Spin symmetry energy and equation of state of spin-polarized neutron star matter

The equation of state (EOS) of spin-polarized nuclear matter (NM) is studied within the Hartree-Fock (HF) formalism using the realistic density-dependent nucleon-nucleon interaction. With a nonzero fraction $\mathrm{\Delta }$ of spin-polarized baryons in NM, the spin- and spin-isospin dependent parts of the HF energy density give rise to the spin symmetry energy that behaves in about the same manner as the isospin symmetry energy, widely discussed in the literature as the nuclear symmetry energy. The present HF study shows a strong correlation between the spin symmetry energy and nuclear symmetry energy over the whole range of baryon densities. The important contribution of the spin symmetry energy to the EOS of the spin-polarized NM is found to be comparable with that of the nuclear symmetry energy to the EOS of the isospin-polarized or asymmetric (neutron-rich) NM. Based on the HF energy density, the EOS of the spin-polarized ($\beta$-stable) $npe\mu$ matter is obtained for the determination of the macroscopic properties of neutron stars (NS). A realistic density dependence of the spin-polarized fraction $\mathrm{\Delta }$ has been suggested to explore the impact of the spin symmetry energy on the gravitational mass $M$ and radius $R$, as well as the tidal deformability of NS. Based on the empirical constrains inferred from a coherent Bayesian analysis of gravitational wave signals of the NS merger GW170817 and the observed masses of the heaviest pulsars, the present study shows the strong impact of the spin symmetry energy $W$, nuclear symmetry energy $S$, and nuclear incompressibility $K$ on the EOS of nucleonic matter in magnetar.

Figure 1

Energy per baryon of the spin-unpolarized (a) symmetric NM and (b) pure neutron matter given by the HF calculation (1) using four versions of the CDM3Yn interaction, in comparison with the BHF results (squares) [23]. The circles and stars are results of the ab initio calculations by Akmal, Pandharipande, and Ravenhall (APR) [24] and the microscopic Monte Carlo (MMC) calculation by Gandolfi et al. [25], respectively.

Figure 2

The same as Fig. 1 but for the fully spin-polarized (a) symmetric NM and (b) pure neutron matter, in comparison with the BHF results (squares) [23].

Figure 3

(a) The HF results obtained with the CDM3Y8 interaction for the nuclear symmetry energy (7) at different spin polarizations of baryons, in comparison with the ab initio results [24, 25] obtained with $\mathrm{\Delta }=0$; the shaded region is the constraint by the HI fragmentation data [31, 32]; the vertical bars are given at 90% confidence level by the Bayesian analysis [33]. (b) The spin symmetry energy (8) given by the CDM3Y8 interaction at different isospin polarizations $\delta$, in comparison with the BHF results for the fully spin-polarized neutron matter [23].

Figure 4

(a) The HF results obtained with four versions of the CDM3Yn interaction for the slope $L$ of the nuclear symmetry energy (9) at different spin polarizations $\mathrm{\Delta }$ of baryons in comparison with the empirical range suggested by Li et al. [34] at the 68% confidence level (the shaded region). (b) The slope $L_{\mathrm{s}}$ of the spin symmetry energy (10) at different isospin asymmetries $\delta$.

Figure 5

(a) Energy per baryon (11) of the fully spin-polarized neutron matter given by the HF calculation using the CDM3Y8 interaction (solid line), and that given by the BHF calculation (squares) [23]. The energy per baryon $E_0$ of the spin-unpolarized symmetric NM, nuclear symmetry energy $S_0$ and spin symmetry energy $W_0$ are shown as dotted, dashed, and dash-dotted lines, respectively. (b) The total baryonic pressure of the fully spin-polarized neutron matter (solid line) in terms of the contributions obtained separately from $E_0$, $S_0$, and $W_0$.

Figure 6

(a) The isospin asymmetry $\delta$ and (b) proton fraction $x_p$ of $\beta$-stable $npe\mu$ matter at different spin polarizations $\mathrm{\Delta }$ of baryons given by the HF calculation (12, 13, 14, 15) using the CDM3Y8 interaction. The circles are $\delta$ and $x_p$ values obtained at the maximum central densities $n_{\mathrm{c}}$, and the thin lines are the corresponding DU thresholds (17).

Figure 7

The total pressure (18) inside the core of NS at different spin polarizations $\mathrm{\Delta }$ given by the HF calculation using the CDM3Y8 interaction and over the range of the mass-energy (${\rho }_{\mathrm{m}}$) and baryon-number ($n_{\mathrm{b}}$) densities. The dark and light shaded regions are the empirical constraints given by the “spectral” EOS inferred from the Bayesian analysis of the GW170817 data at the 50% and 90% confidence levels, respectively [15]. The circles are $P(n_{\mathrm{c}},\mathrm{\Delta })$ values at the corresponding maximum central densities $n_{\mathrm{c}}$.

Figure 8

The tidal deformability parameter (21) given by the EOS of the spin-polarized NS matter obtained with different $\mathrm{\Delta }$ values from the HF calculation using the CDM3Y8 interaction. The vertical bar is the empirical $\mathrm{\Lambda }$ value for NS with $M=1.4M_{\odot }$ inferred from the Bayesian analysis of the GW170817 data at the 90% confidence level [15], and the circles are $\mathrm{\Lambda }$ values obtained at the corresponding maximum central densities $n_{\mathrm{c}}$.

Figure 9

The gravitational mass of NS versus its radius given by the EOS of the spin-polarized NS matter obtained with different $\mathrm{\Delta }$ values from the HF calculation using the CDM3Y8 interaction. The colored contours are the GW170817 constraint for NS with mass $M=1.4M_{\odot }$ [15], and the circles are $M\text{−}R$ values calculated at the corresponding maximum central densities $n_{\mathrm{c}}$. The shaded areas are the observed masses of the second PSR $\mathrm{J}0348+0432$ [40] and millisecond PSR $\mathrm{J}0740+6620$ [41].

Figure 10

Density dependence of the spin polarization $\mathrm{\Delta }$ of baryons inside the core of NS that mimics the distribution of the magnetic field in NS obtained by Fujisawa and Kisaka using the Green's function relaxation method [8], with different maximum $\mathrm{\Delta }$ values.

Figure 11

The same as Fig. 9 but obtained with the density-dependent spin polarization $\mathrm{\Delta }\left(n_{\mathrm{b}}\right)$ of different strengths shown in Fig. 10.

Figure 12

The same as Fig. 7 but obtained with four versions of the CDM3Yn density dependent interaction (2, 3, 4, 5) that are associated with four different values of the nuclear incompressibility $K$. (a) The results obtained for the spin-unpolarized NS matter. (b) The results obtained for the partially spin polarized NS matter with the maximum $\mathrm{\Delta }=0.6$ of the spin-polarization strength shown in Fig. 10.

Figure 13

The same as Fig. 9 but obtained with four versions of the CDM3Yn density dependent interaction (2, 3, 4, 5) that are associated with four different values of the nuclear incompressibility $K$. (a) The results obtained for the spin-unpolarized NS matter. (b) The results obtained for the partially spin-polarized NS matter with the maximum $\mathrm{\Delta }=0.6$ of the spin-polarized fraction of baryons shown in Fig. 10.