# Covariant Density Functional Theory with Localized Exchange Terms

Qiang Zhao gbsn (赵强)  State Key Laboratory of Nuclear Physics and Technology,
School of Physics, Peking University, Beijing 100871, China  Center for Exotic
Nuclear Studies, Institute for Basie Scicence, Daejeon 34126, Korea  Zhengxue
Ren gbsn (任政学)  State Key Laboratory of Nuclear Physics and Technology, School
of Physics, Peking University, Beijing 100871, China  Pengwei Zhao gbsn (赵鹏巍)
<EMAIL_ADDRESS>State Key Laboratory of Nuclear Physics and Technology,
School of Physics, Peking University, Beijing 100871, China  Jie Meng gbsn
(孟杰<EMAIL_ADDRESS>State Key Laboratory of Nuclear Physics and Technology,
School of Physics, Peking University, Beijing 100871, China  School of Physics
and Nuclear Energy Engineering, Beihang University, Beijing 100191, China
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502,
Japan

###### Abstract

A new density-dependent point-coupling covariant density functional PCF-PK1 is
proposed, where the exchange terms of the four-fermion terms are local and are
taken into account with the Fierz transformation. The coupling constants of
the PCF-PK1 functional are determined by empirical saturation properties and
_ab initio_ equation of state and proton-neutron Dirac mass splittings for
nuclear matter as well as the ground-state properties of selected spherical
nuclei. The success of the PCF-PK1 is illustrated with properties of the
infinite nuclear matter and finite nuclei including the ground-state
properties and the Gamow-Teller resonances. In particular, the PCF-PK1
eliminates the spurious shell closures at $Z=58$ and $Z=92$, which exist
commonly in many covariant density functionals without exchange terms.
Moreover, the Gamow-Teller resonances are nicely reproduced without any
adjustable parameters, and this demonstrates that a self-consistent
description for the Gamow-Teller resonances can be achieved with the localized
exchange terms in the PCF-PK1.

##  I Introduction

The development of rare isotope beam facilities has largely extended the
limits of our exploration on the nuclear chart from the stability valley to
the drip lines. In particular, the unstable nuclei with large isospin could
exhibit many unexpected phenomena, which considerably deepen our understanding
of nuclear physics Casten and Sherrill (2000); Tanihata  _et al._ (2013). In
addition, the properties of unstable nuclei also provide essential inputs for
the study of nucleosynthesis in the universe. However, most of the unstable
nuclei are still far beyond the current experimental capacities. Therefore, it
is important to build a unified and consistent nuclear many-body method to
study the unstable nuclei and provide reliable descriptions.

The covariant density functional theory (DFT) is one of the most successful
microscopic theories for a global description of nuclei Men (2016); Meng and
Zhao (2021). It is based on the relativistic quantum field theory and density
functional theory for quantum many-body problems Serot and Walecka (1986);
Ring (1996); Vretenar  _et al._ (2005); Meng  _et al._ (2006). The covariant
DFTs take Lorentz symmetry into account and provide an efficient description
of nuclei with the underlying large scalar and vector fields of the order of a
few hundred MeV, which are hidden in the nonrelativistic DFTs. This is clearly
seen in the nonrelativistic reduction of covariant DFT via a similarity
renormalization method, which provides a possible means to bridge the
relativistic and non-relativistic DFTs Ren and Zhao (2020). The inclusion of
Lorentz symmetry brings several advantages for the covariant DFTs Ring (2012).
For instance, it naturally includes the spin degree of freedom, and
automatically gives the large spin-orbit potential in nuclei. It provides a
consistent treatment of the time-odd fields, which are particularly important
for describing spectroscopic properties associated with nuclear rotations
Afanasjev and Abusara (2010); Meng  _et al._ (2013). Due to these advantages,
the covariant DFTs have attracted a lot of attentions during the past decades,
and have been successfully applied to study quantitatively the static and
dynamic properties of nuclei Ring (1996); Vretenar  _et al._ (2005); Meng  _et
al._ (2006); Nikšić  _et al._ (2011); Ren  _et al._ (2020).

In most of these applications, however, only the direct (Hartree) terms are
considered, and the exchange (Fock) terms are usually neglected. In principle,
at the mean-field level, both the direct and exchange diagrams should be
included. It was assumed that the effects of the exchange terms could be
absorbed in a phenomenological way in the adjustments of the coupling
constants of energy density functionals. However, except for simplicity, there
is no robust physical reason for this assumption. In fact, the contribution of
the pion mesons to the mean fields can be taken into account only via the
exchange terms, due to its negative-parity nature.

The effects of the exchange terms have been discussed within the framework of
the relativistic Hartree-Fock (RHF) approach Bouyssy  _et al._ (1987); Long
_et al._ (2006, 2010). By introducing the exchange terms, the exchange of
$\pi$ mesons as well as the $\rho$-tensor couplings can be taken into account
Long  _et al._ (2007), and the tensor forces are naturally included Jiang  _et
al._ (2015); Wang  _et al._ (2018). It was found that the RHF functionals with
the $\pi$-exchange and $\rho$-tensor couplings could improve the descriptions
of the single-particle energies Long  _et al._ (2008a, 2009); Wang  _et al._
(2013); Li  _et al._ (2016); Liu  _et al._ (2020). Another effect of the
exchange terms was revealed in the description of spin-isospin excitations
with the relativistic random phase approximation, where no additional
parameters are needed with the inclusion of exchange terms Liang  _et al._
(2008a, 2012a), while the Migdal term has to be adjusted with only the direct
terms. However, the bottleneck is that the nonlocal potentials involved in the
meson-exchange RHF framework leads to a surge in the computational cost and,
thus, the application of RHF is usually limited to spherical nuclei and light
deformed ones Ebran  _et al._ (2011). Only recently, the axially deformed RHF
model expanded on Dirac-Woods-Saxon basis is established Geng  _et al._
(2020), which is able to be applied for heavy nuclei but still with a large
demand of computational costs.

On the other hand, the relativistic point-coupling model provides another way
to construct a relativistic energy density functional Nikolaus  _et al._
(1992); Bürvenich  _et al._ (2002); Zhao  _et al._ (2010a), where the meson
exchange is replaced by the corresponding local (contact) interactions between
the nucleons, and the finite-range effects are approximated by local
derivative terms. Such a replacement can be justified since it is known from
the meson-exchange models that the exchanged $\sigma$, $\omega$, and $\rho$
mesons between nucleons are all heavy mesons. In recent years, the
relativistic point-coupling model has attracted more and more attention owing
to the following advantages. First, it is considerably simpler in numerical
applications by avoiding the solution of the Klein-Gordon equations Zhao  _et
al._ (2011) and also the complicated two-body matrix elements of finite-range
in applications going beyond static mean-field theory Nikšić  _et al._ (2011).
Second, it provides more opportunities to investigate its relationship to the
nonrelativistic approaches Sulaksono  _et al._ (2003a); Ren and Zhao (2020).
Third, it is relatively easy to include the exchange terms with the point-
coupling interactions Sulaksono  _et al._ (2003b); Liang  _et al._ (2012b).

In the framework of the relativistic point-coupling models, it is convenient
to use the Fierz transformation to express the exchange terms as
superpositions of the direct terms, such that the exchange terms can be
treated in a similar way as the direct terms. In Ref. Liang  _et al._ (2012b),
by the zero-range reduction and the Fierz transformation, the validity of the
localized exchange terms has been studied, and it retains the simplicity of
point-coupling models and provides proper descriptions of the spin-isospin
excitations and the Dirac masses. Inspired by these findings, the present work
is to propose a practical point-coupling covariant density functional with
localized exchange terms obtained with the Fierz transformation.

The paper is organized as follows. The theoretical framework of the covariant
DFT with localized exchange terms is presented in Sec. II. In Sec. III and IV,
the numerical details as well as the strategy to determine the parameters in
the density functional are given, respectively. Then, the performance of the
new density functional is tested in Sec. V by studying properties of infinite
nuclear matter and finite nuclei including the ground-state properties and the
Gamow-Teller resonance excitations. Finally, the summary is given in Sec. VI.

##  II Theoretical framework

The elementary building blocks of the relativistic point-coupling model are

$\displaystyle(\bar{\psi}\mathcal{O}\Gamma\psi),\qquad\mathcal{O}\in\\{1,\vec{\tau}\\},\qquad\Gamma\in\\{1,\gamma_{\mu},\gamma_{5},\gamma_{5}\gamma_{\mu},\sigma_{\mu\nu}\\},$
(1)

where $\psi$ is the nucleon field, $\vec{\tau}$ is the isospin matrices, and
$\Gamma$ represents the Dirac matrices. Here, the arrows are used to denote
vectors in the isospin space, and the space vectors will be denoted as the
bold types.

