\sqrt{\varepsilon_{-}^{4}+\varepsilon_{\Delta}^{4}}}\big{[}\frac{1-2f(E_{+})}{2E_{+}}-\frac{1-2f(E_{-})}{2E_{-}}\big{]}\Big{\\}}.$

Here $f(E)=[1+\exp(\frac{E}{k_{B}T})]^{-1}$ is the well-known Fermi-Dirac
distribution function under a temperature $T$. Accordingly, the n-p gap
equation is expressed as

$\displaystyle\Delta_{np}$ $\displaystyle=$
$\displaystyle\int\frac{d\textbf{p}}{(2\pi)^{3}}V_{np}\frac{\Delta_{np}}{2}\Big{\\{}\big{[}\frac{1-2f(E_{+})}{2E_{+}}+\frac{1-2f(E_{-})}{2E_{-}}\big{]}$
$\displaystyle+$
$\displaystyle\frac{2\delta\mu^{2}+\frac{(\Delta_{nn}-\Delta_{pp})^{2}}{2}}{\sqrt{\varepsilon_{-}^{4}+\varepsilon_{\Delta}^{4}}}\big{[}\frac{1-2f(E_{+})}{2E_{+}}-\frac{1-2f(E_{-})}{2E_{-}}\big{]}\Big{\\}}.$

In the absence of the n-n and p-p pairings, the quasi-particle energy
$E_{\pm}$ becomes
$E_{\pm}=\sqrt{[(\varepsilon_{n}+\varepsilon_{p})/2]^{2}+\Delta_{np}^{2}}\pm\delta\mu=E_{\Delta}\pm\delta\mu$,
and the gap equation is reduced to a more familiar form for the n-p pairing in
asymmetric nuclear matter:

$\displaystyle\Delta_{np}$ $\displaystyle=$
$\displaystyle\int\frac{d\textbf{p}}{(2\pi)^{3}}V_{np}\frac{\Delta_{np}[1-f(E_{+})-f(E_{-})]}{2E_{\Delta}}.$

Similarly, the n-n and p-p pairing gaps are respectively expressed as

$\displaystyle\Delta_{nn}$ $\displaystyle=$
$\displaystyle\int\frac{d\textbf{p}}{(2\pi)^{3}}V_{nn}\frac{\Delta_{nn}}{2}\Big{\\{}\big{[}\frac{1-2f(E_{+})}{2E_{+}}+\frac{1-2f(E_{-})}{2E_{-}}\big{]}$
$\displaystyle+$
$\displaystyle\frac{\varepsilon_{-}^{2}+\Delta_{np}^{2}(1-\frac{\Delta_{pp}}{\Delta_{nn}})}{\sqrt{\varepsilon_{-}^{4}+\varepsilon_{\Delta}^{4}}}\big{[}\frac{1-2f(E_{+})}{2E_{+}}-\frac{1-2f(E_{-})}{2E_{-}}\big{]}\Big{\\}},$

and

$\displaystyle\Delta_{pp}$ $\displaystyle=$
$\displaystyle\int\frac{d\textbf{p}}{(2\pi)^{3}}V_{pp}\frac{\Delta_{pp}}{2}\Big{\\{}\big{[}\frac{1-2f(E_{+})}{2E_{+}}+\frac{1-2f(E_{-})}{2E_{-}}\big{]}$
$\displaystyle-$
$\displaystyle\frac{\varepsilon_{-}^{2}+\Delta_{np}^{2}(\frac{\Delta_{nn}}{\Delta_{pp}}-1)}{\sqrt{\varepsilon_{-}^{4}+\varepsilon_{\Delta}^{4}}}\big{[}\frac{1-2f(E_{+})}{2E_{+}}-\frac{1-2f(E_{-})}{2E_{-}}\big{]}\Big{\\}},$

The occupation numbers, corresponding to the matrix elements $G_{11}$ and
$G_{22}$, can be calculated by

