# Effects of Finite Temperature and Pairing Correlations in Multi-$\Lambda$
Hypernuclei

Bahruz Suleymanli  Physics Department, Yildiz Technical University, 34220
Esenler, Istanbul, Turkey  Kutsal Bozkurt  Physics Department, Yildiz
Technical University, 34220 Esenler, Istanbul, Turkey  Université Paris-
Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  Elias Khan  Université Paris-
Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France  Institut Universitaire de
France (IUF)  Haşim Güven  Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405
Orsay, France  Physics Department, Yildiz Technical University, 34220 Esenler,
Istanbul, Turkey  Jérôme Margueron  Univ Lyon, Univ Claude Bernard Lyon 1,
CNRS/IN2P3, IP2I Lyon, UMR 5822, F-69622, Villeurbanne, France  International
Research Laboratory on Nuclear Physics and Astrophysics, Michigan State
University and CNRS, East Lansing, MI 48824, USA

###### Abstract

The influence of finite temperatures and pairing correlations on the ground
state properties of multi-$\Lambda$ Ca, Sn and Pb hypernuclei is explored
using finite temperature Hartree–Fock–Bogoliubov approach and contact pairing
interaction. A critical temperature is predicted and is in agreement with the
Bardeen-Cooper-Schrieffer relationship $k_{B}T_{C}^{\Lambda}\approx
0.5\Delta^{T=0}_{\Lambda}$, beyond which pairing correlations drop to zero.
Particle densities, $\Lambda$ single-particle energies, and nuclear radii are
weakly impacted by pairing as well as by finite temperatures. However, other
nuclear properties which are more sensitive to pairing correlations, such as
$\Lambda$ pairing gaps, condensation energies, and abnormal densities are also
more impacted by finite temperature, especially around the critical
temperature. Furthermore, calculations show the occurrence of the pairing re-
entrance effect in the ${}^{280}_{70\Lambda}$Pb hyperon-drip-line
hypernucleus. Our study provides insight into the thermal evolution of
$\Lambda$ pairing, i.e. the emergence and vanishing of pairing correlations in
multi-$\Lambda$ hypernuclei as a function of temperature.

##  I Introduction

Including the temperature effect in nuclear systems can have a large impact on
astrophysical processes [1]. Hot nuclear matter exists only temporary in
compact stars and in finite nuclei since thermal energy can be efficiently
evacuated by neutrino and gamma-ray emissions. The properties of hot matter
may however be different from the ones of cold systems such as neutron stars
(NSs) or finite nuclei and may influence time evolution. For instance, proto-
neutron stars (PNSs) are transient celestial objects for which temperature
impact dynamical evolution [2, 3, 4, 5]. During the first few seconds after
their formation by massive star core-collapse (supernovae) or by binary
neutron star mergers (kilonovae), matter can be heated to temperatures of the
order of a few $10^{11}$ K, which drops down to about $T\lesssim 10^{10}$ K in
less than an hour [6]. During this cooling, hypernuclear matter may temporary
exist in hot PNSs [7, 8], but the actual presence of hyperons in the inner
core of PNSs crucially depends on the hyperon-hyperon interaction [9].
Consequently, the interplay between the hyperon-hyperon interaction and
temperature plays an important role in the understanding of the cooling
properties of PNSs.

The uncertainties in the hyperon-hyperon interaction are however still too
large to allow accurate predictions about hyperon matter. These uncertainties
are largely due to the scarce amount of experimental data, since only a
limited number of double-$\Lambda$ hypernuclei, including
${}^{6}_{\Lambda\Lambda}$He and ${}^{11}_{\Lambda\Lambda}$Be [10, 11, 12] have
been produced and measured in laboratories. In the future, the production of
hyperons at FAIR and JPARC will provide new data which will be employed to
reduce the uncertainties in the actual modeling of hyperon matter [13, 14,
15].

Astrophysical data can also be employed to reduce the model uncertainties. For
instance the presence of hyperons in NSs may impact their cooling, by
influencing, neutrino emission processes. More precisely, two hyperon beta
decays, namely $\Sigma^{-}\rightarrow\Lambda+e^{-}+\bar{\nu}$ and
$\Lambda\rightarrow p+e^{-}+\bar{\nu}$, contributing to the direct URCA
process, have been predicted to be very efficient in the long time
($\gg$years) cooling of NSs [16]. If these processes occur, the surface
temperature would be lower than the observed ones [17, 18, 19]. Therefore, it
is necessary to reduce their efficiency, for instance, when the NS temperature
is below the critical temperature $T_{C}^{\Lambda}$ for $\Lambda$ pairing [20,
18, 21, 22]. If hyperons are present in NSs, the observed surface temperature
is a sign for the existence of pairing among hyperons: in this superfluid
state, the emissivity of strange decay reactions is reduced by a factor of
exp$\left(-\Delta/kT\right)$, where $\Delta$ represents the pairing gap [18].

In conclusion of these first remarks, the composition of dense matter in the
core of NSs and PNSs has observational consequences which can be related to
the properties of hyperon interactions. In addition, the impact of these
interactions on the $\Lambda$ pairing gap is very important for the cooling of
these stellar objects, and several theoretical predictions on $\Lambda$
pairing have been performed in baryonic matter [23, 24, 25, 26, 27, 28, 29,
30, 31] and hypernuclei [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43].