The Lagrangian density of the nuclear system can be constructed as the power
series of these building blocks and their derivatives. The starting point is
an effective Lagrangian density consisting of four parts,

$\displaystyle\mathcal{L}=\mathcal{L}^{\rm free}+\mathcal{L}^{\rm
4f}+\mathcal{L}^{\rm der}+\mathcal{L}^{\rm em},$ (2)

which are the free-nucleon term $\mathcal{L}^{\rm free}$

$\displaystyle\mathcal{L}^{\rm
free}=\bar{\psi}(i\gamma_{\mu}\partial^{\mu}-M)\psi,$ (3)

the four-fermion point-coupling term $\mathcal{L}^{\rm 4f}$

$\displaystyle\mathcal{L}^{\rm 4f}=$
$\displaystyle-\frac{1}{2}\alpha_{S}(\bar{\psi}\psi)(\bar{\psi}\psi)-\frac{1}{2}\alpha_{tS}(\bar{\psi}\vec{\tau}\psi)(\bar{\psi}\vec{\tau}\psi)$
$\displaystyle-\frac{1}{2}\alpha_{V}(\bar{\psi}\gamma_{\mu}\psi)(\bar{\psi}\gamma^{\mu}\psi)-\frac{1}{2}\alpha_{tV}(\bar{\psi}\gamma_{\mu}\vec{\tau}\psi)(\bar{\psi}\gamma^{\mu}\vec{\tau}\psi)$
$\displaystyle-\frac{1}{2}\alpha_{T}(\bar{\psi}\sigma_{\mu\nu}\psi)(\bar{\psi}\sigma^{\mu\nu}\psi)-\frac{1}{2}\alpha_{tT}(\bar{\psi}\sigma_{\mu\nu}\vec{\tau}\psi)(\bar{\psi}\sigma^{\mu\nu}\vec{\tau}\psi)$
$\displaystyle-\frac{1}{2}\alpha_{PS}(\bar{\psi}\gamma_{5}\psi)(\bar{\psi}\gamma_{5}\psi)-\frac{1}{2}\alpha_{tPS}(\bar{\psi}\gamma_{5}\vec{\tau}\psi)(\bar{\psi}\gamma_{5}\vec{\tau}\psi)$
$\displaystyle-\frac{1}{2}\alpha_{PV}(\bar{\psi}\gamma_{5}\gamma_{\mu}\psi)(\bar{\psi}\gamma_{5}\gamma^{\mu}\psi)-\frac{1}{2}\alpha_{tPV}(\bar{\psi}\gamma_{5}\gamma_{\mu}\vec{\tau}\psi)(\bar{\psi}\gamma_{5}\gamma^{\mu}\vec{\tau}\psi),$
(4)

the derivative term $\mathcal{L}^{\rm der}$

$\displaystyle\mathcal{L}^{\rm
der}=-\frac{1}{2}\delta_{S}\partial_{\mu}(\bar{\psi}\psi)\partial^{\mu}(\bar{\psi}\psi),$
(5)

and the electromagnetic interaction term $\mathcal{L}^{\rm em}$

$\displaystyle\mathcal{L}^{\rm
em}=-e\frac{1-\tau_{3}}{2}\bar{\psi}\gamma_{\mu}\psi
A^{\mu}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}.$ (6)

Here, $M$ is the nucleon mass, $e$ is the charge unit, $A_{\mu}$ and
$F_{\mu\nu}$ are respectively the four-vector potential and strength tensor of
the electromagnetic field. The subscripts $S$, $V$, $T$, $PS$, and $PV$ stand
for the scalar, vector, tensor, pseudo-scalar, and pseudo-vector couplings,
respectively. The subscript “$t$” refers to the corresponding isovector
channel. The derivative term is considered only in the isoscalar-scalar
channel, but it is adequate to simulate the finite-range effects of the
effective interaction Nikšić  _et al._ (2008). Higher-order terms or density-
dependent coupling constants are usually implemented to take into account the
in-medium many-body correlations. In this work, we assume density-dependent
coupling constants in the Lagrangian density.

The Hamiltonian of the system can be derived by the Legendre transformation.
With the no-sea approximation, the nucleon field operator is expanded on the
basis of annihilation and creation operators
$\\{c_{\alpha},c^{\dagger}_{\alpha}\\}$ defined by a complete set of Dirac
spinors $\\{\varphi_{\alpha}({\bm{r}})\\}$,

$\displaystyle\psi(\bm{r})=\sum_{\alpha}\varphi_{\alpha}(\bm{r})c_{\alpha},\qquad\qquad\psi^{\dagger}(\bm{r})=\sum_{\alpha}\varphi_{\alpha}^{\dagger}(\bm{r})c^{\dagger}_{\alpha}.$
(7)

In the Hartree-Fock approximation, the ground-state wave function
$|\Phi_{0}\rangle$ is approximated by a Slater determinant,

$\displaystyle|\Phi_{0}\rangle=\prod^{A}_{\alpha=1}c_{\alpha}^{\dagger}|-\rangle,$
(8)

in which $A$ is the number of nucleons and $|-\rangle$ represents the vacuum.
The energy functional of the nuclear system is the expectation value of the
Hamiltonian $H$ in the ground-state Slater determinant,

$\displaystyle E_{\rm CDF}=\langle\Phi_{0}|H|\Phi_{0}\rangle=E_{\rm
kin}+E_{\rm 4f}+E_{\rm der}+E_{\rm em},$ (9)

which contains four parts, i.e., the kinetic $E_{\rm kin}$, four-fermion
interaction $E_{\rm 4f}$, derivative $E_{\rm der}$, and electromagnetic
$E_{\rm em}$ parts.

The kinetic energy part can be readily written as,

$\displaystyle E_{\rm kin}=\int
d^{3}\bm{r}\sum_{\alpha}\bar{\varphi}_{\alpha}(\bm{r})(-i\bm{\gamma}\cdot\bm{\nabla}+M)\varphi_{\alpha}(\bm{r}).$
(10)

For the four-fermion interaction part, both direct (Hartree) $E_{\rm H}$ and
exchange (Fock) $E_{\rm F}$ terms are considered,

$\displaystyle E_{\rm H}=$ $\displaystyle\frac{1}{2}\int
d^{3}\bm{r}\sum_{i;\alpha\beta}\alpha_{i}^{\rm
HF}\left[\bar{\varphi}_{\alpha}(\bm{r})(\mathcal{O}\Gamma)_{i}\varphi_{\alpha}(\bm{r})\right]\left[\bar{\varphi}_{\beta}(\bm{r})(\mathcal{O}\Gamma)^{i}\varphi_{\beta}(\bm{r})\right],$
(11) $\displaystyle E_{\rm F}=$ $\displaystyle-\frac{1}{2}\int
d^{3}\bm{r}\sum_{i;\alpha\beta}\alpha_{i}^{\rm
HF}\left[\bar{\varphi}_{\alpha}(\bm{r})(\mathcal{O}\Gamma)_{i}\varphi_{\beta}(\bm{r})\right]\left[\bar{\varphi}_{\beta}(\bm{r})(\mathcal{O}\Gamma)^{i}\varphi_{\alpha}(\bm{r})\right].$
(12)

Here, the superscript of $\alpha_{i}^{\rm HF}$ is introduced to denote the
coupling constants adopted under the Hartree-Fock approximation, and the index
$i$ should run over all possible coupling channels. The exchange terms can be
expressed as the superposition of the direct terms with the Fierz
transformation Greiner and Müller (2009); Sulaksono  _et al._ (2003b),

$\displaystyle[\bar{\varphi}_{\alpha}(\mathcal{O}\Gamma)_{i}\varphi_{\beta}][\bar{\varphi}_{\beta}(\mathcal{O}\Gamma)^{i}\varphi_{\alpha}]=\sum_{j}\Lambda_{ij}[\bar{\varphi}_{\alpha}(\mathcal{O}\Gamma)_{j}\varphi_{\alpha}][\bar{\varphi}_{\beta}(\mathcal{O}\Gamma)^{j}\varphi_{\beta}]$
(13)

with $\Lambda$ being the Fierz transformation matrix. As a result, the four-
fermion part $E_{\rm 4f}$ can be written as