$\displaystyle n_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{\varepsilon_{n}}{2}\big{[}\frac{1-2f(E_{+})}{2E_{+}}+\frac{1-2f(E_{-})}{2E_{-}}\big{]}$
$\displaystyle-$
$\displaystyle\frac{\varepsilon_{-}^{2}\varepsilon_{n}-2\delta\mu\Delta_{np}^{2}}{2\sqrt{\varepsilon_{-}^{4}+\varepsilon_{\Delta}^{4}}}\big{[}\frac{1-2f(E_{+})}{2E_{+}}-\frac{1-2f(E_{-})}{2E_{-}}\big{]}$

and

$\displaystyle n_{p}$ $\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{\varepsilon_{p}}{2}\big{[}\frac{1-2f(E_{+})}{2E_{+}}+\frac{1-2f(E_{-})}{2E_{-}}\big{]}$
$\displaystyle+$
$\displaystyle\frac{\varepsilon_{-}^{2}\varepsilon_{p}-2\delta\mu\Delta_{np}^{2}}{2\sqrt{\varepsilon_{-}^{4}+\varepsilon_{\Delta}^{4}}}\big{[}\frac{1-2f(E_{+})}{2E_{+}}-\frac{1-2f(E_{-})}{2E_{-}}\big{]}$

The neutron and proton densities are respectively defined as

$\displaystyle\rho_{n}=2\int\frac{d\textbf{p}}{(2\pi)^{3}}n_{n},\ \
\rho_{p}=2\int\frac{d\textbf{p}}{(2\pi)^{3}}n_{p}.$ (17)

Notably, the n-n, p-p, and n-p pairing gaps couple to each other. For
asymmetric nuclear matter at the fixed neutron and proton densities, these gap
equations (4), (6), and (7) should be solved self-consistently with the
densities (10) at give densities and temperatures.

###  II.2 Pairing interaction

In principle, the nucleon-nucleon pairing interaction in nuclear matter
originates from the attractive component of the bare two-body potential and
the three-body force, and this pairing interaction is modified by the nuclear
medium, such as the polarization effect sup2 ; sup3 ; sup4 ; sup5 ; scr1 ;
scr2 ; ulbd . In this research, to obtain qualitative conclusions from the
coexistence of n-n, p-p, and n-p pairs, we adopt the density-dependent contact
interaction developed by Gorrido et al. ddci  to model the pairing potential.
For uniform nuclear matter, the potential takes the form

$\displaystyle
V_{I}(\textbf{r},\textbf{r}^{\prime})=g_{I}\delta(\textbf{r}-\textbf{r}^{\prime}),$
(18)

with the effective coupling constant

$\displaystyle g_{I}=v_{I}[1-\eta_{I}(\rho_{I}/\rho_{0})^{\gamma_{I}}].$ (19)

Here, $v_{I}$, $\eta_{I}$, and $\gamma_{I}$ are adjustable parameters and
$I=0,1$ denote the total isospin of the pairs. For the n-n (p-p) pairing,
$\rho_{I}=\rho_{n}$ ($\rho_{I}=\rho_{p}$) and for the n-p pairing,
$\rho_{I}=\rho_{n}+\rho_{p}$. $\rho_{0}=0.17\text{fm}^{-3}$ represents the
saturation density. Taking suitable values of the parameters, the pairing gap
$\Delta(k_{F})$ can be reproduced as a function of the Fermi momentum
$k_{F}=(3\pi^{2}\rho_{I})^{1/3}$ in the channel $L=0$, $I=1$, $S=0$ (n-n and
p-p) and $k_{F}=(3\pi^{2}\rho_{I}/2)^{1/3}$ in channel $L=0$, $I=0$, $S=1$
(n-p). We would like to emphasize that there is also a kind of n-p pairing in
the channel $L=0$, $I=1$, $S=0$ for the symmetric nuclear matter. In this
channel, the n-p pairing force is approximately the same as the n-n or p-p
pairing force. As will be discussed in Sec. III, even a minor asymmetry will
destroy the n-p pairing in this channel. Therefore, the $I=1$ pairings only
represent neutron-neutron and proton-proton pairings hereafter.

