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

TL;DR

The paper addresses the neutron-star inner-crust crystal phase by performing fully self-consistent 3D Hartree-Fock-Bogoliubov calculations with Bloch boundary conditions to quantify the neutron superfluid fraction in a periodic lattice. Using Skyrme functionals and a non-local separable pairing interaction, it demonstrates that the neutron superfluid density remains large, with a fraction $ ho_S/ig angle ho_nig angle \gtrsim 0.9$ above densities of $0.03 ext{ fm}^{-3}$, approaching the hydrodynamic limit and contradicting strong entrainment predicted by some band-structure studies. The work compares full HFB results with linear-response and hydrodynamic models, highlighting the essential role of the geometric contribution to the superfluid density in reconciling approaches. The findings imply the inner crust alone can supply a sufficient superfluid angular-momentum reservoir to explain pulsar glitches, with robust results against lattice geometry, pairing strength, and mean-field functional choices, thereby refining the understanding of neutron-star crust dynamics.

Abstract

In the most extended layer of the inner crust of neutron stars, nuclear matter is believed to form a crystal of clusters immersed in a superfluid neutron gas. Here we analyze this phase of matter within fully self-consistent Hartree-Fock-Bogoliubov calculations using Skyrme-type energy density functionals for the mean field and a separable interaction in the pairing channel. The periodicity of the lattice is taken into account using Bloch boundary conditions, in order to describe the interplay between band structure and superfluidity. A relative flow between the clusters and the surrounding neutron gas is introduced in a time-independent way. As a consequence, the complex order parameter develops a phase, and in the rest frame of the superfluid one finds a counterflow between neutrons inside and outside the clusters. The neutron superfluid fraction is computed from the resulting current. Our results indicate that at densities above 0.03 fm$^{-3}$, more than 90% of the neutrons are effectively superfluid, independently of the detailed choice of the interaction, cluster charge, and lattice geometry. This fraction is only slightly lower than the one obtained recently within linear response theory on top of the Bardeen-Cooper-Schrieffer approximation, and it approaches the hydrodynamic limit for strong pairing. As a consequence, it is likely that the inner crust alone can provide a sufficient superfluid angular momentum reservoir to explain pulsar glitches.

Figures (8)

• Figure 1: Single-particle (top) and quasi-particle (bottom) band structure for SLy4 in the simple cubic lattice for $\rho_b=0.033\;\text{fm}^3$ and $L=26\;\text{fm}$. $\xi_\alpha=\epsilon_\alpha-\mu$ are HF eigenvalues, first $70$ bands around the Fermi energy are plotted. $E_\alpha$ are HFB eigenvalues, first $90$ positive bands are plotted. The horizontal axis is the Bloch momentum along the high symmetrical path of the simple cubic Brillouin zone Setyawan10. In the quasi-particle band structure one can clearly see the pairing gap, in this case its average value is about $2\;\text{MeV}$.
• Figure 2: Neutron density (colors) and coplanar components of the velocity field (arrows) in three sections through the body-centered cubic cell at $x=x_0$ (position of the central cluster, bottom), $x=x_0+L/4$ (plane between the clusters, center) and $x=x_0+L/2$ (cell boundary, top), computed for SLy4, proton number $Z=20$, chemical potential $\mu_n=9\;\text{MeV}$, baryon density $\rho_b=0.033\;\text{fm}^{-3}$, bcc cell extension $L=33 \;\text{fm}$, velocity $\bm{\mathrm{v}}_N/c=10^{-3}\hat{\bm{\mathrm{z}}}$.
• Figure 3: Simple cubic lattice section at $y=z=L/2$ of neutron and proton densities. Results for interactions SLy4 (top) and BSk24 (bottom) are shown for the same $\rho_b=0.043\;\text{fm}^{-3}$, cell extension $L=24\;\text{fm}$ for $Z=20$ and $L=30\;\text{fm}$ for $Z=40$. The corresponding neutron chemical potentials are $\mu_n^{\text{SLy}}=10\;\text{MeV}$ and $\mu_n^{\text{BSk}}=9.3\;\text{MeV}$.
• Figure 4: Simple cubic lattice section at $y=z=L/2$ of neutron and proton mean-field potentials $U$ (top) and microscopic effective masses $m^*$ (bottom), for the same parameters as in Fig. \ref{['fig:densities3']} for the case $L=24 \;\text{fm}$, $Z=20$.
• Figure 5: Simple cubic lattice section at $y=z=L/2$ of the neutron pairing gap at the local Fermi momentum as a function of the pair c.o.m. position, for the same parameters as in Fig. \ref{['fig:meanfield3']}.