$\displaystyle E_{\rm 4f}=E_{\rm H}+E_{\rm F}=\frac{1}{2}\int
d\bm{r}\sum_{i;\alpha\beta}\alpha_{i}\left[\bar{\varphi}_{\alpha}(\bm{r})(\mathcal{O}\Gamma)_{i}\varphi_{\alpha}(\bm{r})\right]\left[\bar{\varphi}_{\beta}(\bm{r})(\mathcal{O}\Gamma)^{i}\varphi_{\beta}(\bm{r})\right],$
(14)

where $\alpha_{i}$ is defined as

$\displaystyle\alpha_{i}=\sum_{j}C_{ij}\alpha_{j}^{\rm HF},\qquad
i,j=\\{S,tS,V,tV,T,tT,PS,tPS,PV,tPV\\}$ (15)

with the matrix $C$

$\displaystyle
C=1-\Lambda^{T}=\frac{1}{16}\left(\begin{array}[]{rrrrrrrrrr}14&-6&-8&-24&-24&-72&-2&-6&8&24\\\
-2&18&-8&8&-24&24&-2&2&8&-8\\\ -2&-6&20&12&0&0&2&6&4&12\\\
-2&2&4&12&0&0&2&-2&4&-4\\\ -1&-3&0&0&20&12&-1&-3&0&0\\\
-1&1&0&0&4&12&-1&1&0&0\\\ -2&-6&8&24&-24&-72&14&-6&-8&-24\\\
-2&2&8&-8&-24&24&-2&18&-8&8\\\ 2&6&4&12&0&0&-2&-6&20&12\\\
2&-2&4&-4&0&0&-2&2&4&12\\\ \end{array}\right).$ (26)

Note that the rank of the matrix $C$ is 5, so once the coupling constants
$\alpha_{i}$ in five channels are known, one can immediately obtain the ones
in the remain channels. In the present work, the coupling constants in the
channels of $S$, $tS$, $V$, $tV$, and $T$ are regarded as the free parameters
to be determined, so the ones in other channels read

$\displaystyle\alpha_{tT}$
$\displaystyle=\frac{1}{18}(-\alpha_{S}+3\alpha_{tS}+2\alpha_{V}-6\alpha_{tV}+6\alpha_{T}),$
(27a) $\displaystyle\alpha_{PS}$
$\displaystyle=\frac{1}{3}(-\alpha_{S}-6\alpha_{tS}-4\alpha_{V}+12\alpha_{tV}-12\alpha_{T}),$
(27b) $\displaystyle\alpha_{tPS}$
$\displaystyle=\frac{1}{9}(-4\alpha_{S}+3\alpha_{tS}+8\alpha_{V}-24\alpha_{tV}-12\alpha_{T}),$
(27c) $\displaystyle\alpha_{PV}$
$\displaystyle=\frac{1}{3}(2\alpha_{S}+3\alpha_{tS}+2\alpha_{V}+3\alpha_{tV}+6\alpha_{T}),$
(27d) $\displaystyle\alpha_{tPV}$
$\displaystyle=\frac{1}{9}(2\alpha_{S}+3\alpha_{tS}+5\alpha_{V}-6\alpha_{tV}+6\alpha_{T}).$
(27e)

With the redefined coupling constants $\alpha_{i}$, the exchange terms are now
treated in a similar way to the direct ones.

The derivative $E_{\rm der}$ and electromagnetic $E_{\rm em}$ parts can be
respectively written as

$\displaystyle E_{\rm der}=-\frac{1}{2}\int
d^{3}\bm{r}\sum_{\alpha\beta}\delta_{S}\bm{\nabla}\left[\bar{\varphi}_{\alpha}(\bm{r})\varphi_{\alpha}(\bm{r})\right]\bm{\nabla}\left[\bar{\varphi}_{\beta}(\bm{r})\varphi_{\beta}(\bm{r})\right],$
(28)

and

$\displaystyle E_{\rm em}=\frac{1}{2}\iint
d^{3}\bm{r}d^{3}\bm{r}^{\prime}\sum_{\alpha\beta}\frac{e^{2}}{4\pi}\left[\bar{\varphi}_{\alpha}(\bm{r})\gamma_{\mu}\frac{1-\tau_{3}}{2}\varphi_{\alpha}(\bm{r})\right]\frac{1}{|\bm{r}-\bm{r}^{\prime}|}\left[\bar{\varphi}_{\beta}(\bm{r}^{\prime})\gamma^{\mu}\frac{1-\tau_{3}}{2}\varphi_{\beta}(\bm{r}^{\prime})\right].$
(29)

Here, for simplicity reason, the exchange terms are neglected.

Pairing correlations are essential for open-shell nuclei, so the pairing
energy, which depends on the pairing density, should also be included in the
energy density functional,

$\displaystyle E_{\rm pair}=\frac{1}{2}\rm{Tr}[\Delta\kappa].$ (30)

Here, $\kappa$ is the pairing density (see below), and $\Delta$ is the pairing
field, whose matrix elements can be written as

$\displaystyle\Delta_{ab}=\frac{1}{2}\sum_{cd}\left\langle
ab\left|V^{pp}\right|cd\right\rangle\kappa_{cd},$ (31)

with $V^{pp}$ being the pairing force. In this work, a separable form of the
finite-range interaction Gogny D1S is adopted for the pairing force Tian  _et
al._ (2009),

$\displaystyle
V^{pp}\left(\bm{r}_{1},\bm{r}_{2},\bm{r}_{1}^{\prime},\bm{r}_{2}^{\prime}\right)=-G\delta\left(\bm{R}-\bm{R}^{\prime}\right)P(\bm{r})P\left(\bm{r}^{\prime}\right)\frac{1}{2}\left(1-P^{\sigma}\right),$
(32)

in which $\bm{R}=(\bm{r}_{1}+\bm{r}_{2})/2$, $\bm{r}=\bm{r}_{1}-\bm{r}_{2}$,
and $P(\bm{r})$ has a Gaussian expression

$\displaystyle P(\bm{r})=\frac{1}{\left(4\pi
a^{2}\right)^{3/2}}e^{-r^{2}/4a^{2}}.$ (33)

The constants $G$ and $a$ are two parameters for the pairing density
functional.

The variation of the energy density functional results in the relativistic
Hartree-Fock-Bogoliubov (RHFB) equation,

$\displaystyle\left(\begin{array}[]{cc}\hat{h}-\lambda&\hat{\Delta}\\\
-\hat{\Delta}^{*}&-\hat{h}^{*}+\lambda\end{array}\right)\left(\begin{array}[]{l}U_{\alpha}\\\
V_{\alpha}\end{array}\right)=E_{\alpha}\left(\begin{array}[]{l}U_{\alpha}\\\
V_{\alpha}\end{array}\right),$ (40)

where $\hat{h}$ is the single-particle Dirac Hamiltonian, $\hat{\Delta}$ is
the pairing field, $\lambda$ is the Fermi energy, $E_{\alpha}$ is the
quasiparticle energy, and $U_{\alpha}$ and $V_{\alpha}$ are the quasiparticle
wavefunctions. The RHFB equation gives a unified and self-consistent treatment
of the mean field and the pairing field Kucharek and Ring (1991); Gonzalez-
Llarena  _et al._ (1996); Meng (1998).

Here, the single-particle Hamiltonian $\hat{h}$ reads

$\displaystyle\hat{h}=\bm{\alpha}\cdot\bm{p}+\beta\left(M+S\right)+V-\bm{\alpha}\cdot\bm{V}-i\beta\bm{\alpha}\cdot\bm{T}^{0}$
(41)

which is coupled with the scalar, vector, and tensor potentials

$\displaystyle S$
$\displaystyle=\alpha_{S}\rho_{S}+\alpha_{tS}\tau_{3}\rho_{tS}+\delta_{S}\Delta\rho_{S},$
(42) $\displaystyle V^{0}$
$\displaystyle=\alpha_{V}\rho_{V}+\alpha_{tV}\tau_{3}\rho_{tV}+e\frac{1-\tau_{3}}{2}A^{0},$
(43) $\displaystyle\bm{V}$
$\displaystyle=\alpha_{V}\bm{j}_{V}+\alpha_{tV}\tau_{3}\bm{j}_{tV}+e\frac{1-\tau_{3}}{2}\bm{A},$
(44) $\displaystyle\bm{T}^{0}$
$\displaystyle=2\alpha_{T}\bm{j}^{0}_{T}+2\alpha_{tV}\tau_{3}\bm{j}^{0}_{tT},$
(45)

via various densities and currents

$\displaystyle\rho_{S}$
$\displaystyle=\sum_{\alpha>0}\bar{V}_{\alpha}V_{\alpha},\quad$
$\displaystyle\rho_{tS}$
$\displaystyle=\sum_{\alpha>0}\bar{V}_{\alpha}\tau_{3}V_{\alpha},$ (46)
$\displaystyle\rho_{V}$
$\displaystyle=\sum_{\alpha>0}\bar{V}_{\alpha}\gamma^{0}V_{\alpha},\quad$
$\displaystyle\rho_{tV}$
$\displaystyle=\sum_{\alpha>0}\bar{V}_{\alpha}\gamma^{0}\tau_{3}V_{\alpha},$
(47) $\displaystyle\bm{j}_{V}$
$\displaystyle=\sum_{\alpha>0}\bar{V}_{\alpha}\bm{\gamma}V_{\alpha},\quad$
$\displaystyle\bm{j}_{tV}$
$\displaystyle=\sum_{\alpha>0}\bar{V}_{\alpha}\bm{\gamma}\tau_{3}V_{\alpha},$
(48) $\displaystyle\bm{j}^{0}_{T}$
$\displaystyle=\sum_{\alpha>0}\bar{V}_{\alpha}i\gamma^{0}\bm{\gamma}V_{\alpha},\quad$
$\displaystyle\bm{j}^{0}_{tT}$
$\displaystyle=\sum_{\alpha>0}\bar{V}_{\alpha}i\gamma^{0}\bm{\gamma}\tau_{3}V_{\alpha}.$
(49)

