Superfluid fraction in the slab phase of the inner crust of neutron stars

An analysis of the slab phase as it is expected in the innermost layer of neutron-star crusts is performed within the Hartree-Fock-Bogoliubov framework. We take the periodicity of the slabs into account using Bloch boundary conditions, in order to well describe the interplay between the band structure and superfluidity. We introduce a relative flow between the slabs and the surrounding neutron gas in a time-independent way. This induces a nontrivial phase of the complex order parameter, leading to a counterflow between neutrons inside and outside the slabs. With the resulting current, we compute the actual neutron superfluid fraction. For the latter our results are slightly larger than previous ones obtained in normal band theory, suggesting that normal band theory overestimates the entrainment effect.

Figure 1

Number densities $\rho$ for neutrons and protons at different chemical potentials ${\mu }_n$ and cell extensions $L$.

Figure 2

Mean-field potential $U$ for neutrons and protons at different cell extensions $L$ for ${\mu }_n=12$ MeV.

Figure 3

Pairing gap $\mathrm{\Delta }$ for neutrons at different chemical potentials ${\mu }_n$ and cell extensions $L$. The HFB gap is taken at the local Fermi momentum as in Eq. (41) while the LDA one is obtained by solving the BCS gap equation for uniform matter (see text for details).

Figure 4

Phase $\varphi$ of the neutron pairing gap computed as in Eq. (45) at different chemical potentials ${\mu }_n$ and slab velocities $v_N$ for $L=24\mathrm{fm}$.

Figure 5

Particle currents $\mathit{\rho }$ for neutrons and protons computed as in Eq. (34) for different chemical potentials ${\mu }_n$ and $L=24\mathrm{fm}$ at slab velocity $v_N/c={10}^{-3}$.

Figure 6

Neutron superfluid fraction (blue dashes) and Leggett upper bound (orange line) as functions of the average gap obtained with an artificially scaled strength of the pairing interaction. The explored values for the pairing coupling constant are in the interval $g/g_{\text{phys}}\in [0.79,1.8]$, while the other parameters ${\mu }_n=12$ MeV, $L=24$ fm, and $v_N/c={10}^{-3}$ are kept constant.