The long-standing controversy about the isospin dependence of the effective Dirac mass in ab initio calculations of asymmetric nuclear matter is clarified by solving the relativistic Brueckner-Hartree-Fock equations in the full Dirac space. The symmetry energy and its slope parameter at the saturation density are $E_{\text{sym}}\left({\rho }_0\right)=33.1$ MeV and $L=65.2$ MeV, in agreement with empirical and experimental values. Further applications predict the neutron star radius $R_{1.4M_{\odot }}\approx 12$ km and the maximum mass of a neutron star $M_{\text{max}}\le 2.4M_{\odot }$.

More and more halo nuclei or candidates have been identified or suggested in experiments in recent years. It was declared that the halo structure of ${}^{22}\mathrm{Al}$ is revealed by the large isospin asymmetry in ${}^{22}\mathrm{Si}/{}^{22}\mathrm{O}$ mirror Gamow-Teller transitions [ Phys. Rev. Lett. 125, 192503 (2020)]. We highlight that a significant mirror asymmetry already exists between wave functions of the likely unbound nucleus ${}^{22}\mathrm{Si}$ and the doubly magic nucleus ${}^{22}\mathrm{O}$, which largely explains the observed asymmetry in the transitions. Furthermore, these transitions involve only the $1^{+}$ excited states of the daughter nuclei ${}^{22}\mathrm{Al}$ and ${}^{22}\mathrm{F}$. The $1^{+}$ state of ${}^{22}\mathrm{Al}$ cannot be considered a halo state due to its proton-unbound nature. An analysis of the spin parity suggests that a weakly bound $2s_{1/2}$ valence proton in the ground state of ${}^{22}\mathrm{Al}$ is improbable. To investigate the shell structure for the ground state of ${}^{22}\mathrm{Al}$, we employ the state-of-the-art deformed and triaxial relativistic Hartree-Bogoliubov theories in continuum. We find that a small $s$-wave component of $5\%$ appears for the weakly bound valence proton in ${}^{22}\mathrm{Al}$ only when triaxial deformation is considered. While the examination of densities and rms radii indicates that this small $s$-wave component is insufficient to form a discernible proton halo in ${}^{22}\mathrm{Al}$, slightly larger $2s_{1/2}$ occupations have been reported in other recent theoretical results. The question of how many low-$\ell$ components are sufficient to form a proton halo in the presence of the Coulomb barrier remains open. Thus, future measurements of reaction or interaction cross sections and momentum distributions of breakup fragments are highly desirable to verify whether ${}^{22}\mathrm{Al}$ is a halo nucleus.

Nucleon effective masses in neutron-rich matter are studied with the relativistic Brueckner-Hartree-Fock (RBHF) theory in the full Dirac space. The neutron and proton effective masses for symmetric nuclear matter are 0.80 times rest mass, which agrees well with the empirical values. In neutron-rich matter, the effective mass of the neutron is found to be larger than that of the proton, and the neutron-proton effective mass splittings at the empirical saturation density are predicted as $0.187\alpha$ with $\alpha$ being the isospin asymmetry parameter. The result is compared to other ab initio calculations and is consistent with the constraints from the nuclear reaction and structure measurements, such as the nucleon-nucleus scattering, the giant resonances of ${}^{208}\mathrm{Pb}$, and the Hugenholtz–Van Hove theorem with systematics of nuclear symmetry energy and its slope. The predictions of the neutron-proton effective mass splitting from the RBHF theory in the full Dirac space might be helpful to constrain the isovector parameters in phenomenological density functionals.

Starting from the relativistic realistic nucleon-nucleon ($NN$) interactions, the relativistic Brueckner-Hartree-Fock (RBHF) theory in the full Dirac space is employed to study neutron star properties. First, the one-to-one correspondence relation for gravitational redshift and mass is established and used to infer the masses of isolated neutron stars by combining gravitational redshift measurements. Next, the ratio of the moment of inertia $I$ to mass times radius squared $M{R}^2$ as a function of the compactness $M/R$ is obtained, and is consistent with the universal relations in the literature. The moment of inertia for $1.338M_{\odot }$ pulsar PSR J0737$-$3039A $I_{1.338M_{\odot }}$ is predicted to be $1.356\times {10}^{45}, 1.381\times {10}^{45}$, and $1.407\times {10}^{45}\mathrm{g}{\mathrm{cm}}^2$ by the RBHF theory in the full Dirac space with $NN$ interactions Bonn A, B, and C, respectively. Finally, the quadrupole moment of neutron star is calculated under the slow-rotation and small-tidal-deformation approximation. The equations of state constructed by the RBHF theory in the full Dirac space, together with those by the projection method and momentum-independence approximation, conform to universal $I$-Love-$Q$ relations as well. By combing the tidal deformability from GW170817 and the universal relations from relativistic ab initio methods, the moment of inertia of a neutron star with 1.4 solar mass is also deduced as $I_{1.4M_{\odot }}=1.{22}_{-0.25}^{+0.40}\times {10}^{45}\mathrm{g}{\mathrm{cm}}^2$.