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

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.

💡 Research Summary

In the outermost region of a neutron star’s inner crust, nuclear matter is thought to arrange itself into a crystalline lattice of proton‑rich clusters immersed in a sea of superfluid neutrons, with a relativistic electron background ensuring charge neutrality and β‑equilibrium. The fraction of neutrons that participate in the superfluid component (the superfluid density ρS relative to the total neutron density ⟨ρn⟩) is a key microscopic parameter for models of pulsar glitches, because it determines how much angular momentum can be stored in the crustal superfluid and later transferred to the rigid crust.

Previous theoretical approaches have yielded conflicting results. Band‑theory calculations that ignore pairing predict a very strong entrainment effect, i.e., a small superfluid fraction, whereas hydrodynamic treatments that assume strong pairing predict a much larger fraction. Early attempts to include pairing within the Bardeen‑Cooper‑Schrieffer (BCS) approximation suggested that the pairing gap has little impact on the superfluid density, but more recent Hartree‑Fock‑Bogoliubov (HFB) studies in simplified geometries indicated that the superfluid fraction actually grows with the size of the gap. Moreover, linear‑response calculations based on BCS missed a “geometric contribution” that becomes important when the pairing field acquires a spatially varying phase in the presence of flow.

In this work the authors perform fully self‑consistent three‑dimensional HFB calculations for the crystal phase, treating neutrons as a superfluid and protons as a normal fluid. The mean‑field part is described by Skyrme energy‑density functionals (SLy4 and BSk24), while the pairing interaction is taken to be separable and non‑local, with a Gaussian form factor f(k)=exp(−k²/k0²). This choice yields a momentum‑dependent pairing gap Δ(k,k′) that can be expressed in Wigner phase space as Δ(Q,x)=f(Q)Δ0(x) e^{iϕ(x)}, where ϕ(x) is the phase of the order parameter. The lattice periodicity is enforced through Bloch boundary conditions; the single‑particle momentum k is split into a reciprocal‑lattice vector and a continuous Bloch momentum k_b confined to the first Brillouin zone. Consequently, the HFB Hamiltonian becomes a block matrix that must be diagonalized for each k_b, leading to a large set of neutron bands (thousands in practice) and their associated quasiparticle eigenvalues.

A relative flow between the clusters (normal component) and the neutron superfluid is introduced by a Galilean boost of the mean‑field Hamiltonian, h(v)=h−ℏ k·v δ_{kk′}. This generates a non‑zero momentum density j_n and, crucially, forces the pairing field to acquire a spatially varying phase ϕ(x). Within the two‑fluid (Andreev‑Bashkin) framework, the normal fluid velocity v_N coincides with the cluster (and proton) velocity, while the superfluid velocity v_S is given by v_S=(ℏ/2m)∇ϕ. By computing the neutron current from the self‑consistent HFB solution and using the relation ⟨ρ_n⟩=(⟨ρ_n⟩−ρ_S)v_N (valid when the superfluid phase is periodic and thus v_S=0), the superfluid density ρ_S can be extracted directly.

The calculations are performed for both simple‑cubic (SC) and body‑centered‑cubic (BCC) lattices, with a range of proton numbers Z and cell volumes V that satisfy β‑equilibrium (µ_n=µ_p+µ_e) and charge neutrality (ρ_e=⟨ρ_p⟩). Because protons occupy almost flat bands, Z can only take discrete values determined by the band structure, whereas neutrons populate a dense set of bands that depend sensitively on the Bloch momentum. The authors therefore explore several representative Z values and corresponding equilibrium cell sizes, finding that the resulting superfluid fraction is remarkably insensitive to these details.

The main quantitative result is that for baryon densities ρ_b≳0.03 fm⁻³ the superfluid fraction exceeds 90 % of the total neutron density, regardless of the Skyrme functional, the cluster charge, or the lattice geometry. This high fraction persists even when a finite relative flow is imposed, indicating that the system behaves essentially as a hydrodynamic superfluid in the strong‑pairing regime. As the pairing strength is increased, ρ_S approaches the Leggett upper bound, confirming that the HFB framework correctly captures the limit where Cooper pairs are tightly bound compared to the lattice spacing. The authors also show that the superfluid fraction obtained here is only slightly lower than that derived from linear‑response theory built on top of BCS, once the missing geometric contribution is included.

These findings have important astrophysical implications. A superfluid fraction above 90 % implies that the inner crust alone can store enough angular momentum (of order 10⁴⁴ g cm² s⁻¹) to account for the magnitude of observed pulsar glitches, without invoking additional reservoirs in the core. Moreover, the work demonstrates that a fully self‑consistent HFB treatment, which naturally incorporates the phase dynamics of the pairing field, is essential for reliable estimates of entrainment and superfluid density in non‑uniform nuclear matter. The methodology presented—combining realistic Skyrme mean fields, a separable pairing interaction, Bloch boundary conditions, and a Galilean boost—provides a robust platform for future studies of transport, collective modes, and vortex dynamics in neutron‑star crusts.