Background: The inner crust of neutron stars consists of a Coulomb lattice of neutron-rich nuclei, immersed in a sea of superfluid neutrons with background relativistic electron gas. A proper quantum-mechanical treatment for such a system under a periodic potential is the band theory of solids. The effect of band structure on the effective mass of dripped neutrons, the so-called entrainment effect, is currently in a debatable situation, and it has been highly desired to develop a microscopic nuclear band theory taking into account neutron superfluidity in a fully self-consistent manner.

Purpose: The main purpose of the present work is twofold: (1) to develop a formalism of the time-dependent self-consistent band theory, taking fully into account nuclear superfluidity, based on time-dependent density-functional theory (TDDFT) extended for superfluid systems, and (2) to quantify the effects of band structure and superfluidity on the entrainment phenomenon, applying the formalism to the slab phase of the inner crust of neutron stars.

Methods: The fully self-consistent time-dependent band theory, proposed in a previous work [Phys. Rev. C 105, 045807 (2022)], is extended to superfluid systems. To this end, a superfluid TDDFT with a local treatment of pairing, known as time-dependent superfluid local density approximation, is formulated in the coordinate space with a Skyrme-type energy density functional, adopting the Bloch's boundary condition. A real-time method is employed to extract the collective masses of a slab and of protons, which in turn quantify the conduction neutron number density and the neutron effective mass, i.e., the entrainment effect.

Results: Static calculations have been performed for a range of baryon number density ($n_{\text{b}}=0.04\text{–}0.07{\mathrm{fm}}^{-3}$) under the $\beta$-equilibrium condition with and without superfluidity, for various interslab spacings. From the results, we find that the system gains energy through the formation of Cooper pairs for all densities examined, which supports the existence of superfluidity in the inner crust of cold neutron stars. From a response of the system to an external potential, we dynamically extract the collective masses of a slab and of protons immersed in neutron superfluid. The obtained results show the collective mass of a slab is substantially reduced by 57.5%–82.5% for $n_{\text{b}}=0.04\text{–}0.07{\mathrm{fm}}^{-3}$, which corresponds to an enhancement of conduction neutron number density and, thus, to a reduction of the neutron effective mass, which we call the anti-entrainment effect. A comparison of the results with and without superfluidity reveals that superfluidity slightly enhances the anti-entrainment effects for the slab phase of neutron-star matter. We discuss a novel phenomenon associated with superfluidity, that is, quasiparticle resonances in the inner crust, which are absent in normal systems.

Conclusions: Our fully self-consistent, microscopic, superfluid band theory calculations based on (TD)DFT showed that the effective mass of dripped neutrons is reduced by about 20%–40% for $n_{\text{b}}=0.04\text{–}0.07{\mathrm{fm}}^{-3}$ because of the band-structure effects, and superfluidity slightly $\mathit{enhances}$ the reduction.

• Received 1 August 2023
• Accepted 7 May 2024

DOI:https://doi.org/10.1103/PhysRevC.109.065804