pith. sign in

arxiv: 2403.07766 · v1 · pith:YVQBSEURnew · submitted 2024-03-12 · ⚛️ nucl-th · astro-ph.HE

Gapless superfluidity in neutron stars: Thermal properties

Valentin Allard , Nicolas Chamel This is my paper

Pith reviewed 2026-05-24 03:06 UTC · model grok-4.3

classification ⚛️ nucl-th astro-ph.HE
keywords gapless superfluidityneutron starsspecific heatnuclear energy-density functionalquasiparticle spectrumsuperfluid velocitycooling simulationsnucleon superfluidity
0
0 comments X

The pith

Self-consistent nuclear energy-density functional theory predicts a gapless superfluid regime for nucleons in neutron stars.

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

The paper examines how temperature and currents affect superfluid neutrons and superconducting protons inside mature neutron stars using self-consistent time-dependent nuclear energy-density functional theory. It shows that this approach allows nucleons to remain superfluid, with a finite order parameter, while the quasiparticle excitation spectrum has no energy gap. In this regime the specific heat loses its exponential suppression at low temperatures and can reach values comparable to the normal phase. The specific heat depends essentially only on a dimensionless effective superfluid velocity, which permits simple analytical approximations suitable for cooling calculations.

Core claim

Within the self-consistent time-dependent nuclear energy-density functional theory, nucleons can be superfluid with a finite order parameter even when the quasiparticle energy spectrum has no gap. The resulting specific heat at low temperatures is comparable to that of the normal phase rather than being exponentially suppressed. This behavior is essentially universal when expressed in terms of a dimensionless effective superfluid velocity, allowing general approximate analytical formulas for the specific heat.

What carries the argument

Gapless superfluid regime in which the order parameter stays finite while the quasiparticle spectrum lacks an energy gap, governed by a dimensionless effective superfluid velocity.

If this is right

  • Specific heat in the gapless regime is not exponentially suppressed at low temperatures.
  • Specific heat values can become comparable to those in the normal phase.
  • The dependence on the dimensionless effective superfluid velocity makes the specific-heat behavior essentially universal.
  • Approximate analytical formulas derived for the specific heat can be directly inserted into neutron-star cooling simulations.

Where Pith is reading between the lines

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

  • Cooling curves for young neutron stars may show faster temperature decline than expected from gapped superfluid models.
  • Similar gapless regimes could appear in other flowing fermionic superfluids once the same functional-theory approach is applied.
  • The analytical formulas reduce computational cost when modeling thermal evolution across a wide range of stellar densities and velocities.

Load-bearing premise

The self-consistent time-dependent nuclear energy-density functional theory remains quantitatively reliable for the gapless regime inside mature neutron stars at the densities and temperatures relevant to cooling.

What would settle it

A measurement or simulation showing that the low-temperature specific heat of superfluid nuclear matter either follows or deviates sharply from the gapless-regime formulas derived from the dimensionless effective superfluid velocity.

Figures

Figures reproduced from arXiv: 2403.07766 by Nicolas Chamel, Valentin Allard.

Figure 1
Figure 1. Figure 1: FIG. 1. Density of quasiparticle states [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Same as Figure 1 for effective superfluid velocities ranging from Landau’s velocity [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Low-temperature specific heat (69) in subcritical regime (normalized by the specific heat [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Relative deviation (in percents) between the specific heat given by the “exact” expres [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Low-temperature specific heat in gapless regime (86) (normalized by the specific heat in [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Relative deviation (in percents) between the specific heat given by the “exact” expres [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
read the original abstract

The interior of mature neutron stars is expected to contain superfluid neutrons and superconducting protons. The influence of temperature and currents on superfluid properties is studied within the self-consistent time-dependent nuclear energy-density functional theory. We find that this theory predicts the existence of a regime in which nucleons are superfluid (the order parameter remains finite) even though the energy spectrum of quasiparticle excitations exhibits no gap. We show that the disappearance of the gap leads to a specific heat that is not exponentially suppressed at low temperatures as in the BCS regime but can be comparable to that in the normal phase. Introducing some dimensionless effective superfluid velocity, we show that the behavior of the specific heat is essentially universal and we derive general approximate analytical formulas for applications to neutron-star cooling simulations.

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

2 major / 2 minor

Summary. The manuscript applies self-consistent time-dependent nuclear energy-density functional theory to examine how temperature and superfluid currents affect nucleon superfluidity in neutron-star interiors. It reports a gapless superfluid regime in which the pairing order parameter remains finite while the quasiparticle excitation spectrum has a vanishing minimum, yielding a specific heat that is not exponentially suppressed at low T and can approach normal-phase values. A dimensionless effective superfluid velocity is introduced, and approximate analytical formulas for the specific heat are derived for use in cooling simulations.

