Generalized skyrmion crystals with applications to neutron stars

In this article we study properties of isospin asymmetric nuclear matter in the generalized ${\mathcal{L}}_{0246}$-Skyrme model. This is achieved by canonically quantizing the isospin collective degrees of freedom of the recently found skyrmion multiwall crystal. We obtain, for the first time, an equation of state from the Skyrme model which interpolates between infinite isospin asymmetric nuclear matter and finite isospin symmetric atomic nuclei. This enables us to describe neutron stars with crusts within the Skyrme framework. Furthermore, we observe that the symmetry energy tends to a constant value at zero density, which can be identified with the asymmetry coefficient in the semiempirical mass formula for atomic nuclei. The symmetry energy also reveals a cusp in its structure below the nuclear saturation point $n_0$ at $n_{*}\sim 3n_0/4$. This cusp density point $n_{*}$ can be interpreted as the nuclear density whereby the infinite crystalline multiwall configuration undergoes a phase transition to a finite isolated multiwall configuration. Both of these observations are observed to be generic features of skyrmion crystals that tend asymptotically to somewhat isolated skyrmion configurations in the zero density limit. We find that the resulting neutron stars from our study agree quite well with recent NICER/LIGO observational data.

Figure 1

${\mathcal{L}}_{0246}$-Skyrme multiwall crystal at a fixed baryon density $n_B<n_0$. The isobaryon density is depicted in (a) and isosurface plots of the $\sigma$ field, where the vacuum $(\sigma =+0.9)$ is colored red and the antivacuum $(\sigma =-0.9)$ blue, are shown in (b).

Figure 2

The classical static energy per baryon $M_B/B$ as a function of the nuclear density $n_B$. The nuclear density at which at the cusp in the symmetry energy appears is labeled by $n_{*}$. This corresponds to the density at which the infinite crystalline multiwall solution begins transitioning to an isolated multiwall configuration.

Figure 3

The nuclear symmetry energy $S_N$ as a function of the baryon density $n_B$, exhibiting the cusp structure detailed in the text at $n_{*}\sim 3n_0/4$.

Figure 4

Plot of the particle number densities $n_i$ as functions of the baryon density $n_B$. The particle number densities are normalized such that the total number density is ${\sum }_i{n}_i=1$. The transition between isospin asymmetric infinite matter and symmetric finite matter at the cusp density $n_{*}$ is now manifest.

Figure 5

Comparison between the classical isospin symmetric crystal, the pure neutron crystal, and the $\beta$-equilibrated asymmetric crystal with the Maxwell construction applied.

Figure 6

Mass-radius curves for neutron stars obtained from the multiwall crystal EoS with (blue curve) and without (red curve) the Maxwell construction. The maximal mass $M_{\max }$ obtained from the MC multiwall crystal EoS is also shown.

Figure 7

Plots at $M_{\max }=2.0971M_{\odot }$ of the pressure $p$, energy density $\rho$, metric function $B(r)$ and equations of state $\rho =\rho (p)$. The blue curve is for the crystal EoS with the Maxwell construction applied, removing any negative pressure from the system, whereas the red curve is for the “true” crystal EoS.