Figure 1:  The density-dependent contact pairing interaction with parameters
calibrated to the calculated pairing gaps. The dots represent the pairing gaps
in Ref. shen ; sdbaldo , whereas the lines correspond to the calculation from
the effective pairing interaction. The left (right) panel is relate to the
isospin triplet (singlet) channel.

In addition to the polarization effect, the self-energy effect of the medium
quenches the pairing gaps shen ; sh2 . Because the self-energy effect for
nuclear pairing remais an open question in asymmetric nuclear matter, we adopt
the calculated pairing gaps shen ; sdbaldo  under the Hartree-Fock approaches
to calibrate the parameters presented in Fig.1. It should be noted that the
self-energy sh2  and polarization ulbd  effects should be included to obtain a
more reliable pairing interaction. As is well known that, to avoid the
ultraviolet divergence, an energy cut is required for the contact interaction.
Here, we fix the energy at approximately $80$ MeV for both cases. The left
(right) panel corresponds to the $I=1$ ($I=0$) pairings.

###  II.3 Thermodynamics

Now, we are in a position to determine the key thermodynamic quantities.
Because the occupation of the quasi-particle states is given by the Fermi-
Dirac distribution function, the entropy of the system is obtained from

$\displaystyle
S=-2k_{B}\sum_{\textbf{p}}\sum_{i}\big{[}f(E_{i})lnf(E_{i})+\overline{f}(E_{i})ln\overline{f}(E_{i})\big{]},$
(20)

where $\overline{f}(E_{i})=1-f(E_{i})$ and $i=\pm$. The internal energy of the
superfluid state is expressed as

$\displaystyle
U=2\sum_{\textbf{p}}\big{[}\varepsilon_{n}n^{n}+\varepsilon_{p}n^{p}\big{]}+\sum_{\textbf{p}}\big{[}g_{nn}\nu_{nn}^{2}+g_{pp}\nu_{pp}^{2}+2g_{np}\nu_{np}^{2}\big{]},$

The factor $2$ corresponds to the spin summation. The first term of Eq. (14)
includes the kinetic energy of the quasi-particle, as a function of the
pairing gap and chemical potential. The BCS mean-field interaction among the
particles in the condensate is embodied in the second term of Eq. (14). It
should be noted that for asymmetric nuclear matter, the n-n and p-p pairing
interactions can be different, i.e., $g_{nn}\neq g_{pp}$, owing to
$\rho_{n}\neq\rho_{p}$. Accordingly, the thermodynamic potential can be given
as

$\displaystyle\Omega=U-TS.$ (22)

Once the contact pairing interaction is adopted, the pairing gap is momentum
independent. Therefore, the thermodynamic potential can be obtained in a
simple form:

$\displaystyle\Omega$ $\displaystyle=$ $\displaystyle
2\frac{\Delta_{np}^{2}}{g_{np}}+\frac{\Delta_{nn}^{2}}{g_{nn}}+\frac{\Delta_{pp}^{2}}{g_{pp}}+\int\frac{d\textbf{p}}{(2\pi)^{3}}$
(23) $\displaystyle\times$
$\displaystyle\Big{\\{}\varepsilon_{n}+\varepsilon_{p}-\sum_{i=\pm}\big{[}E_{i}+2k_{B}Tln(1+e^{\frac{-E_{i}}{k_{B}T}})\big{]}\Big{\\}}.$

Her, We Consider the property
$f(\omega)lnf(\omega)+\overline{f}(\omega)ln\overline{f}(\omega)=-\frac{\omega}{k_{B}T}-ln(1+e^{-\omega/(k_{B}T)})$.
The gap equations (4), (6), and (7) and the densities of Eq. (10) can be
equivalently expressed as

$\displaystyle\frac{\partial\Omega}{\partial\Delta_{np}}$ $\displaystyle=$
$\displaystyle 0,\ \ \frac{\partial\Omega}{\partial\Delta_{nn}}=0,\ \
\frac{\partial\Omega}{\partial\Delta_{pp}}=0,$ $\displaystyle\rho_{n}$
$\displaystyle=$ $\displaystyle-\frac{\partial\Omega}{\partial\mu_{n}},\ \ \ \
\rho_{p}=-\frac{\partial\Omega}{\partial\mu_{p}}.$ (24)

It should be noted that the solution of these equations corresponds to the
global minimum of the free energy $F=\Omega+\mu_{n}\rho_{n}+\mu_{p}\rho_{p}$,
which is the essential quantity that describes the thermodynamics of
asymmetric nuclear matter.