Significance. If the central numerical predictions hold, the work supplies a concrete mechanism by which superfluidity can persist without an excitation gap, directly affecting the thermal evolution of mature neutron stars. The reported universality with respect to the effective velocity and the closed-form approximations constitute a practical strength for phenomenological modeling. Credit is due for the explicit construction of the effective-velocity scaling and the attempt to connect the EDF solutions to observable cooling curves.

major comments (2)
  1. [Numerical results and discussion of the gapless regime] The central claim that a finite order parameter coexists with a gapless spectrum rests on the quantitative accuracy of the mean-field EDF once the quasiparticle minimum reaches zero. No benchmark against ab-initio pairing calculations, fluctuation-corrected theories, or known stability limits in the gapless regime is presented; this directly undermines in the reported specific-heat values being comparable to the normal phase.
  2. [Analytical approximations and applications to cooling] The derivation of the approximate analytical formulas for the specific heat (presented after the introduction of the dimensionless velocity) assumes the EDF spectrum remains reliable throughout the gapless window. Without an explicit check that dynamical instabilities or beyond-mean-field corrections remain negligible when the gap closes under superflow, the formulas cannot be regarded as robust for cooling simulations.
minor comments (2)
  1. [Abstract] The abstract states the central prediction but does not indicate which specific EDF parametrization is employed; this information should be added for reproducibility.
  2. [Figures] Figure captions and axis labels for the specific-heat plots should explicitly state the range of the effective superfluid velocity used, to allow direct comparison with the analytical formulas.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the potential implications of our results. We address the two major comments below, clarifying the scope of the mean-field EDF calculations while acknowledging their limitations. Revisions have been made to strengthen the discussion of these limitations.

read point-by-point responses

  1. Referee: [Numerical results and discussion of the gapless regime] The central claim that a finite order parameter coexists with a gapless spectrum rests on the quantitative accuracy of the mean-field EDF once the quasiparticle minimum reaches zero. No benchmark against ab-initio pairing calculations, fluctuation-corrected theories, or known stability limits in the gapless regime is presented; this directly undermines in the reported specific-heat values being comparable to the normal phase.

    Authors: Our manuscript is confined to the predictions of self-consistent time-dependent nuclear EDF theory, a mean-field framework routinely applied to superfluidity in neutron-star matter. Direct benchmarks against ab-initio pairing calculations or fluctuation-corrected theories for the gapless regime under superflow are not available in the literature, and performing them lies outside the present scope. The Skyrme EDF employed has been validated against finite-nucleus data and is expected to capture the qualitative onset of gapless superfluidity. In the revised manuscript we have added an explicit paragraph in the discussion section stating that the reported specific-heat values are mean-field predictions and that beyond-mean-field effects could modify the quantitative results near the gap-closing point. revision: partial

  2. Referee: [Analytical approximations and applications to cooling] The derivation of the approximate analytical formulas for the specific heat (presented after the introduction of the dimensionless velocity) assumes the EDF spectrum remains reliable throughout the gapless window. Without an explicit check that dynamical instabilities or beyond-mean-field corrections remain negligible when the gap closes under superflow, the formulas cannot be regarded as robust for cooling simulations.

    Authors: The approximate analytical formulas are obtained by parametrizing the numerically computed EDF specific heat versus the dimensionless effective velocity; they are therefore intended for use inside the same mean-field model. Our time-dependent simulations remain stable for the velocities and temperatures examined, providing indirect support for the absence of immediate dynamical instabilities within the explored window. We agree that a comprehensive stability analysis against all possible modes would be desirable. The revised manuscript now includes a dedicated sentence specifying the assumptions and the intended applicability range of the formulas for cooling simulations. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation from independent EDF equations

full rationale

The paper obtains its gapless superfluid regime and specific-heat formulas by solving the self-consistent time-dependent nuclear energy-density functional equations numerically and then deriving approximate analytic expressions from those solutions. No step reduces a prediction to a fitted input by construction, no self-citation is invoked as a uniqueness theorem or to smuggle an ansatz, and the central claim is not a renaming of a known result. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are listed. The central claim rests on the domain assumption that the chosen EDF accurately captures pairing and quasiparticle spectra at neutron-star densities.

axioms (1)
  • domain assumption Self-consistent time-dependent nuclear energy-density functional theory is an adequate microscopic description for superfluid neutrons in the interior of mature neutron stars.
    The entire prediction is obtained within this framework.

pith-pipeline@v0.9.0 · 5652 in / 1141 out tokens · 26146 ms · 2026-05-24T03:06:19.739350+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

52 extracted references · 52 canonical work pages

  1. [1]