We briefly review these predictions. Based on a G-matrix, $\Lambda$ pairing
gap has been predicted to be about a few $100$ keV [23] for densities
$n_{\Lambda}\approx n_{\mathrm{sat}}$, where $n_{\mathrm{sat}}$ is the
saturation density of symmetric matter, $n_{\mathrm{sat}}\approx 0.155\pm
0.005$ fm-3 [24], and the BCS relation obtained in the weak-coupling
approximation $k_{B}T_{C}\approx 0.57\Delta^{T=0}$ has been found to be well
satisfied [25, 26, 27]. Moreover, superfluidity has been predicted to occur at
about $4n_{\mathrm{sat}}$ in beta-equilibrated matter, using one-boson-
exchange potentials for the $\Lambda\Lambda$ interaction [28, 29].
Phenomenological nuclear interactions have also been employed to predict the
properties of $\Lambda$ pairing in nuclear matter. For instance, a
relativistic mean field (RMF) model has predicted the pairing gap to be 810
keV at the baryon density of 0.349 fm-3 [30]. Other models consider
$\sigma^{*}$ and $\varphi$ mesons (with strange quark) added on top of the
other (without strange quark) mesons: the pairing gap is predicted to be
reduced down to 282 keV at the total baryon density of 0.364 fm-3 [31]. These
results illustrate the variability among different predictions.

One can also employ hypernuclei as a benchmark for $\Lambda\Lambda$
interactions. For instance, a study based on the relativistic Thomas-Fermi
approach and applied to hypernuclei, has shown an interesting effect of the
temperature: the $\Lambda$ radii increases faster as function of the
temperature than the nucleon radii, since it is easier to excite the small
amount of hyperons than the large number of nucleons [32]. Still in finite
nuclei, Skyrme-Hartree-Fock model [33, 34, 35], generalized liquid drop model
[36, 37], and beyond-mean-field SHF model [38, 39, 40], have been employed
within the BCS approximation. Microscopic calculations applied to
multi-$\Lambda$ hypernuclei have emphasized the connection between pairing in
$\Lambda$ channel and the $\Lambda\Lambda$ interaction [41, 42, 43]. Using
different methods to calibrate $\Lambda$ pairing strength, these studies have
predicted the properties of multi-$\Lambda$ hypernuclei in their ground-state.
However, there are no calculation in hypernuclei which explicitly treat
temperature and $\Lambda$ pairing together.

The main purpose of the present study is to understand the effect of
temperature and $\Lambda$ pairing in hypernuclei, in view of the future
production of a wealth of hypernuclei at FAIR and JPARC. It is also expected
to impact hyperon matter in astrophysics. In this work, we extend our previous
analysis of $\Lambda$ pairing in multistrange hypernuclei [41] by
incorporating the finite temperature effect. To do so, we implement the finite
temperature Hartree-Fock-Bogoliubov (FT-HFB) approach [44], where we consider
Skyrme-type energy density functionals in the nucleon channel. To account for
the $\Lambda$N interaction, we consider the density functionals DF-NSC89, DF-
NSC97a, and DF-NSC97f, adjusted to reproduce Brueckner Hartree-Fock (BHF)
results based on Nijmegen interactions NSC89, NSC97a and NSC97f [45, 46]. To
describe the $\Lambda\Lambda$ interaction, we use the empirical prescription
EmpC [47], which is adjusted to reproduce the binding energy of
${}^{6}_{\Lambda\Lambda}$He.

In practice, we consider spherical symmetry in this first study, aiming at
having a qualitative understanding of $\Lambda$ pairing at finite temperature.
We have therefore specifically chosen hypernuclei with closed proton and
neutron shells, such as
${}_{-S\Lambda}^{40-\mathrm{S}}\mathrm{Ca},{}_{-\mathrm{S}\Lambda}^{132-\mathrm{S}}\mathrm{Sn}$,
and ${}_{-\mathrm{S}\Lambda}^{208-\mathrm{S}}\mathrm{Pb}$, known for their
semi-magic properties. Moreover, the structure of hyperon-drip-line
hypernuclei is also examined, particularly those with open neutron shells
located near closed neutron shells, such as ${}^{56,58,62,64}_{20\Lambda}$Ca,
${}^{168,170,174,176}_{40\Lambda}$Sn, and
${}^{274,276,280,282}_{70\Lambda}$Pb. Our study focuses on investigating the
impact of the temperature on the low-energy properties of multi-$\Lambda$
hypernuclei.

The present manuscript is organized as follows. In Sec. II, we provide a brief
overview of the FT-HFB approach applied to multi-$\Lambda$ hypernuclei. In
Sec. III, we explore the interplay between temperature and $\Lambda$ pairing,
analyzing their impact on the low-energy properties. Conclusions are given in
Sec. IV.

##  II Finite-temperature Hartree–Fock–Bogoliubov Approach for Multi-$\Lambda$
Hypernuclei

The present model is based on the FT-HFB approach, which is described in
details in Ref. [44]. It is a non-relativistic model for nucleons
$\mathrm{N}=(p,n)$ and $\Lambda$ hyperons, which is employed to describe
multi-$\Lambda$ hypernuclei.