In addition, the pairing density reads
$\kappa=\sum\limits_{\alpha>0}V^{\ast}_{\alpha}U^{T}_{\alpha}$. According to
the no-sea approximation, the sum of $\alpha$ runs over all the quasiparticle
states with positive energies in the Fermi sea. For a system with time-
reversal invariance, the spatial components of the currents $\bm{j}_{V}$,
$\bm{j}_{tV}$ in Eqs. (48) and (49) vanish. One should also note that the
$\sigma^{ij}$ ($i,j=1,2,$ and $3$) component of the tensor channel, the
pseudo-scalar and pseudo-vector channels give no contribution in infinite
nuclear matter and the ground state of finite nuclei in the present framework.

The center-of-mass (c.m.) correction energy $E_{\rm c.m.}$ should be taken
into account for finite nuclei due to the breaking of the translational
symmetry. It is estimated with a microscopic method Bender  _et al._ (2000);
Long  _et al._ (2004); Zhao  _et al._ (2009)

$\displaystyle E_{\rm c.m.}=-\frac{\langle\bm{P}^{2}_{\rm c.m.}\rangle}{2MA},$
(50)

where $\bm{P}_{\rm c.m.}=\sum_{i}^{A}\bm{p}_{i}$ is the total momentum in the
c.m. frame.

##  III Numerical Details

The localized RHFB equation (40) is solved in the space of harmonic oscillator
wave functions Nikšić  _et al._ (2014). The harmonic oscillator basis in the
spherical and Cartesian frame are respectively used for spherical and deformed
calculations, and the oscillator frequencies are taken as
$\hbar\omega_{0}=70A^{-1/3}$ MeV. In the present work, it includes 30 major
shells for the spherical calculations, and 24 major shells for the deformed
cases. It has been checked that for the spherical cases, by increasing the
major shells from $N_{f}=30$ to $N_{f}=32$, the binding energy, the charge
radius, and the neutron skin thickness of 208Pb change by $0.0006\%$,
$0.0004\%$, and $0.05\%$, respectively. For the deformed cases, the variation
of the binding energy of 240U is within $0.007\%$ from $N_{f}=24$ to
$N_{f}=26$.

In this work, the least-square fit with the penalty function

$\displaystyle\chi^{2}(\bm{p})=\sum^{N}_{i}\left(\frac{\mathcal{O}^{\rm
exp.}_{i}-\mathcal{O}^{\rm
cal.}_{i}(\bm{p})}{\Delta\mathcal{O}_{i}}\right)^{2},$ (51)

is employed to determine the parameters in the density functional. Here, the
vector $\bm{p}$ represents the ensemble of the parameters, $\mathcal{O}^{\rm
exp.}_{i}$ are the experimental data or pseudo-data from the _ab initio_
calculations, $\mathcal{O}^{\rm cal.}_{i}(\bm{p})$ are the predicted values
from the covariant DFT, and $\Delta\mathcal{O}_{i}$ denotes the adopted
weights for the selected observables. The multi-parameter fitting is carried
out by the _lmfit_ package in python with the Levenberg-Marquardt algorithm
Newville  _et al._ (2014).

##  IV Strategy of the parametrization

By taking into account the localized exchange terms with the Fierz
transformation, there are totally 11 coupling constants in the Lagrangian
density (2) to be determined, i.e., $\alpha_{S}$, $\alpha_{V}$, $\alpha_{tS}$,
$\alpha_{tV}$, $\alpha_{T}$, $\alpha_{tT}$, $\alpha_{PS}$, $\alpha_{tPS}$,
$\alpha_{PV}$, $\alpha_{tPV}$, and $\delta_{S}$. However, due to the relations
given in Eqs. (27a) - (27e), only 6 of them are free parameters, which are
$\alpha_{S}$, $\alpha_{V}$, $\alpha_{tS}$, $\alpha_{tV}$, $\alpha_{T}$, and
$\delta_{S}$. The first four coupling constants $\alpha_{S}$, $\alpha_{V}$,
$\alpha_{tS}$, and $\alpha_{tV}$ are assumed to be density-dependent,

$\displaystyle\alpha_{i}(\rho)=\alpha_{i}(\rho_{\rm
sat.})f_{i}(x),\quad\text{for}\quad i=S,V,tS,\text{ and }tV,$ (52)

where $\rho_{\rm sat.}$ is the saturation density and $x=\rho/\rho_{\rm
sat.}$. The density-dependent function $f_{i}(x)$ is taken as the following
ansatz Typel and Wolter (1999),

$\displaystyle
f_{i}(x)=a_{i}\frac{1+b_{i}(x+d_{i})^{2}}{1+c_{i}(x+d_{i})^{2}}.$ (53)

It should satisfy $f_{i}(1)=1$ by definition, and another constraint
$f^{\prime\prime}_{i}(0)=0$ is imposed as in Ref. Typel and Wolter (1999) to
further reduce the number of parameters. Therefore, three free parameters are
needed for the density dependence in each channel. Here, we take
$\alpha_{i}(\rho_{\rm sat.})$, $a_{i}$, and $d_{i}$ as the free parameters. In
addition, the coupling constants $\alpha_{T}$ and $\delta_{S}$ are assumed to
be density independent.

In the following, the parameters involved in the energy density functional
will be determined by using the bulk properties of infinite nuclear matter and
the ground state of finite nuclei step by step.

###  IV.1 Infinite nuclear matter

First, we determine the coupling constants in the isoscalar ($S$ and $V$)
channels as well as their density dependence, i.e., $\alpha_{S}(\rho)$ and
$\alpha_{V}(\rho)$. The 6 parameters are determined by the properties of the
symmetric nuclear matter, as listed in Table 1, which include the saturation
density, energy per nucleon at saturation, Dirac mass and compression modulus
at the saturation density, as well as the energies per nucleon at two
densities respectively below and above the saturation density Akmal  _et al._
(1998). These properties are solely determined by the isoscalar channels.

Table 1: The properties of symmetric nuclear matter used to determine the coupling constants in the isoscalar channels. They include the saturation density $\rho_{\rm sat.}$, energy per nucleon at saturation $E/A|_{\rho_{\rm sat.}}$, Dirac Mass $M^{*}_{D}$ and the compression modulus $K$ at the saturation density, as well as the energies per nucleon $E/A$ at two additional densities $\rho=0.04\text{ fm}^{-3}\text{ and }0.56\text{ fm}^{-3}$, which are taken from Ref. Akmal  _et al._ (1998).  Nuclear matter properties | Values
---|---
$\rho_{\rm sat.}$ (fm-3) | 0.152, 0.154, 0.156, 0.158, 0.160
$E/A|_{\rho_{\rm sat.}}$ (MeV) | -16.00, -16.02, -16.04, -16.06, -16.08, -16.10, -16.12, -16.14, -16.16
$M^{*}_{D}/M$ | 0.58, 0.70, 0.80
$K$ (MeV) | 230
$E/A|_{\rho=0.04\text{ fm}^{-3}}$ (MeV) | -6.48
$E/A|_{\rho=0.56\text{ fm}^{-3}}$ (MeV) | 34.39

The properties of the symmetric nuclear matter are known only in empirical
regions. For the saturation density $\rho_{\rm sat.}$ and the corresponding
energy per nucleon $E/A|_{\rho_{\rm sat.}}$, 5 and 9 values in the empirical
regions are respectively adopted for the fitting, since they have pronounced
influences on the properties of finite nuclei. The Dirac mass $M_{D}^{*}$ is
closely related to the spin-orbit energy splittings in finite nuclei. It is
usually known to be around $0.58M$ Nikšić  _et al._ (2008); Zhao  _et al._
(2010b). For the present density functional, however, a larger Dirac mass
becomes possible because the tensor couplings also contribute to the spin-
orbit potential. Therefore, three values $0.58$, $0.70$, and $0.80$ are chosen
for $M_{D}^{*}/M$ in the fitting. The compression modulus $K$ is taken as
$K=230$ MeV, which is consistent with the notion that the experimental
excitation energy of the isoscalar giant monopole resonances could be
reproduced with this incompressibility Garg and Colò (2018). In addition, the
equation of state (EOS) at two densities $\rho=0.04$ fm-3 and $\rho=0.56$ fm-3
obtained by the _ab initio_ variational calculations of Akmal, Pandharipande,
and Ravenhall (hereafter APR) Akmal  _et al._ (1998) is used to further
constrain the density dependence of $\alpha_{S}(\rho)$ and $\alpha_{V}(\rho)$.

