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arxiv: 2512.18549 · v2 · pith:SSAURH7Fnew · submitted 2025-12-21 · ⚛️ nucl-th · astro-ph.HE· cond-mat.quant-gas

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

Giorgio Almirante , Theodora Kaskitsi , Michael Urban This is my paper

Pith reviewed 2026-05-25 07:47 UTC · model grok-4.3

classification ⚛️ nucl-th astro-ph.HEcond-mat.quant-gas
keywords neutron starsinner crustsuperfluid fractionHartree-Fock-BogoliubovSkyrme energy density functionalBloch boundary conditionspulsar glitchespairing
0
0 comments X

The pith

Above 0.03 fm^{-3} density, over 90% of neutrons in the inner crust are superfluid, independent of interaction details.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper performs fully self-consistent Hartree-Fock-Bogoliubov calculations with Skyrme functionals and separable pairing to model the crystal lattice of nuclear clusters immersed in a neutron gas. It imposes Bloch boundary conditions and a time-independent relative flow to generate a phase gradient in the order parameter, from which the superfluid fraction is extracted via the resulting current in the superfluid rest frame. The calculations find that this fraction exceeds 90% at densities above 0.03 fm^{-3} and remains high across choices of interaction, cluster charge, and lattice geometry, approaching the hydrodynamic limit for strong pairing. This result implies the inner crust alone supplies enough superfluid angular momentum to account for observed pulsar glitches.

Core claim

Within the crystal phase of the inner crust, a relative flow between clusters and the surrounding neutron gas induces a phase in the complex order parameter; the neutron superfluid fraction extracted from the counterflow current exceeds 90% above 0.03 fm^{-3}, is only slightly below the linear-response BCS value, and is nearly independent of the Skyrme parametrization, cluster charge, and lattice geometry.

What carries the argument

Time-independent relative flow between clusters and neutron gas under Bloch boundary conditions in HFB calculations, which develops a phase in the pairing order parameter whose induced current defines the superfluid fraction.

If this is right

  • The inner crust alone can supply the superfluid angular momentum reservoir needed to explain pulsar glitches.
  • The superfluid fraction approaches the hydrodynamic limit when pairing is strong.
  • The result is robust against variations in Skyrme interaction, cluster charge, and lattice geometry.
  • The value lies only slightly below that obtained from linear response on top of BCS theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar Bloch-HFB setups could be applied to the pasta phases at higher densities to check whether the high superfluid fraction persists.
  • Glitch rise-time observations might be used to place bounds on the effective pairing strength inside the crust.
  • The near-independence from microscopic details suggests that simplified hydrodynamic models of the crust could be calibrated directly to the 90% figure for glitch modeling.

Load-bearing premise

A time-independent relative flow combined with Bloch boundary conditions produces a phase in the order parameter that accurately quantifies the superfluid fraction through the resulting current.

What would settle it

A calculation or simulation at densities above 0.03 fm^{-3} that yields a superfluid fraction below 80% when the relative flow is replaced by a fully time-dependent dynamical treatment would contradict the central result.

Figures

Figures reproduced from arXiv: 2512.18549 by Giorgio Almirante, Michael Urban, Theodora Kaskitsi.

Figure 1
Figure 1. Figure 1: FIG. 1. Single-particle ( [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Simple cubic lattice section at [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Neutron density ( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Simple cubic lattice section at [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Simple cubic lattice section at [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Neutron density ( [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Simple cubic lattice section at [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Superfluid fraction [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript presents fully self-consistent Hartree-Fock-Bogoliubov (HFB) calculations of the inner crust of neutron stars modeled as a crystal of nuclear clusters immersed in a superfluid neutron gas. Skyrme-type energy density functionals are used for the mean field together with a separable pairing interaction. Lattice periodicity is incorporated via Bloch boundary conditions. A time-independent relative flow is imposed between the clusters and the neutron gas, causing the complex order parameter to acquire a phase; the superfluid fraction is then extracted from the resulting neutron current evaluated in the superfluid rest frame. The central claim is that above 0.03 fm^{-3} more than 90% of the neutrons are effectively superfluid, with this fraction independent of the detailed choice of interaction, cluster charge, and lattice geometry. The result lies close to recent linear-response calculations on top of BCS and approaches the hydrodynamic limit for strong pairing, implying that the inner crust alone can supply a sufficient superfluid angular-momentum reservoir to account for pulsar glitches.