###  II.1 Formalism

Table 1: Parameters corresponding to the energy density and effective mass of the $\Lambda$ hyperons. Functional | $\alpha_{1}$ | $\alpha_{2}$ | $\alpha_{3}$ | $\alpha_{4}$ | $\alpha_{5}$ | $\alpha_{6}$ | $\alpha_{7}$ | $\alpha_{8}$ | $\alpha_{9}$ | $\mu_{1}$ | $\mu_{2}$ | $\mu_{3}$ | $\mu_{4}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
DF-NSC89 + EmpC | 327 | 1159 | 1163 | 335 | 1102 | 1660 | 22.81 | 0 | 0 | 1 | 1.83 | 5.33 | 6.07
DF-NSC97a + EmpC | 423 | 1899 | 3795 | 577 | 4017 | 11061 | 21.12 | 0 | 0 | 0.98 | 1.72 | 3.18 | 0
DF-NSC97f + EmpC | 384 | 1473 | 1933 | 635 | 1829 | 4100 | 33.25 | 0 | 0 | 0.93 | 2.19 | 3.89 | 0

The total Hamiltonian reads

$\widehat{H}=\sum_{i}\widehat{T}_{i}+\sum_{i,j}\widehat{H}^{\prime}_{ij},$ (1)

where $\widehat{T}$ represents the kinetic energy operators,
$\widehat{H}^{\prime}$ denotes the interaction operator with
$(i,j)=(\mathrm{N},\mathrm{N})$, $(\mathrm{N},\Lambda)$, and
$(\Lambda,\Lambda)$. The total energy can be expressed as

$E=\int\\!\\!d^{3}r\sum_{i,j}\epsilon_{ij},$ (2)

with the following contributions: the hyperon-nucleon energy-density

$\displaystyle\epsilon_{\mathrm{\Lambda N}}$ $\displaystyle=$
$\displaystyle-(\alpha_{1}-\alpha_{2}\rho_{\mathrm{N}}+\alpha_{3}\rho_{\mathrm{N}}^{2})\rho_{\Lambda}\rho_{\mathrm{N}}$
(3)
$\displaystyle+(\alpha_{4}-\alpha_{5}\rho_{\mathrm{N}}+\alpha_{6}\rho_{\mathrm{N}}^{2})\rho_{\Lambda}^{5/3}\rho_{\mathrm{N}},$

the hyperon-hyperon energy-density

$\displaystyle\epsilon_{\mathrm{\Lambda\Lambda}}=\frac{\hbar^{2}}{2m_{\Lambda}}\tau_{\Lambda}-(\alpha_{7}-\alpha_{8}\rho_{\Lambda}+\alpha_{9}\rho_{\Lambda}^{2})\rho_{\Lambda}^{2}\,,$
(4)

and the nucleon-nucleon energy-density, where $q$ stands for neutrons or
protons,

$\displaystyle\epsilon_{\mathrm{NN}}$ $\displaystyle=$
$\displaystyle\frac{\hbar^{2}}{2m_{N}}\tau_{\mathrm{N}}+\frac{1}{2}t_{0}\left[\left(1+\frac{x_{0}}{2}\right)\rho_{\mathrm{N}}^{2}-\left(x_{0}+\frac{1}{2}\right)\sum_{q}\rho_{q}^{2}\right]$
(5)
$\displaystyle+\frac{t_{1}}{4}\left\\{\left(1+\frac{x_{1}}{2}\right)\left[\rho_{\mathrm{N}}\tau_{\mathrm{N}}+\frac{3}{4}(\nabla\rho_{\mathrm{N}})^{2}\right]-\left(x_{1}+\frac{1}{2}\right)\right.$
$\displaystyle\left.\times\sum_{q}\left[\rho_{q}\tau_{q}+\frac{3}{4}\left(\nabla\rho_{q}\right)^{2}\right]\right\\}$
$\displaystyle+\frac{t_{2}}{4}\left\\{\left(1+\frac{x_{2}}{2}\right)\left[\rho_{\mathrm{N}}\tau_{\mathrm{N}}-\frac{1}{4}(\nabla\rho_{\mathrm{N}})^{2}\right]+\left(x_{2}+\frac{1}{2}\right)\right.$
$\displaystyle\left.\times\sum_{q}\left[\rho_{q}\tau_{q}-\frac{1}{4}\left(\nabla\rho_{q}\right)^{2}\right]\right\\}$
$\displaystyle-\frac{1}{16}\left(t_{1}x_{1}+t_{2}x_{2}\right)J_{\mathrm{N}}^{2}+\frac{1}{16}\left(t_{1}-t_{2}\right)\sum_{q}J_{q}^{2}$
$\displaystyle+\frac{1}{12}t_{3}\rho_{\mathrm{N}}^{\gamma}\left[\left(1+\frac{x_{3}}{2}\right)\rho_{\mathrm{N}}^{2}-\left(x_{3}+\frac{1}{2}\right)\sum_{q}\rho_{q}^{2}\right]$
$\displaystyle+\frac{1}{2}W_{0}\left(J_{\mathrm{N}}\nabla\rho_{\mathrm{N}}+\sum_{q}J_{q}\nabla\rho_{q}\right)+\epsilon_{\mathrm{Coul}}\,.$

The parameters $\alpha_{1-9}$ in Eqs. (3) and (4) are provided in Table 1 for
the following functionals: DF-NSC89, DF-NSC97a, and DF-NSC97f [45, 46] with
EmpC prescription for $\alpha_{7}$ [47]. The density $\rho_{\Lambda}$
represents the $\Lambda$ density, and the nucleon density is
$\rho_{\mathrm{N}}=\rho_{p}+\rho_{n}$. In infinite nuclear matter, the kinetic
energy densities $\tau_{\Lambda}$ and $\tau_{\mathrm{N}}$ are analytical
functions of the matter density:
$\tau_{i}=\frac{3}{5}(6\pi^{2}/g_{i})^{2/3}\rho_{i}^{5/3}$, where $g_{i}=4(2)$
for $i=\mathrm{N}(\Lambda)$. The parameters $t_{0-3}$, $x_{0-3}$, $\gamma$,
and $W_{0}$ are given by the Skyrme interaction and $J$ represents the spin-
current densities of nucleons. The term $\epsilon_{\mathrm{Coul}}$ corresponds
to the Coulomb energy density. It is worth noting that in Eqs. (3) and (4), we
employ energy density functionals, which have been adjusted to reproduce
Brueckner-Hartree-Fock predictions, incorporating both nucleons and $\Lambda$
hyperons [45, 46], as well as the bond energy of ${}^{6}_{\Lambda\Lambda}$He.