By fitting to the selected target properties of the symmetric nuclear matter,
we have determined totally 135 sets of density-dependent coupling constants
for $\alpha_{S}(\rho)$ and $\alpha_{V}(\rho)$, which respectively correspond
to the 135 combinations for the fitting targets. The fitting precision is very
high, for instance, the relative deviations are within 0.005% for the target
saturation properties. The target energies per nucleon at density 0.04 fm-3
and 0.56 fm-3 are also reproduced rather well, as seen in Fig. 1, where the
calculated EOSs of symmetric nuclear matter given by the 135 sets of coupling
constants are depicted. Apart from the two fitted points in the EOS, one can
see that the other points in the EOS given by the _ab initio_ variational
calculations (APR) Akmal  _et al._ (1998) are also well reproduced by the
obtained coupling constants. Therefore, one should not exclude any set of the
coupling constants for $\alpha_{S}(\rho)$ and $\alpha_{V}(\rho)$ at this
stage.

Figure 1: (Color online). The equation of state for symmetric nuclear matter
as a function of the baryon density calculated with the 135 optimized sets of
the coupling constants $\alpha_{S}(\rho)$ and $\alpha_{V}(\rho)$ in comparison
with the predictions by the _ab initio_ variational calculations (APR) Akmal
_et al._ (1998) (solid circles). The two points at the density 0.04 fm-3 and
0.56 fm-3 shown with larger filled circles have been used in the determination
of the coupling constants.

In the second step, the density-dependent coupling constants
$\alpha_{tS}(\rho)$ in the isovector-scalar ($tS$) channel are determined by
the Dirac mass splitting of the pure neutron matter $\Delta
M^{*}_{D}=M^{*(p)}_{D}-M^{*(n)}_{D}$. This quantity depends only on the $tS$
channel as long as the coupling constants in the $S$ and $V$ channels are
fixed. As there are no experimental data for the Dirac mass splittings, the
results given by the Dirac-Brueckner-Hartree-Fock (DBHF) calculations van
Dalen  _et al._ (2007) are employed as pseudo-data for the fitting. Note that
in the DBHF calculations the scattering equations are solved with positive
energy states only. Recently, by including the negative energy states, the
scattering equations are solved in the full Dirac space for both symmetric and
asymmetric nuclear matter, namely the relativistic Brueckner Hartree-Fock
(RBHF) theory Wang  _et al._ (2021, 2022). The values from the DBHF
calculations are slightly larger than the RBHF results but qualitatively agree
with the RBHF results. In the future, more results from the RBHF calculations
can provide more and better guidance for the optimization of the density
functional Shen  _et al._ (2019). The density-dependent coupling constants
$\alpha_{tS}(\rho)$ are determined by freezing $\alpha_{S}(\rho)$ and
$\alpha_{V}(\rho)$ as determined in last step. In Fig. 2, the Dirac mass
splitting given by the 135 optimized sets of the coupling constants
$\alpha_{S}(\rho)$, $\alpha_{V}(\rho)$, and $\alpha_{tS}(\rho)$ are depicted
as a function of the nucleon density. One can see that the Dirac mass
splittings can be fitted quite nicely for all the 135 sets of the coupling
constants.

Figure 2: (Color online). Proton-neutron Dirac mass splitting as a function of
the baryon density in pure neutron matter given by the 135 optimized sets of
the coupling constants $\alpha_{S}(\rho)$, $\alpha_{V}(\rho)$, and
$\alpha_{tS}(\rho)$. The solid circles represent the results from the DBHF
calculations van Dalen  _et al._ (2007).

In the third step, the coupling constant at the saturation density in the
isovector-vector ($tV$) channel $\alpha_{tV}(\rho_{\rm sat.})$ is solely
determined by the symmetry energy $J$. In the present work, the symmetry
energy $J$ is taken as $J=33$ MeV, which is also consistent with results of
the _ab initio_ calculations Akmal  _et al._ (1998); Baldo  _et al._ (2004);
Wang  _et al._ (2022). Freezing $\alpha_{S}(\rho)$, $\alpha_{V}(\rho)$, and
$\alpha_{tS}(\rho)$ at the values found in steps 1 and 2, the coupling
constant $\alpha_{tV}(\rho_{sat.})$ is determined by reproducing $J=33$ MeV
within 0.001%. Note that due to the relatively large uncertainties of the
density-dependence of the symmetry energy, the density-dependence of
$\alpha_{tV}(\rho)$ cannot be well constrained, so they are left to be
determined by the observables of finite nuclei.

###  IV.2 Finite Nuclei

Apart from the density-dependence of $\alpha_{tV}(\rho)$, the coupling
constants $\alpha_{T}$ and $\delta_{S}$ are also optimized by fitting to the
bulk properties of spherical nuclei, because they only affect the properties
of finite nuclei.

Figure 3: (Color online). Nuclei whose binding energies (a) and charge radii
(b) are used in the fitting of the present density functionals.

The binding energies of 60 spherical nuclei Wang  _et al._ (2017) and the
charge radii of 44 ones Angeli and Marinova (2013) are selected as the
observables to be fitted, and they are shown in Figs. 3(a) and 3(b),
respectively. The two-neutron separation energies of 18O, 42Ca, 50Ca, 134Sn,
210Pb and the two-proton separation energies of 18Ne, 42Ti, 50Ti, 134Te, 210Po
are also fitted to achieve a reasonable description of the major shell gaps.
In addition, the empirical proton pairing gaps of 92Mo, 136Xe, 144Sm, and the
neutron ones of 122Sn, 124Sn, 208Pb obtained with the three-point formula are
also employed to constrain the paring strength $G$ in Eq. (32). The adopted
weights $\Delta\mathcal{O}_{i}$ for the binding energies, charge radii, two-
nucleon separation energies, and pairing gaps, are 1.0 MeV, 0.01 fm, 0.07 MeV,
and 0.05 MeV, respectively. Keeping the parameters as found with the nuclear
matter properties, all the remaining parameters including $a_{tV}$, $d_{tV}$,
$\alpha_{T}$, $\delta_{S}$, and $G$ are fitted by minimizing the $\chi^{2}$ in
Eq. (51) for the observables of finite nuclei.

Figure 4: (Color online). The root-mean-square (RMS) deviations for the
binding energies versus those for the charge radii given by the 135 optimized
density functionals. The shaded area denotes the region with the $rms$
deviations for the charge radius larger than 0.02 fm. The symbols and colors
represent different Dirac masses and saturation densities of the corresponding
density functionals, respectively. The new density functional PCF-PK1 is
represented with the star.

In such a way, 135 sets of the covariant density functional with localized
exchange terms have been obtained. In Fig. 4, the root-mean-square (RMS)
deviations for the binding energies versus those for the charge radii given by
the obtained 135 density functionals are depicted. The functionals with the
$rms$ deviations for the charge radii beyond $0.02$ fm, as shown in the shaded
region, are excluded since most of the popular covariant density functionals
in the market could describe the charge radii on the same level Long  _et al._
(2004); Lalazissis  _et al._ (2005); Zhao  _et al._ (2010b); Roca-Maza  _et
al._ (2011). For the rest of the density functionals, the one that has the
least $rms$ deviation for the binding energies, as marked with the star, is
chosen to be the most optimized density functional. Hereafter, this density
functional will be called PCF-PK1, whose parameters are listed in Table 2.

Note that not all parameters in Table 2 are independent, in fact, it includes
14 independent parameters in the mean-field channel, 10 of which are
determined by the empirical saturation properties and the pseudo-data obtained
from the _ab initio_ calculations of infinite nuclear matter Akmal  _et al._
(1998); van Dalen  _et al._ (2007). The remaining 4 parameters ($a_{tV}$,
$d_{tV}$, $\alpha_{T}$, and $\delta_{S}$), as well as the pairing strength $G$
in the pairing channel are fitted to the masses and charge radii of finite
nuclei.