##  III RESULTS AND DISCUSSION

The numerical calculations in this study focus on the coexistence of three
different types of pairs in isospin asymmetric nuclear matter with total
density $\rho=\rho_{n}+\rho_{p}$ and isospin asymmetry
$\beta=(\rho_{n}-\rho_{p})/\rho$. We adopt the effective contact pairing
interaction at zero temperature. Fig.2 illustrates the pairing gaps as a
function of asymmetry $\beta$ at the total density $\rho=0.068\text{fm}^{-3}$,
at which both the $I=1$ and $I=0$ pairing interactions are most attractive.
The thick lines correspond to the results of the coexistence of three types of
pairings, which include $\Delta_{nn}\neq 0,\Delta_{np}\neq 0,\Delta_{pp}\neq
0$. In the symmetric matter, neutrons and protons share the same Fermi
surface, i.e., $k_{Fn}=k_{Fp}=k_{F}$, and the region near the Fermi surface
contributes dominantly to the pairing gaps. Two neutrons and two protons near
the Fermi surface can form a n-n pair and a p-p pair or two n-p pairs. Because
the n-p pairing strength is significantly stronger than that of n-n and p-p,
the nucleons prefer to form n-p pair instead of n-n (p-p) pair. Equivalently,
the n-p pairings severely suppress the n-n and p-p pairings for $\beta=0$. As
illustraed in Fig.2, the n-n (p-p) gap disappears in symmetric case. In
asymmetric nuclear matter, the dominant region, which contributes
significantly to the n-n (p-p) pairing gap, is located at the neutron (proton)
Fermi momentum $k_{Fn}$ ($k_{Fp}$), whereas the region for n-p pairing is
between $k_{Fp}$ and $k_{Fn}$ (the average Fermi surface related to the
average chemical potential of neutrons and protons). The split between neutron
and proton Fermi surfaces separates the dominant regions for n-n, p-p, and n-p
pairings, which enables the n-n and p-p pairing. And this discrepancy between
$k_{Fp}$ and $k_{Fn}$ increases with the increasing isospin asymmetry.
Therefore the n-n and p-p pairing gaps increase with $\beta$.

Figure 2:  (Color online) The n-n, p-p, n-p pairing gaps as a function of the
isospin asymmetry $\beta$, at the total density $\rho=0.068\text{fm}^{-3}$.
The thick and thin lines correspond to the coexistence of three types of
pairings and single pairings, respectively. The dashed, short-dashed, and
solid lines are related to the n-n, p-p, and n-p pairings, respectively.

In addition, the results for single pairing, i.e., $\Delta_{nn}\neq
0,\Delta_{pp}=\Delta_{np}=0$, $\Delta_{np}\neq 0,\Delta_{nn}=\Delta_{pp}=0,$,
or $\Delta_{pp}\neq 0,\Delta_{nn}=\Delta_{np}=0,$ are depicted as thin lines
in Fig. 2 for comparison. Owing to the suppression from the mismatched Fermi
surfaces, n-p pairing gaps decrease with $\beta$ and disappear at certain
asymmetries for both the single pairing and the coexistence of three types of
pairings. In the calculation of the coexistence of three types of pairings,
$\Delta_{nn}$ and $\Delta_{pp}$ coincide with the results obtained from the
single pairing calculation when the n-p pairing vanishes. In fact, if
$\Delta_{np}=0$ the coupled equations (17) degenerates into two groups of
completely independent equations, which are the gap equation for $\Delta_{nn}$
with the neutron density and the gap equation for $\Delta_{pp}$ with proton
density.