The HFB equations in coordinate representation are derived by taking the
variation of the energy functional expressed in Eq. (2) with respect to the
density matrices:

$\left(\begin{array}[]{cc}h-\lambda&\tilde{h}\\\
\tilde{h}&-h+\lambda\end{array}\right)\left(\begin{array}[]{l}U_{k}\\\
V_{k}\end{array}\right)\\\ =E_{k}\left(\begin{array}[]{c}U_{k}\\\
V_{k}\end{array}\right)$ (6)

where
$h\left(\mathbf{r}\sigma,\mathbf{r}^{\prime}\sigma^{\prime}\right)=\frac{\delta
E}{\delta\rho\left(\mathbf{r}\sigma,\mathbf{r}^{\prime}\sigma^{\prime}\right)}$
is the Hamiltonian containing the self-consistent field with particle density
$\rho(r,r^{\prime})=\frac{1}{4\pi}\sum_{k}(2j_{k}+1)V_{k}(r)V_{k}^{*}(r^{\prime})$
and the chemical potential $\lambda$, and
$\tilde{h}\left(\mathbf{r}\sigma,\mathbf{r}^{\prime}\sigma^{\prime}\right)=\frac{\delta
E}{\delta\tilde{\rho}\left(\mathbf{r}\sigma,\mathbf{r}^{\prime}\sigma^{\prime}\right)}$
is the pairing field with pairing density
$\tilde{\rho}(r,r^{\prime})=-\frac{1}{4\pi}\sum_{k}(2j_{k}+1)V_{k}(r)U_{k}^{*}(r^{\prime})$.
The fields $U_{k}$ and $V_{k}$ are the components of the radial HFB wave
function and $E_{k}$ is the quasiparticle energy. It should be noted that is
that the considered model incorporates two distinct types of HFB mean fields
potentials:

$V_{\Lambda}=\frac{\partial\left(\epsilon_{\Lambda\mathrm{N}}+\epsilon_{\Lambda\Lambda}\right)}{\partial\rho_{\Lambda}}-\left(\frac{m_{\Lambda}}{m_{\Lambda}^{*}\left(\rho_{\mathrm{N}}\right)}-1\right)\frac{\left(3\pi^{2}\right)^{2/3}}{2m_{\Lambda}}\rho_{\Lambda}^{2/3},$
(7)

and

$\displaystyle V_{N}$ $\displaystyle=$
$\displaystyle\frac{\partial\left(\epsilon_{\Lambda\mathrm{N}}+\epsilon_{\mathrm{NN}}\right)}{\partial\rho_{\mathrm{N}}}+\frac{\partial}{\partial\rho_{\mathrm{N}}}\left(\frac{m_{\Lambda}}{m_{\Lambda}^{*}\left(\rho_{\mathrm{N}}\right)}\right)$
(8)
$\displaystyle\times\left(\frac{\tau_{\Lambda}}{2m_{\Lambda}}-\frac{3}{5}\frac{\left(3\pi^{2}\right)^{2/3}}{2m_{\Lambda}}\rho_{\Lambda}^{5/3}\right)\,.$

The $\Lambda$ effective mass reads

$\frac{m_{\Lambda}^{*}\left(\rho_{\mathrm{N}}\right)}{m_{\Lambda}}=\mu_{1}-\mu_{2}\rho_{\mathrm{N}}+\mu_{3}\rho_{\mathrm{N}}^{2}-\mu_{4}\rho_{\mathrm{N}}^{3},$
(9)

where the parameters $\mu_{1-4}$ are given in Table 1. The $\Lambda\Lambda$
pairing interaction is defined as a volume-type contact interaction, similar
to the one used in Ref. [41]:

$V_{\Lambda_{\text{pair}}}=V_{\Lambda_{0}}\delta(\bm{r}_{1}-\bm{r}_{2})\,,$
(10)

where $V_{\Lambda_{0}}$ is the $\Lambda\Lambda$ pairing interaction strength.

All equations presented so far, including Eq. (6), are identical at finite
temperature [44, 48] provided the densities are generalised at finite
temperature according to the following expressions:

$\displaystyle\rho_{T}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\sum_{k}\left(2j_{k}+1\right)$ (11)
$\displaystyle\times\left[V_{k}^{*}(r)V_{k}(r)\left(1-f_{k}\right)+U_{k}^{*}(r)U_{k}(r)f_{k}\right]\,,$
$\displaystyle\tau_{T}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\sum_{k}\left(2j_{k}+1\right)$ (12)
$\displaystyle\times\left\\{\left[\left(\frac{dV_{k}}{dr}-\frac{V_{k}}{r}\right)^{2}+\frac{l_{k}\left(l_{k}+1\right)}{r^{2}}V_{k}^{2}\right]\left(1-f_{k}\right)\right.$
$\displaystyle+{\left.\left[\left(\frac{dU_{k}}{dr}-\frac{U_{k}}{r}\right)^{2}+\frac{l_{k}\left(l_{k}+1\right)}{r^{2}}U_{k}^{2}\right]f_{k}\right\\}}\,,$
$\displaystyle J_{T}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\sum_{k}\left(2j_{k}+1\right)\left[j_{k}\left(j_{k}+1\right)-l_{k}\left(l_{k}+1\right)-\frac{3}{4}\right]$
(13)
$\displaystyle\left\\{V_{k}^{2}\left(1-f_{k}\right)+U_{k}^{2}f_{k}\right\\},$

where $f_{k}=\left[1+\exp\left(E_{k}/k_{B}T\right)\right]^{-1}$ is the Fermi
distribution, $k_{B}$ is the Boltzmann constant, and $T$ is the temperature.
The low-energy properties in the equilibrium state at finite temperature for
multi-$\Lambda$ hypernuclei are obtained from the solution of Eq. (6),
injecting Eqs. (11)-(13) into Eqs. (2)-(9).