Significance. If the numerical result holds, the work is significant for neutron-star astrophysics. It supplies a quantitative, largely model-independent estimate of the superfluid reservoir in the inner crust and thereby strengthens the case that this layer can explain observed glitch sizes without additional contributions from the core. The methodological combination of fully self-consistent HFB with Bloch boundary conditions for the periodic lattice constitutes a clear advance over simpler approximations, and the reported approach to the hydrodynamic limit for strong pairing provides an internal consistency check. The independence from microscopic details is a noteworthy feature of the findings.

major comments (1)
  1. [Abstract and §3 (computational method)] Abstract and computational-setup description: the extraction of the superfluid fraction from the neutron current that arises after the order parameter acquires a phase under the imposed time-independent relative flow (with Bloch boundary conditions) is load-bearing for both the >90% value and the independence claim. The manuscript should supply an explicit test or derivation (e.g., comparison to the uniform-matter limit or variation of flow velocity) demonstrating that residual entrainment, lattice-induced artifacts, or pairing-channel dependence do not alter the current-to-density ratio across the scanned parameter space.
minor comments (2)
  1. [Figure captions] Figure captions should explicitly list the Skyrme parametrizations, cluster charges, and lattice geometries corresponding to each curve or symbol so that the independence statement can be verified at a glance.
  2. [Results discussion] A short reference or one-line derivation of the hydrodynamic-limit expression for the superfluid fraction would help readers confirm that the numerical results indeed approach this limit for strong pairing.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of our work and the constructive suggestion. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract and §3 (computational method)] Abstract and computational-setup description: the extraction of the superfluid fraction from the neutron current that arises after the order parameter acquires a phase under the imposed time-independent relative flow (with Bloch boundary conditions) is load-bearing for both the >90% value and the independence claim. The manuscript should supply an explicit test or derivation (e.g., comparison to the uniform-matter limit or variation of flow velocity) demonstrating that residual entrainment, lattice-induced artifacts, or pairing-channel dependence do not alter the current-to-density ratio across the scanned parameter space.

    Authors: We agree that an explicit validation of the extraction procedure would strengthen the manuscript. While the reported near-independence from the choice of Skyrme functional (which varies both the mean-field and pairing channels), cluster charge, and lattice geometry, together with the close agreement to linear-response BCS results and the approach to the hydrodynamic limit, already indicates that lattice artifacts and residual entrainment do not dominate, we acknowledge the value of a direct test. In the revised version we will add (i) a comparison to the uniform-matter limit obtained by suppressing the lattice modulation and (ii) a short check confirming that the current-to-density ratio remains stable under small variations of the imposed flow velocity within the linear regime. These additions will be placed in §3 or an appendix. revision: yes

Circularity Check

0 steps flagged

No circularity: superfluid fraction from direct numerical HFB current extraction

full rationale

The paper computes the superfluid fraction by solving the HFB equations with Skyrme functionals and separable pairing under Bloch boundary conditions, imposing a time-independent relative flow, allowing the order parameter to develop a phase, and extracting the fraction from the resulting neutron current in the superfluid rest frame. This is a standard numerical procedure using external functionals; the reported >90% value above 0.03 fm^{-3} (independent of interaction, charge, geometry) emerges from parameter scans rather than reducing by construction to any fitted input or self-citation chain within the paper's own equations. No self-definitional, fitted-input, or uniqueness-imported steps are present in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on established nuclear energy-density functionals and pairing models from prior literature; the novel element is the numerical implementation with flow and Bloch conditions. No new free parameters are introduced in this study.

axioms (1)
  • domain assumption Bloch boundary conditions accurately capture the periodicity of the nuclear lattice and the interplay between band structure and superfluidity.
    Invoked in the abstract to describe the crystal phase.

pith-pipeline@v0.9.0 · 5788 in / 1496 out tokens · 45940 ms · 2026-05-25T07:47:50.520393+00:00 · methodology

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