Table 2: Parameters of the relativistic point-coupling functional PCF-PK1 with localized exchange terms. The nucleon mass $M$ is taken 939.0 MeV and the saturation density $\rho_{\rm sat.}$ is 0.156 fm-3. It includes 14 independent parameters, only 4 of which ($a_{tV}$, $d_{tV}$, $\alpha_{T}$, and $\delta_{S}$) are fitted to finite nuclei, and the other 10 are derived from an adjustment to empirical saturation properties and _ab initio_ calculations of infinite nuclear matter Akmal  _et al._ (1998); van Dalen  _et al._ (2007). The strength and width parameters of the separable pairing force are also listed in the last row. $i$ |  $\alpha_{i}(\rho_{\rm sat.})$ [fm2] | $a_{i}$ | $b_{i}$ | $c_{i}$ | $d_{i}$
---|---|---|---|---|---
S | -6.315494 | 1.425216 | 0.3185493 | 0.592876 | 0.7498207
V | 3.533254 | 0.8075938 | 0.0605804 | 0.03692602 | 3.004506
tS | -1.980515 | 2.328043 | 0.06178842 | 0.5704278 | 0.7644323
tV | 2.975891 | 2.546886 | 0.3459254 | 1.611987 | 0.4547352
$\alpha_{T}$ [fm2] | 3.373807 |  $\delta_{S}$ [fm4] | -0.665808 |  |
$G$ [MeV$\cdot$fm3] | 657.5419 |  $a$ [fm] | 0.644 |  |

In this work, calculations with the density functional PC-PK1 Zhao  _et al._
(2010b), DD-PC1 Nikšić  _et al._ (2008), and DD-ME$\delta$ Roca-Maza  _et al._
(2011) are also performed for comparison, in which the separable pairing force
with $G=728$ MeV$\cdot$fm-3 is used. Note that since the pairing force is not
the ones used in the determination of the three density functionals, we obtain
slightly different results for each density functional as compared to those
given in Refs. Nikšić  _et al._ (2008); Roca-Maza  _et al._ (2011); Zhao  _et
al._ (2010b).

Figure 5: (Color online). Deviations of the theoretical binding energies
obtained with PCF-PK1, PC-PK1, DD-PC1, and DD-ME$\delta$ from the experimental
values Wang  _et al._ (2017) for nuclei used in the fitting of PCF-PK1. The
shaded region corresponds to deviations within $\pm 2$ MeV. The $rms$
deviations $\Delta=\sqrt{\sum_{i}^{N}(\mathcal{O}_{i}^{\rm
exp.}-\mathcal{O}_{i}^{\rm cal.})^{2}/N}$ for each density functional are also
listed.  Figure 6: (Color online). Same as Fig. 5, but for charge radii. The
experimental data are taken from Angeli and Marinova (2013).

Differences between the theoretical binding energies given by PCF-PK1 and the
experimental ones for the selected nuclei are shown in Fig. 5 in comparison
with the results of PC-PK1, DD-PC1, and DD-ME$\delta$. It can be seen that
PCF-PK1 provides a good description for the binding energy, which is better
than DD-PC1 and DD-ME$\delta$, and even slightly better than PC-PK1. It should
be kept in mind that DD-PC1 is fitted to deformed nuclei, so it seriously
overestimates the binding energies of the spherical closed-shell nuclei, which
leads to a large $rms$ deviation. Apart from the binding energies, the charge
radii are also calculated with PCF-PK1 for the selected nuclei, and the
deviations from the experimental values are shown in Fig. 6 in comparison with
the ones given by PC-PK1, DD-PC1, and DD-ME$\delta$. It is found that the PCF-
PK1 gives a good quantitative description for charge radii, and it is
comparable to the results given by the other density functionals. In addition,
the selected pairing gaps, two-neutron separation energies, and two-proton
separation energies are also well reproduced by the PCF-PK1 functional, and
the corresponding $rms$ deviations are 0.15 MeV, 0.74 MeV, and 0.46 MeV,
respectively.

##  V Results and Discussions

In this section, the performance of the new density functional PCF-PK1 is
illustrated with the calculations for the bulk properties of nuclear matter,
the ground-state properties of spherical and deformed nuclei, and the Gamow-
Teller resonances (GTR). The main advantages of PCF-PK1 on the effective
nucleon mass for nuclear matter, the removal of spurious shell closures at
$Z=58$ and $92$, and the self-consistent descriptions for the Gamow-Teller
resonances will be focused.

###  V.1 Effective nucleon mass of nuclear matter

The saturation properties of the symmetric nuclear matter including the
saturation density $\rho_{\rm sat.}$, the binding energy per nucleon
$E/A|_{\rho_{\rm sat.}}$, the Dirac mass $M^{*}_{D}$, the Landau mass
$M^{*}_{L}$, the compression modulus $K$, the symmetry energy $J$ and its
slope $L$ are calculated with the PCF-PK1 density functional, and the obtained
results are listed in Table 3 in comparison with the ones given by the PC-PK1,
DD-PC1, DD-ME$\delta$, and PKO2 Long  _et al._ (2008b) functionals, as well as
the corresponding empirical values.

The main difference of PCF-PK1 from other functionals is the Dirac mass
$M^{*}_{D}/M$. The value of $M^{*}_{D}/M$ given by PCF-PK1 is 0.8, and the
corresponding Landau mass $M^{*}_{L}/M=[pdp/dE]_{p=p_{F}}/M$ is 0.85. They are
significantly larger than the ones given by other density functionals
including PKO2, where the Fock terms are taken into account. It is known that
there is a strong correlation between the Dirac mass $M^{*}_{D}$ and the
tensor couplings, which are responsible for the proper size of the spin-orbit
splitting in finite nuclei Furnstahl  _et al._ (1998). In the present work,
due to the inclusion of the tensor couplings, the Dirac mass is enlarged to
give reasonable spin-orbit splittings. Moreover, the large Landau mass given
by the PCF-PK1 implies a large single-particle level density around the Fermi
energy in finite nuclei. Note that the PKO2 provides a much larger Landau
mass, but a similar Dirac mass, than the other functionals except for PCF-PK1.
This is due to the momentum dependence of the self-energies introduced by the
finite-range exchange terms in PKO2, while in the present localized exchange
terms, there is no momentum dependence for the self-energies.

Table 3: The saturation properties for nuclear matter obtained by the PCF-PK1, PC-PK1, DD-PC1, DD-ME$\delta$, and PKO2 in comparison with the empirical data.  | Empirical | PCF-PK1 | PC-PK1 | DD-PC1 | DD-ME$\delta$ | PKO2
---|---|---|---|---|---|---
$\rho_{\rm sat}$ (fm-3) |  $0.155\pm 0.005$ Margueron  _et al._ (2018) | 0.156 | 0.154 | 0.152 | 0.152 | 0.151
$E/A|_{\rho_{\rm sat.}}$ (MeV) |  $-15.8\pm 0.3$ Margueron  _et al._ (2018) | -16.10 | -16.12 | -16.06 | -16.12 | -16.00
$M^{*}_{D}/M$ |  | 0.80 | 0.59 | 0.58 | 0.61 | 0.60
$M^{*}_{L}/M$ |  | 0.85 | 0.65 | 0.64 | 0.67 | 0.74
$K$ (MeV) |  $230\pm 20$ Margueron  _et al._ (2018) | 230 | 238 | 230 | 219 | 250
$J$ (MeV) |  $31.7\pm 3.2$ Oertel  _et al._ (2017) | 33.0 | 35.6 | 33.0 | 32.4 | 32.5
$L$ (MeV) |  $58.7\pm 28.1$ Oertel  _et al._ (2017) | 78.4 | 112.7 | 70.2 | 52.9 | 75.9

The PCF-PK1 results for the saturation properties $\rho_{\rm sat.}$,
$E/A|_{\rho_{\rm sat.}}$, $K$, and $J$ are excellently consistent with the
empirical values, since they are used to in the fitting procedure of the
functional. The slope $L$ of the symmetry energy at the saturation density is
an important quantity that characterizes the density dependence of the
symmetry energy. The PCF-PK1 predicts the slope $L$ to be $78.4$ MeV, which is
also in agreement with the corresponding empirical value Oertel  _et al._
(2017).

Figure 7: (Color online). Equation of states for the symmetric nuclear matter
and the pure neutron matter calculated with the PCF-PK1, PC-PK1, DD-PC1, and
DD-ME$\delta$. The _ab initio_ variational calculations Akmal  _et al._ (1998)
are shown as solid circles and diamonds for the symmetric nuclear matter and
the pure neutron matter, respectively.