###  II.2 Numerical Strategy

Table 2: $\Lambda\Lambda$ pairing interaction strength and corresponding hyperisotopic chain for each functional. Functional | $\begin{array}[]{c}\text{ V${}_{\Lambda_{0}}$}\\\ \left(\mathrm{MeV}\cdot\mathrm{fm}^{3}\right)\end{array}$ | $\begin{array}[]{c}\text{Hyperisotopic chain}\end{array}$
---|---|---
DF-NSC89+EmpC | -139 |  ${}^{40-S}_{-S\Lambda}$Ca $(-S=6-20)$
DF-NSC97a+EmpC | -148 |  ${}^{40-S}_{-S\Lambda}$Ca $(-S=6-20)$
DF-NSC97f+EmpC | -180 |  ${}^{40-S}_{-S\Lambda}$Ca $(-S=6-20)$
DF-NSC89+EmpC | -158 |  ${}^{132-S}_{-S\Lambda}$Sn $(-S=18-40)$
DF-NSC97a+EmpC | -145 |  ${}^{132-S}_{-S\Lambda}$Sn $(-S=18-40)$
DF-NSC97f+EmpC | -180 |  ${}^{132-S}_{-S\Lambda}$Sn $(-S=18-40)$
DF-NSC89+EmpC | -184 |  ${}^{208-S}_{-S\Lambda}$Pb $(-S=58-70)$
DF-NSC97a+EmpC | -180 |  ${}^{208-S}_{-S\Lambda}$Pb $(-S=58-70)$
DF-NSC97f+EmpC | -220 |  ${}^{208-S}_{-S\Lambda}$Pb $(-S=58-70)$

The FT-HFB equations for multi-$\Lambda$ hypernuclei presented hereabove are
solved in coordinate representation and considering spherical symmetry. The
SLy5 Skyrme interaction is fixed for the nucleon sector since it accurately
reproduces stable and exotic nuclei properties [49]. As usual, the spin-orbit
term in the $\Lambda$ mean field channel is neglected, see Ref. [50] for
instance.

Following the approach suggested in Ref. [41], the strength of the
$\Lambda\Lambda$ pairing interaction $V_{\Lambda_{0}}$ is adjusted to
reproduce the maximum value of the theoretical BCS prediction in uniform
matter calculated in Ref. [28], see Tab. 2. It should be noted that the
pairing gap at T=0 in nuclei are expected to be close to the one of uniform
matter. For instance, in the considered $-S$ intervals displayed in Tab. 2,
the average pairing gap $\bar{\Delta}_{\Lambda}$ (running over a set of
hypernuclei changing the number of hyperons while keeping the number of
protons and neutrons) is close to the value in uniform matter [41].

We employ the Numerov method to solve the FT-HFB equations and we consider the
Dirichlet boundary condition for vanishing wave functions. The wave-functions
are obtained numerically for a spherical box with a radius of 30 fm and a
resolution of 0.1 fm. The self-consistent solution is obtained by an iterative
process, which is repeated until the total energy converges within an accuracy
of about $10^{-8}$ MeV.

##  III Results at finite temperatures

In this section, we investigate how the pairing gaps, condensation energies,
normal and pairing densities, and radii of multi-$\Lambda$ hypernuclei evolve
as function of the temperature, considering different numbers of $\Lambda$
hyperons $(-S)$ present in the system. We focus on studying the hyper-isotopes
${}^{40-S}_{-S\Lambda}$Ca, ${}^{132-S}_{-S\Lambda}$Sn, and
${}^{208-S}_{-S\Lambda}$Pb, being magic in the nucleon sector. Additionally,
in cases where the hypernuclei are close to the hyperon-drip-line,
investigations have also explored hypernuclei with neutron open-shell
configurations at neutron numbers lying between $\pm 2$ and $\pm 4$ from the
neutron magic number.

###  III.1 $\Lambda$ Pairing Gaps and Binding Energies

Figure 1: Evolution of $\Lambda$ pairing gap with increasing temperature for
${}^{40-S}_{-S\Lambda}$Ca (a), ${}^{132-S}_{-S\Lambda}$Sn (b), and
${}^{208-S}_{-S\Lambda}$Pb (c) hypernuclei using DF-NSC97a+EmpC functional.

We first study the $\Lambda$ pairing gap ($\Delta_{\Lambda}$) as function of
the temperature in the following multi-$\Lambda$ hyper-isotopes:
${}^{40-S}_{-S\Lambda}$Ca, ${}^{132-S}_{-S\Lambda}$Sn, and
${}^{208-S}_{-S\Lambda}$Pb. The results for $\Delta_{\Lambda}(T)$ are shown in
Fig. 1, using the DF-NSC97a+EmpC functional.