Compared to single pairing, the critical isospin asymmetry, where
$\Delta_{np}$ vanishes, is enhanced by the existence of n-n and p-p pairs, as
demonstrated in Fig. 2. Unfortunately, this conclusion cannot be considered as
definite, as the effective pairing interaction is simply obtained from the
pairing gaps under the Hartree-Fock approximation. In addition, the effective
n-p pairing interaction can be significantly reduced by the nucleon-nucleon
correlation beyond the Hartree-Fock approaches sh2 . Owing to the complexity
of the nuclear many-body medium effects, the exact effective pairing
interaction remais an open problem. To eliminate the uncertainty of the
effective pairing strength, we adjust the effective neutron-proton pairing
interaction artificially to obtain the qualitative conclusion. The results
obtained are presented in Fig. 3. The solid and dashed lines correspond to the
results obtained from the coexistence of the three types of pairings and the
single pairing, respectively. For the effective interaction obtained from Ref.
sdbaldo , $g_{np}/g_{nn}=1.3837$. If we reduce the n-p pairing strength
$g_{np}$, the enhancement of the n-p pairing from the existence of the n-n
(p-p) pairs is reduced. When $g_{np}/g_{nn}$ is under a certain value, the
existence of n-n (p-p) pairing might suppress the n-p pairing eventually. An
interesting property is that if $g_{np}\simeq g_{nn}$, $\Delta_{np}$ decreases
rapidly with $\beta$. As mentioned in Sec. II (B), the channel $L=0$, $I=1$,
$S=0$ embodies n-n, p-p, and n-p pairings, and the pairing interactions are
approximately the same for the asymmetric case. A negligible asymmetry can
destroy the n-p pairing in the $L=0$, $I=1$, $S=0$ channel. Therefore, in
general, the $I=1$ pairing solely refers to the n-n and p-p pairings.

Figure 3:  The n-p pairing gaps as a function of isospin asymmetry at total
density $\rho=0.068\text{fm}^{-3}$ for different n-p pairing strengths,
$g_{np}/g_{nn}=1.3837$, $1.2$, $1.12$, $1.01$. The solid and dashed lines
correspond to the coexistence of three types of pairings and single pairing,
respectively.

One straightforward way to understand the enhancement of n-p pairing from the
existing n-n and p-p pairs is to investigate the n-p pairing probabilities
near the average Fermi surface (related to the average chemical potentials of
the neutron and proton). The results obtained are depicted in Fig. 4, in the
case where total density $\rho=0.068\text{fm}^{-3}$ and isospin asymmetry
$\beta=0.3$. The n-p pairing strength is set to be $g_{np}/g_{nn}=1.3837$. For
the single n-p pairing, the pairing is forbidden in a window around the
average Fermi surface owing to the absence of protons. Once the n-n and p-p
pairings are included, the dispersion of neutron and proton Fermi surfaces can
provide the kinematical phase space near the average Fermi surface for the
occurence of the n-p pairing phenomena. This is a positive mechanism, such
that the existence of n-n and p-p pairs enhances the n-p pairing.

Figure 4:  The n-p pairing probabilities as a function of $k$ near the average
Fermi surface with the total density $\rho=0.068\text{fm}^{-3}$ and isospin
asymmetry $\beta=0.3$. Here, $k=p/\hbar$ is the wave number. The pairing
strength is set to be $g_{np}/g_{nn}=1.3837$. The solid and dashed lines
correspond to the coexistence of three types of pairings and single pairing,
respectively.

Another effect of the existence of n-n and p-p pairs is that a n-n pair and a
p-p pair ought to be broken up to form two n-p pairs. Exclusively, when the
pairing energy of n-n and p-p pairs is smaller than that of two n-p pairs, the
existence of n-n and p-p pairs can enhance the n-p pairing. The pairing energy
is related to the pairing strength directly. As presented in Fig. 5, when the
n-p pairing strength is insufficient, the n-p pairing probability is
suppressed significantly by n-n and p-p pairs.