Figure 7 depicts the EOSs of the symmetric nuclear matter and the pure neutron
matter calculated with the PCF-PK1, PC-PK1, DD-PC1, and DD-ME$\delta$ in
comparison with the _ab initio_ variational calculations (APR) Akmal  _et al._
(1998). All density functionals give similar results that are consistent with
the APR ones at lower densities, in particular, below the saturation density.
At higher densities, the DD-PC1 and DD-ME$\delta$ provide the EOSs for both
symmetric nuclear matter and pure neutron matter as soft as the APR ones. Note
that the EOSs of DD-ME$\delta$ is fitted to the Bruckner-Hartree-Fock (BHF)
calculations Baldo  _et al._ (2004) which are not much different from the APR
results. The DD-PC1 is not fitted to the APR EOS of pure neutron matter, but
it becomes soft at higher densities due to the exponential decrease of the
coupling constant of the $tV$ channel. The PC-PK1 gives much stiffer EOSs for
both symmetric nuclear matter and pure neutron matter at higher densities.
This is a common feature for most nonlinear density functionals, in which a
polynomial density dependence of the coupling constants is introduced.

The PCF-PK1 gives the EOS of the symmetric nuclear matter closed to the APR
one, because the APR EOS of the symmetric nuclear matter has been employed in
the fitting of the PCF-PK1 functional. The EOS of pure neutron matter given by
PCF-PK1 is consistent with the APR one at the densities below 0.2 fm-3, while
it is stiffer at higher densities. It should be noted that the existing
predictions for neutron matter vary largely among different theories due to
the unclear isospin $T=3/2$ component of the three-body force Pieper (2003);
Hammer  _et al._ (2013); Zhao and Gandolfi (2016). Therefore, the APR EOS of
pure neutron matter is not adopted in the fitting of the PCF-PK1 functional.

###  V.2 Removal of the spurious shell closures at $Z=58$ and $92$

In this part, we present single-particle levels, binding energies, two-proton
shell gaps, and neutron-skin thicknesses of selected spherical isotopes and
isotones obtained with the density functional PCF-PK1.

Figure 8: (Color online). The single-particle energies of 132Sn and 208Pb
calculated by PCF-PK1, PC-PK1, DD-PC1, and DD-ME$\delta$ in comparison with
the data Isakov  _et al._ (2002).

To examine the effect of the enhanced Dirac mass and Landau mass, the single-
particle spectrum calculated by PCF-PK1 are given in Fig. 8 for 132Sn and
208Pb, in comparison with those obtained by PC-PK1, DD-PC1, and DD-ME$\delta$.
The experimental values are extracted from the single-nucleon separation
energies or excitation energies Isakov  _et al._ (2002). In general, the PCF-
PK1 results are in good agreement with the experimental values around the
Fermi levels. Due to the larger Landau mass, the PCF-PK1 provides higher level
densities around the Fermi levels as compared with the results given by the
other density functionals, and this also makes the PCF-PK1 results closer to
the experimental values. Moreover, from the energy gap between the proton
$1g_{7/2}$ and $2d_{5/2}$ levels in 132Sn and the one between the proton
$1h_{9/2}$ and $2f_{7/2}$ levels in 208Pb, one can clearly see the spurious
shell closures at $Z=58$ and $Z=92$ predicted by all density functionals
except PCF-PK1. The removal of the spurious shell closures at $Z=58$ and
$Z=92$ can be also seen from the binding energies and the two-proton shell
gaps.

Figure 9: (Color online). Deviations of the binding energies of Ca, Ni, Sn,
and Pb isotopes obtained by PCF-PK1, PC-PK1, DD-PC1, and DD-ME$\delta$ from
the experimental values Wang  _et al._ (2017). The shaded regions represent
the deviations within $\pm 2$ MeV.

The binding energies of Ca, Ni, Sn, and Pb isotopes calculated with PCF-PK1
are depicted in Fig. 9 in terms of the deviations from the experimental data
Wang  _et al._ (2017) in comparison with the results of PC-PK1, DD-PC1, and
DD-ME$\delta$. The PCF-PK1 results are in good agreement with the experimental
data, especially for Ca isotopes, the discrepancies are less than 2 MeV. On
the neutron-deficient sides for the Ni, Sn, and Pb isotopes, the PCF-PK1
results underestimate the binding energies by up to 4 MeV. However, for these
nuclei, due to the soft potential energy surface, the dynamical correlation
energy could provide more binding and, thus, improve the descriptions Lu  _et
al._ (2015); Yang  _et al._ (2021).

Figure 10: (Color online). Same as Fig. 9, but for $N=20$, $50$, $82$, and
$126$ isotones.

In Fig. 10, the deviations from the experimental data for the binding energies
of $N=20$, $50$, $82$, and $126$ isotones calculated with the PCF-PK1 are
depicted in comparison with the results of PC-PK1, DD-PC1, and DD-ME$\delta$.
The PCF-PK1 well reproduces the binding energies of these isotones within 2
MeV for most nuclei. In particular for the $N=126$ isotones, the isospin
dependence of the binding energies along the isotonic chain is improved
greatly by the PCF-PK1. For the other density functionals, with the increasing
mass number, the binding energy is more and more overestimated, which leads to
the so-called spurious shell closure $Z=92$ at $A=218$.

Figure 11: (Color online). Calculated two-proton shell gaps of $N=82$ and
$126$ isotones as functions of the proton number $Z$ by the PCF-PK1, PC-PK1,
DD-PC1, and DD-ME$\delta$. The experimental values extracted from Ref. Wang
_et al._ (2017) are shown for comparison.

To further discuss the spurious shell closures, in Fig. 11, the two-proton
shell gaps for the $N=82$ and $N=126$ isotones given by PCF-PK1 are shown in
comparison with the results of PC-PK1, DD-PC1, and DD-ME$\delta$ as well as
the available experimental data Wang  _et al._ (2017). The two-proton shell
gaps is defined as

$\displaystyle\delta_{2p}(Z,N)=S_{2p}(Z,N)-S_{2p}(Z+2,N),$ (54)

where $S_{2p}(Z,N)$ is the two-proton separation energy. For the isotones with
$N=82$, except for the nucleus with $Z=58$, all density functionals can well
reproduce the experimental data. However, only the PCF-PK1 results agree with
the data at $Z=58$, and all the other density functionals overestimate the
shell gaps at $Z=58$. Similar behaviors can also be found along the $N=126$
isotones, where all density functionals predict a substantial shell gap at
$Z=92$ except PCF-PK1. It should be noted that a recent experiment on the
short-lived isotope 223Np disproves the existence of a $Z=92$ subshell closure
Sun  _et al._ (2017). Therefore, one could conclude that the spurious shell
closures at $Z=58$ and $Z=92$ can be well eliminated with the new density
functional PCF-PK1. It is also worthwhile to mention that these spurious shell
closures are also eliminated with the recent density functional DD-LZ1 without
Fock terms, which is guided by the pseudo-spin symmetry restoration Wei  _et
al._ (2020).

Figure 12: (Color online). Calculated neutron-skin thicknesses of Sn isotopes
(a) and 208Pb (b) by PCF-PK1, PC-PK1, DD-PC1, and DD-ME$\delta$ in comparison
with data, which are taken from Ref. Krasznahorkay  _et al._ (1994) for Sn
isotopes, and for 208Pb, deduced from parity-violating electron scattering
(PREX-2) PREX Collaboration  _et al._ (2021), dispersive optical model
analysis (DOM) Pruitt  _et al._ (2020), coherent pion photoproduction
($\gamma$, $\pi^{0}$) Crystal Ball at MAMI and A2 Collaboration  _et al._
(2014), electric dipole polarizability (p,p’) Roca-Maza  _et al._ (2013),
antiprotonic atoms ($\bar{p}$) Kłos  _et al._ (2007), and pygmy dipole
resonances (PDR) LAND Collaboration  _et al._ (2007).

We present in Fig. 12 the neutron-skin thicknesses of Sn isotopes and 208Pb
obtained by PCF-PK1, PC-PK1, DD-PC1, and DD-ME$\delta$ in comparison with the
experimental data. For Sn isotopes, similar to other density-dependent
functionals, the PCF-PK1 well describes the neutron-skin thicknesses within
the experimental errors. Note that the neutron-skin thickness is highly
related to the symmetry energy, and the nonlinear density functionals, such as
PC-PK1, usually provide higher symmetry energies and, thus, lead to larger
neutron-skin thicknesses.