As the temperature increases, the pairing gaps $\Delta_{\Lambda}$ decrease
until temperature reaches the critical temperature $T_{C}^{\Lambda}$ for
$\Lambda$ pairing, which is studied in more details in the following. At low
temperature, the pairing gap is only weakly reduced but it drops down only
when the temperature is close to the $T_{C}^{\Lambda}$. Note that for hyper-
nuclei with $-S$ close to magic numbers, the pairing gap drops to zero even
before the critical temperature is reached, at about $k_{B}T=0.4$ MeV. This
trend is visible in Fig. 2, illustrating the influence of strangeness on
finite temperature across various functionals for ${}^{40-S}_{-S\Lambda}$Ca
hypernuclei. In the three panels of Fig. 1 the largest temperature is set to
the maximal critical temperature $T_{C}^{\Lambda}$ for the considered
hyperisotopic chain (e.g. 0.54 MeV in the ${}^{40-S}_{-S\Lambda}$Ca case).

Figure 2: Impact of strangeness on finite temperature for
${}^{40-S}_{-S\Lambda}$Ca hypernuclei.

In ${}^{40-S}_{-S\Lambda}$Ca, the $\Lambda$ pairing gap $\Delta_{\Lambda}$ is
maximum at $-S=6$ in the studied range $-S=6-20$. Similarly, for
${}^{132-S}_{-S\Lambda}$Sn, the maximum pairing gap is obtained at $-S=28$ in
the range $-S=18-40$, and for ${}^{208-S}_{-S\Lambda}$Pb, it occurs at $-S=64$
in the range $-S=58-70$. The pairing gaps are quenched for the following magic
numbers: $-S=8$, $18$, $20$, $34$, $40$, $58$, and $70$. This sequence is
different from the usual one in non-strange nuclei since the spin-orbit has a
negligible impact in the $\Lambda$ channel, involving a different shell
structure.

It should be noted that qualitatively similar results are obtained for DF-
NSC89+EmpC and DF-NSC97f+EmpC functionals. This can be seen in Fig. 3 for
${}^{46}_{6\Lambda}$Ca, ${}^{160}_{28\Lambda}$Sn, and ${}^{272}_{64\Lambda}$Pb
hypernuclei. Comparing $\Delta_{\Lambda}$ values for all three functionals,
the DF-NSC97a+EmpC functional exhibits the largest $\Lambda$ pairing gap value
across almost all temperatures. While a larger density of states around the
Fermi level generally contributes to enhanced pairing gap, for the hypernuclei
we are looking at, the increased number of bound single-particle states
expands the available states for pairing interactions. This leads to a larger
pairing gap, driven by the deeper $N\Lambda$ potential of the DF-NSC97a+EmpC
functional compared to DF-NSC89+EmpC and DF-NSC97f+EmpC [10, 11].

Figure 3: Evolution of $\Lambda$ pairing gap with increasing temperature for
${}^{46}_{6\Lambda}$Ca (a), ${}^{160}_{28\Lambda}$Sn (b), and
${}^{272}_{64\Lambda}$Pb (c) hypernuclei. The plot depicts the results
obtained from three functionals: DF-NSC89+EmpC (solid line), DF-NSC97a+EmpC
(dashed line), and DF-NSC97f+EmpC (dotted line).

Table 3 displays the values of the pairing gaps in the ground-state
$\Delta_{\Lambda}^{T=0}$ and the critical temperature $T_{C}^{\Lambda}$ for a
set of hypernuclei. We find that the BCS relation in uniform matter, namely
$k_{B}T_{C}\approx 0.5\Delta_{\Lambda}^{T=0}$, is well reproduced for finite
hypernuclei. Note that the BCS relation is almost independent of the
functional, since all used functionals impact both $\Delta_{\Lambda}^{T=0}$
and $T_{C}^{\Lambda}$.

Table 3: Pairing gap in the ground-state $\Delta_{\Lambda}^{T=0}$ and critical temperatures $T_{C}^{\Lambda}$ for different functionals and for a set of hypernuclei, which are the ones with maximum pairing gap, see Fig. 1. Functional | Hypernucleus | $\begin{array}[]{c}\text{ $\Delta_{\Lambda}^{T=0}$}\\\ \left(\mathrm{MeV}\right)\end{array}$ | $\begin{array}[]{c}\text{ k${}_{\mathrm{B}}$T${}_{\mathrm{C}}^{\Lambda}$}\\\ \left(\mathrm{MeV}\right)\end{array}$
---|---|---|---
DF-NSC89+EmpC |  ${}^{46}_{6\Lambda}$Ca | 0.82 | 0.46
DF-NSC97a+EmpC |  ${}^{46}_{6\Lambda}$Ca | 1.04 | 0.54
DF-NSC97f+EmpC |  ${}^{46}_{6\Lambda}$Ca | 0.98 | 0.48
DF-NSC89+EmpC |  ${}^{160}_{28\Lambda}$Sn | 0.84 | 0.43
DF-NSC97a+EmpC |  ${}^{160}_{28\Lambda}$Sn | 0.82 | 0.44
DF-NSC97f+EmpC |  ${}^{160}_{28\Lambda}$Sn | 0.82 | 0.43
DF-NSC89+EmpC |  ${}^{272}_{64\Lambda}$Pb | 0.69 | 0.39
DF-NSC97a+EmpC |  ${}^{272}_{64\Lambda}$Pb | 0.76 | 0.44
DF-NSC97f+EmpC |  ${}^{272}_{64\Lambda}$Pb | 0.71 | 0.38

The effects of finite temperatures and pairing correlations on binding energy
can also be assessed by estimating the condensation energy, defined as
E${}_{\text{cond}}$ = E${}_{\text{HF}}$ \- E${}_{\text{HFB}}$. Fig. 4 shows
E${}_{\text{cond}}$ for a set of ${}^{132-S}_{-S\Lambda}$Sn hypernuclei, using
DF-NSC89+EmpC, DF-NSC97a+EmpC and DF-NSC97f+EmpC functionals. The behavior of
the condensation energy closely mirrors the trend observed in the $\Lambda$
pairing gap of Fig. 1.