Figure 5: The pairing probabilities vs $k$ near the average Fermi surface at
the total density $\rho=0.068\text{fm}^{-3}$ with isospin asymmetry
$\beta=0.15$. Here $k=p/\hbar$ is the wave number. The dashed, short-dashed,
and solid lines correspond to the n-n, pp, and n-p pairings, respectively. The
n-p pairing strength $g_{np}/g_{nn}$ is set to be $1.3837$ ($1.12$) in left
(right) panel.

In the calculations of this study, the temperature is set to be zero. However,
for asymmetric nuclear matter, the temperature can also disperse the neutron
and proton Fermi surfaces, which will eventually reduce the suppression of
Fermi surface mismatch at low temperature. At high temperature, the
temperature will destroy all types of pairings. Once the temperature is
included, the enhanced and reduced effects on n-p paring from the existence of
n-n and p-p pairings should be weakened.

In finite nuclei, the n-p pairing might be suppressed by the strong spin-orbit
splitting sp1 ; sp2 . However, in nuclei where the spin-splitting becomes
small, the coexistence of three types of pairings may occur. Understanding the
enhanced and reduced effects on n-p paring owing to the existence of n-n and
p-p pairings could be beneficial in elucidating the n-p pairing in $N\approx
Z$ nuclei. For asymmetric nuclei, the interplay between n-n and n-p pairings
might be the same as that in asymmetric nuclear matter.

##  IV SUMMARY

In this study, we investigated the coexistence of n-n, p-p, and n-p pairings
in isospin asymmetric nuclear matter with an effective density-dependent
contact pairing interaction. The three types of pairings cannot coexist in
symmetric nuclear matter, only n-p pairs can survive when the n-p pairing
strength is stronger than that of the n-n and p-p pairs, whereas the n-n and
p-p pairs are preferred if the n-n and p-p pairing interactions become strong.
Furthermore, n-n, p-p, and n-p pairs can coexist in isospin asymmetric nuclear
matter when the n-p pairing interaction is stronger than n-n and p-p pairs.

Compared to the single pairing calculation (gap equation with only one kind of
nucleon pair), the results indicate two effects of the existence of n-n and
p-p pairs. On the one hand, the existence of n-n and p-p pairs can disperse
the neutron and proton Fermi surfaces, which increase the phase-space overlap
between neutrons and protons and eventually enhance the n-p pairing near the
average Fermi surface. This positive mechanism can reduce the suppression
owing to the mismatched Fermi surface of neutrons and protons in the isospin
asymmetric nuclear matter. On the other hand, a n-n pair and a p-p pair should
be broken up to form two n-p pairs. In this process, the pairing interaction
plays a crucial role. The final results are determined by these two effects.
In isospin asymmetric nuclear matter, the existence of n-n and p-p pairs can
enhance the n-p pairing when the n-p pairing strength is significantly
stronger than that of n-n and p-p pairs. However, the existence of n-n and p-p
pairs would reduce the n-p pairing probability when the n-p pairing
interaction decreases in strength. Moreover, when the n-p pairing strength
becomes approximately that of n-n and p-p pairs, the n-p pairing rapidly
disappears with the isospin asymmetries.

In this paper, the gap solution is only thermodynamically stable. The Cooper
pair momentum should also be included in the future to avoid dynamic
instability ins1 ; ins2 . In addition, in future works, the pairing
interaction should be calibrated to the pairing gaps, including the
polarization correction and the correlation effect. As a prospect, this
interesting coexistence of the three types of pairings would should also be
applied to the studies on pairing correlations in finite nuclei.

## Acknowledgments

This work is supported by National Natural Science Foundation of China (No.
11975282, 11775276, 11435014, 11505241), the Strategic Priority Research
Program of Chinese Academy of Sciences, Grant No. XDB34000000, the Youth
Innovation Promotion Association of Chinese Academy of Sciences (Grant No.
Y2021414, Y201871).

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