The neutron-skin thickness of 208Pb has also attracted a lot of attentions
both theoretically and experimentally. Similar to the results of Sn isotopes,
the three density-dependent functionals, i.e., PCF-PK1, DD-PC1, and DD-
ME$\delta$, provides similar neutron-skin thicknesses for 208Pb, while the
nonlinear PC-PK1 provides a relatively larger value. However, it should be
mentioned that the corresponding experimental data have still a large
uncertainty. The data deduced from the dispersive optical model analysis
Pruitt  _et al._ (2020), coherent pion photoproduction Crystal Ball at MAMI
and A2 Collaboration  _et al._ (2014), electric dipole polarizability Roca-
Maza  _et al._ (2013), antiprotonic atoms Kłos  _et al._ (2007), and pygmy
dipole resonances (PDR) LAND Collaboration  _et al._ (2007) are in general
consistent with each other, but smaller than the recent data from the parity-
violating electron scattering (PREX-2). Recent studies find that the
theoretical predictions of the electric dipole polarizability that are
consistent with the PREX-2 measurement systematically overestimate the
corresponding values extracted from the direct measurements of the
distribution of electric dipole strength Piekarewicz (2021); Reinhard  _et
al._ (2021). There is yet no solution to this problem.

###  V.3 Self-consistent descriptions of the Gamow-Teller resonances

Figure 13: (Color online). Transition strength distributions of Gamow-Teller
resonances in 48Ca, 90Zr, and 208Pb as functions of the excitation energy
calculated with PCF-PK1, PKO2, and DD-ME2. A Lorentzian smearing parameter
$\Gamma=1$ MeV is used. The experimental peak energies Anderson  _et al._
(1985); Bainum  _et al._ (1980); Wakasa  _et al._ (1997); Horen  _et al._
(1980); Akimune  _et al._ (1995) are denoted with arrows. See text for more
details.

Apart from the ground-state properties, the Gamow-Teller resonances are also
studies with the new density functional PCF-PK1. In particular, the transition
strength distributions of Gamow-Teller resonances of 48Ca, 90Zr, and 208Pb are
calculated with PCF-PK1 with the random-phase approximation (RPA), and the
results are depicted in Fig. 13. Note that for the PCF-PK1 density functional,
the $tPS$ and $tPV$ channels appear in the particle-hole (_p-h_) residual
interaction with the strengths determined by Eqs. (27c) and (27e), though they
automatically vanish in the ground-state level. For comparison, the RPA
calculations based on the relativistic Hartree (RH) and RHF approaches have
also been performed with DD-ME2 Lalazissis  _et al._ (2005) and PKO2 Long  _et
al._ (2008b), respectively. For the RH calculations with DD-ME2, the pseudo-
vector pion-nucleon coupling with strength $f^{2}/4\pi=0.08$ is included in
the _p-h_ residual interaction and a Landau-Migdal term with the adjustable
strength $g^{\prime}=0.52$ has to be employed to describe the data Paar  _et
al._ (2008); Liang  _et al._ (2012c). For the RHF calculations with PKO2,
however, as discussed in Ref. Liang  _et al._ (2008b), the experimental data
can be well reproduced without any additional parameters. For the present
localized RHF calculations with PCF-PK1, one can also see that the observed
excitation energies for 48Ca, 90Zr, and 208Pb are reproduced nicely without
any adjustments. This demonstrates that, similar to the previous RHF and RPA
calculations Liang  _et al._ (2008b), a self-consistent description for the
Gamow-Teller resonances can be achieved with the present localized exchange
terms in PCF-PK1.

###  V.4 Ground-state properties of deformed nuclei

Figure 14: (Color online). Panels (c) and (d): Deviations of the calculated
binding energies from the experimental data Wang  _et al._ (2017) for Yb and U
isotopes obtained by PCF-PK1, PC-PK1, DD-PC1, and DD-ME$\delta$. The solid
(open) symbols denote the values with (without) the rotational correction
energies. Panel (c) and (d): Ground-state deformations of Yb and U isotopes
given by PCF-PK1, PC-PK1, DD-PC1, and DD-ME$\delta$ in comparison with the
data Pritychenko  _et al._ (2016).

The ground-state binding energies and quadrupole deformations for the well-
deformed Yb and U isotopes calculated with the density functional PCF-PK1, and
the results are compared with the ones by PC-PK1, DD-PC1, and DD-ME$\delta$ as
well as the data. In Figs. 14(a) and 14(b), the deviations of the calculated
binding energies from the experimental data Wang  _et al._ (2017) are
depicted. Without the rotational correction energies, the PCF-PK1
systematically underestimates the binding energies about 3—4 MeV for both Yb
and U isotopes; similar to PC-PK1. For DD-ME$\delta$, it also underestimates
the binding energies in most cases, but the deviations exhibit a clear isospin
dependence behavior. In particular for the lighter U isotopes, the DD-
ME$\delta$ even overestimates the binding energies. The DD-PC1 can well
describe the binding energies with the deviations less than 1 MeV, because
almost all nuclei calculated here were used in the fitting procedure of DD-
PC1.

One should keep in mind that for deformed nuclei, due to the rotational
symmetry breaking, the correction energies associated with the restoration of
the rotational symmetry should be considered in addition to the mean-field
energies. Therefore, similar to Ref. Zhao  _et al._ (2010a), here the
rotational correction energies are calculated with the cranking approximation
Girod and Grammaticos (1979). After taking into account the rotational
correction energies, the calculated results by PCF-PK1 and PC-PK1 are in good
agreement with the experimental binding energies for both Yb and U isotopes,
and the discrepancies are within 1 MeV. For DD-ME$\delta$, the deviations from
the data for the Yb isotopes are slightly larger because of the apparent
isospin dependence. Moreover, the rotational correction energies have not been
calculated for the nuclei from 230U to 236U with DD-ME$\delta$, because they
are predicted to be spherical nuclei in the calculations.

In Figs. 14(c) and 14(d), it depicts the quadrupole deformations of the ground
states for the Yb and U isotopes obtained by PCF-PK1, PC-PK1, DD-PC1, and DD-
ME$\delta$ in comparison with data Pritychenko  _et al._ (2016). Generally
speaking, the deformations are well reproduced by all density functionals
except for the DD-ME$\delta$ results of U isotopes from 230U to 236U. As seen
in Fig 11, the DD-ME$\delta$ predicts a large spurious shell closure at
$Z=92$, and this results in a spherical shape for the ground states of
230-236U.

##  VI Summary

In summary, a new density-dependent point-coupling covariant density
functional PCF-PK1 has been developed, in which the exchange terms of the
four-fermion terms are taken into account with the Fierz transformation. The
new density functional PCF-PK1 contains 14 independent parameters in the mean-
field channel, where 10 parameters are determined by the empirical saturation
properties of nuclear matter and pseudo-data obtained from the _ab initio_
calculations. The remain 4 parameters are optimized by fitting to the selected
observables of 60 spherical nuclei including the binding energies, charge
radii, and two-nucleon separation energies. The performance of PCF-PK1 is
illustrated with properties of the infinite nuclear matter and finite nuclei
including the ground-state properties and the Gamow-Teller resonances.

For nuclear matter, the most prominent feature of the PCF-PK1 results is the
large Dirac mass ($0.80M$) and Landau mass ($0.85M$), which is associated with
the high level densities around the Fermi surface in finite nuclei. It should
be noted that the large Dirac mass ($0.80M$) here does not worsen the
description of the spin-orbit splittings in finite nuclei due to the inclusion
of the tensor couplings in the functional, which also contribute to the spin-
orbit potential.

For the spherical nuclei, the PCF-PK1 results can reproduce the experimental
binding energies and charge radii quite well. The results of two-proton shell
gaps of the $N=82$ and $126$ isotones illustrate that the PCF-PK1 eliminates
the spurious shell closures at $Z=58$ and $92$, which commonly exist in many
relativistic density functionals.

Apart from the ground-states properties, the Gamow-Teller resonances of 48Ca,
90Zr, and 208Pb have also been calculated with the relativistic random-phase
approximation. Without any adjustable parameters, the PCF-PK1 reproduces the
experimental excitation energies quite well. This clearly demonstrates that a
self-consistent description for the Gamow-Teller resonances can be achieved
with present localized exchange terms.

For the deformed nuclei, the reliability of PCF-PK1 is illustrated by taking
Yb and U isotopes as examples. The quadrupole deformations are well reproduced
by the PCF-PK1. After taking into account the rotational correction energies,
the PCF-PK1 results reproduce the experimental binding energies for the Yb and
U isotopes within 1 MeV.

## Acknowledgments

The authors thank P. Ring for helpful discussions and suggestions and H.Z.
Liang for providing the relativistic RPA code. This work is supported by the
National Key R&D Program of China (Contracts No. 2018YFA0404400 and
2017YFE0116700), the National Natural Science Foundation of China (Grants No.
12070131001, 11875075, 11935003, 11975031, and 12141501), the China
Postdoctoral Science Foundation under Grant No. 2020M670013, the IBS grant
funded by the Korean government No. IBS-R031-D1 (Q.Z.), and the High-
performance Computing Platform of Peking University.

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