Figure 4: Temperature-dependent binding energy differences of HF and HFB
approaches for ${}^{132-S}_{-S\Lambda}$Sn hypernuclei with different
functionals. The plot depicts the results obtained from three functionals: DF-
NSC89+EmpC (solid line), DF-NSC97a+EmpC (dashed line), and DF-NSC97f+EmpC
(dotted line).

The peak value of the condensation energy nearly reaches $3$ MeV at $T=0$.
With increasing temperature, the E${}_{\text{HF}}$ \- E${}_{\text{HFB}}$ value
experiences a consistent reduction across all hypernuclei, mirroring the
pattern displayed in Fig. 1. For instance, as the temperature increases to
$k_{B}T=0.3$ MeV, the peak of the condensation energy decresases to
approximately 2 MeV. At a specific temperature threshold E${}_{\text{HF}}$ =
E${}_{\text{HFB}}$ across the whole $-S$ range.

###  III.2 Normal and abnormal pairing densities

Figure 5: Normal density profiles of ${}^{46}_{6\Lambda}$Ca,
${}^{160}_{28\Lambda}$Sn, and ${}^{272}_{64\Lambda}$Pb hypernuclei at finite
temperatures using DF-NSC97a+EmpC functional.

Fig. 5 displays the density profiles for typical three hyper-nuclei,
${}^{46}_{6\Lambda}$Ca, ${}^{160}_{28\Lambda}$Sn, and
${}^{272}_{64\Lambda}$Pb, for $T=0$ and $T=T_{C}^{\Lambda}$. The impact of
temperature on the normal densities is very small in the range of temperature
bounded by the critical temperature. The $\Lambda$ single-particle spectrum
has also been analyzed. In the considered hypernuclei, shifts in energy levels
of up to a few hundred keV are observed, as temperature increases. Therefore,
there isn’t an important impact of the temperature of the $\Lambda$ single-
particle spectrum.

Figure 6: Correlation between temperature and $\Lambda$ hyperon radii in
hypernuclei ${}^{46}_{6\Lambda}$Ca, ${}^{160}_{28\Lambda}$Sn, and
${}^{272}_{64\Lambda}$Pb using DF-NSC97a+EmpC functional. Corresponding
$k_{B}T_{C}^{\Lambda}$ values of these hypernuclei are listed in Table 3.

The $\Lambda$ hyperon radius is also analyzed. Fig. 6 illustrates this in the
hypernuclei ${}^{46}_{6\Lambda}$Ca, ${}^{160}_{28\Lambda}$Sn, and
${}^{272}_{64\Lambda}$Pb using the DF-NSC97a+EmpC functional. Small effect of
temperature are found. Across all examined hyper-nuclei, the modification in
radii remains around 0.3% as temperature changes from 0 to $T_{C}^{\Lambda}$.
Similarly, this change remains relatively small as temperature extends from 0
to $2T_{C}^{\Lambda}$ for these hyper-nuclei. For instance, in the case of
${}^{40-S}_{-S\Lambda}$Ca hyper-isotopes, the radius increase is roughly 0.6%,
while for ${}^{132-S}_{-S\Lambda}$Sn hyper-isotopes, it reaches around 3%. In
the scenario of $\Lambda$ hyperon-rich ${}^{208-S}_{-S\Lambda}$Pb, this
increase amounts to approximately 4%. In conclusion, we found only weak impact
of temperature on the $\Lambda$ hyperon radius. Beyond 2TC, the sudden
increase of the radii is due to the creation of a neutron gas, which is an
spurious effect of using a finite-size box [32].

Figure 7: Abnormal pairing density profiles for ${}^{46}_{6\Lambda}$Ca (a),
${}^{160}_{28\Lambda}$Sn (b), and ${}^{272}_{64\Lambda}$Pb (c) hypernuclei at
finite temperatures using DF-NSC97a+EmpC functional.

While the impact of temperature on normal density profiles has been found to
be negligible, the effect of temperature on the abnormal pairing density
profiles is much stronger, as shown in Fig. 7 for ${}^{46}_{6\Lambda}$Ca,
${}^{160}_{28\Lambda}$Sn, and ${}^{272}_{64\Lambda}$Pb hypernuclei. Note that
all the other hyper-nuclei that we have studied, see Fig. 1, show similar
results. As expected, the reduction of the abnormal pairing density scales
with the one of the pairing gaps shown in Fig. 1.

###  III.3 Pairing correlations around the hyperon-drip-line

Among the ${}^{40-S}_{-S\Lambda}$Ca, ${}^{132-S}_{-S\Lambda}$Sn, and
${}^{208-S}_{-S\Lambda}$Pb hypernuclei that we examined in detail above,
${}^{60}_{20\Lambda}$Ca, ${}^{172}_{40\Lambda}$Sn, and
${}^{278}_{70\Lambda}$Pb lie the closest to the hyperon-drip-line at $T=0$:
their $\Lambda\Lambda$ separation energies is very close to 0. These drip-line
systems can contribute to conditions where the pairing re-entrance effect (the
pairing gap becomes non-zero above a given temperature) is predicted [51, 52].
However, as seen from Fig. 1, even at finite temperatures,
$\Delta_{\Lambda}=0$ for these hypernuclei. This is not the case when
considering open-shell neutron cases. For this purpose, calculations were
performed for ${}^{56,58,62,64}_{20\Lambda}$Ca,
${}^{168,170,174,176}_{40\Lambda}$Sn, and ${}^{274,276,280,282}_{70\Lambda}$Pb
hypernuclei using all three functionals. Mixed-type pairing interaction was
used in the neutron channel for these hypernuclei, with the pairing strength
$V_{n0}=-265$ MeV$\cdot$fm3, such that the pairing gap $\Delta_{n}\approx 0.6$
MeV in ${}^{280}_{70\Lambda}$Pb, see Fig. 8. For the $\Lambda$ channel, we
used the pairing strengthes given in Tab. 2. Pairing re-entrance occurs in
${}^{280}_{70\Lambda}$Pb using DF-NSC97f+EmpC functional, with the two
predicted critical temperatures $k_{B}T_{c1}=0.19$ MeV and $k_{B}T_{c2}=0.41$
MeV. It should be noted that at $k_{B}T_{c2}=0.41$ MeV, using the DF-
NSC97f+EmpC functional yields chemical potentials of -0.32 MeV for $\Lambda$
hyperons, -2.83 MeV for neutrons, and -7.21 MeV for protons in
${}^{280}_{70\Lambda}$Pb. However, for all other neutron open-shell
hypernuclei we investigated, $\Delta_{\Lambda}$ remains constant at 0 across
all temperatures. The reason for the pairing re-entrance phenomenon specific
to ${}^{280}_{70\Lambda}$Pb, is that in this hypernucleus using DF-
NSC97f+EmpC, the energy difference between the last occupied 3s level and the
first unoccupied 1h level is smaller compared to other similar
${}^{274,276,278,282}_{70\Lambda}$Pb hypernuclei, as shown in Table 4. As the
temperature changes, the 1h level, which is for this hypernucleus empty at
$T=0$, begins to fill, and pairing properties begin to emerge at temperatures
greater than $k_{B}T=0.19$ MeV.

Figure 8: Comparison of the evolution of $\Lambda$ and neutron pairing gaps with increasing temperature for the ${}^{280}_{70\Lambda}$Pb hypernucleus. The red line represents the $\Lambda$ pairing gap calculated using the DF-NSC97f+EmpC functional, while the blue line represents the neutron pairing gap using a pairing strength of $V_{n0}=-265$ MeV$\cdot$fm3. Table 4: Energy difference between the highest occupied 3s level and the lowest unoccupied 1h level in ${}^{274,276,278,280,282}_{70\Lambda}$Pb hypernuclei at $T=0$. Functionals | Energy difference $(\mathrm{MeV})$
---|---
|  ${}^{274}_{70\Lambda}$Pb |  ${}^{276}_{70\Lambda}$Pb |  ${}^{278}_{70\Lambda}$Pb |  ${}^{280}_{70\Lambda}$Pb |  ${}^{282}_{70\Lambda}$Pb
DF-NSC89+EmpC | 1.87 | 1.79 | 1.81 | 1.80 | 1.76
DF-NSC97a+EmpC | 3.21 | 2.87 | 3.03 | 3.56 | 3.41
DF-NSC97f+EmpC | 1.68 | 1.68 | 1.59 | 1.53 | 1.62

##  IV Conclusions

In conclusion, the interplay between pairing and finite temperature for
multi-$\Lambda$ hypernuclei has been studied using the FT-HFB apparoach. We
have considered three different functionals for the $N\Lambda$ channel, DF-
NSC89, DF-NSC97a, and DF-NSC97f, all adjusted from microscopic Brueckner-
Hartree-Fock predictions. For the $\Lambda\Lambda$ channel, the empirical
prescription EmpC has been used, which is adjusted to reproduce the
experimental bond energy in ${}^{6}_{\Lambda\Lambda}$He.

The pairing gaps for individual hyper-nuclei show similar trends as the
temperature approach the critical temperature $T_{C}^{\Lambda}$, which is
found to be about $0.54$ MeV and to scale with respect to
$\Delta_{\Lambda}(T=0)$ as suggested by the BCS relation. Among the studied
functionals, DF-NSC97a+EmpC predicts the largest pairing gaps.

We have also investigated the effect of temperature on the binding energy, on
the single-particle spectra, on the normal density and on the $\Lambda$
hyperon radius. There is no effect of temperature for $T\lesssim
T_{C}^{\Lambda}$. As expected, the abnormal pairing densities and the
condensation energy are impacted by the temperature, and shrinks to zero for
$T=T_{C}^{\Lambda}$. Calculations also explored hyperon-drip-line hypernuclei.
A pairing re-entrance effect was observed in the neutron open-shell hyperon
drip-line hypernucleus ${}^{280}_{70\Lambda}$Pb, in the $\Lambda$ channel.

In summary, this study clarifies the combined effects of temperature and
pairing correlations in hyper-nuclei. Our main findings is to show that the
usual BCS correlation between $T_{C}^{\Lambda}$ and $\Delta_{\Lambda}(T=0)$
remains valid for hyper-nuclei.

###### Acknowledgements.

This work is supported by the Scientific and Technological Research Council of
Turkey (TÜBİTAK) under project number MFAG-118F098 and the Yildiz Technical
University under project number FBI-2018-3325 and FBA-2021-4229. This work is
also supported by the LABEX Lyon Institute of Origins (ANR-10-LABX-0066) of
the Université de Lyon and the IN2P3 masterproject MAC.

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