    Setting ∆ q = 0 in Eq

    Normal phase In the normal phase, nucleons obviously remain at rest in the normal fluid frame therefore we must have Vq = 0. Setting ∆ q = 0 in Eq. (67) and substituting in Eq. (69) leads to the classical result (see, e.g., Ref. [44] in the context of metals) 16 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 / (0) q 0 1 2 3 4q( , q(T = 0, q))/ (q) N (0) 0 Lq (BCS) 0.25 ...

    work page

  2. [2]

    (65) in Eq

    Superfluid phase: general case Substituting Eq. (65) in Eq. (69) yields the general expression of the specific heat in the superfluid phase for arbitrary effective superfluid velocities: c(q) V (T ≪ T (0) cq , Vq) ≈ kBD(q) N (0) 4ℏkF qβVq Z +∞ β(∆q−ℏkF qVq) dx s x β + ℏkF qVq 2 − ∆2 q x2 sech2 x 2 − kBD(q) N (0) 4ℏkF qβVq Z +∞ β(∆q+ℏkF qVq) dx s x β − ℏkF...

    work page

  3. [3]

    (69) using Eq

    Superfluid phase: BCS limit Vq = 0 In the superfluid phase and in the absence of superflows, the specific heat can be easily calculated from Eq. (69) using Eq. (67) leading to: c(q) V (T ≪ T (0) cq , Vq = 0) ≈ 3 √ 2 π3/2 T (0) cq T π eγ !5/2 exp − T (0) cq T π eγ ! c(q) N (T ) . (73) The exponential suppression of the specific heat at low temperatures is ...

    work page

  4. [4]

    with effective superfluid velocities below Landau’s critical velocity

    Superfluid phase: subcritical regime 0 ≤ Vq < VLq Let us now consider the more general situation of stationary mixtures in presence of superflows in the subcritical regime, i.e. with effective superfluid velocities below Landau’s critical velocity. In this case, we notice that δ+ q > 0 and δ− q < 0. For sufficiently low tem- peratures T ≪ T (0) cq , we ha...

    work page

  5. [5]

    Setting Vq = VLq = ∆(0) q /(ℏkF q), along with ∆q = ∆(0) q , in Eqs

    Superfluid phase: onset of gapless regime Vq = VLq The specific heat is amenable to an analytical approximation when the effective superfluid velocity is equal to Landau’s critical velocity. Setting Vq = VLq = ∆(0) q /(ℏkF q), along with ∆q = ∆(0) q , in Eqs. (71) and (72) leads to c(q) V /c(q) N ≈ 3 4π2 Z +∞ 0 dx x2 sech2 x 2 vuut eγ π T T (0) cq x + 1 !...

    work page

  6. [6]

    Superfluid phase: gapless regime VLq < Vq ≤ V(0) cq For effective superfluid velocities larger than VLq, ∆ q no longer remains constant but decreases with increasing Vq and vanishes at the critical velocity V(0) cq . The specific heat (71) 22 can be equivalently written as c(q) V /c(q) N ≈ 3 4π2 Z +∞ −δ− q dx vuut eγ π T T (0) cq VLq Vq x + 1 !2 − ∆q ∆(0)...

    work page

  7. [7]

    Chamel, J

    N. Chamel, J. Astrophys. Astron. 38, 43 (2017)

    work page 2017

  8. [8]

    A. Y. Potekhin, J. A. Pons, and D. Page, Space Sci. Rev. 191, 239 (2015). 26

    work page 2015

  9. [9]

    A. B. Migdal, Nucl. Phys. 13, 655 (1959)

    work page 1959

  10. [10]

    Sedrakian and J

    A. Sedrakian and J. W. Clark, Eur. Phys. J. A 55, 167 (2019)

    work page 2019

  11. [11]

    Antonopoulou, B

    D. Antonopoulou, B. Haskell, and C. M. Espinoza, Rep. Prog. Phys. 85, 126901 (2022)

    work page 2022

  12. [12]

    D. Page, M. Prakash, J. M. Lattimer, and A. W. Steiner, Phys. Rev. Lett. 106, 081101 (2011)

    work page 2011

  13. [13]

    P. S. Shternin, D. G. Yakovlev, C. O. Heinke, W. C. G. Ho, and D. J. Patnaude, Mon. Not. R. Astron. Soc. 412, L108 (2011)

    work page 2011

  14. [14]

    K. G. Elshamouty, C. O. Heinke, G. R. Sivakoff, W. C. G. Ho, P. S. Shternin, D. G. Yakovlev, D. J. Patnaude, and L. David, Astrophys. J. 777, 22 (2013)

    work page 2013

  15. [15]

    Posselt and G

    B. Posselt and G. G. Pavlov, Astrophys. J. 864, 135 (2018)

    work page 2018

  16. [16]

    M. J. P. Wijngaarden, W. C. G. Ho, P. Chang, C. O. Heinke, D. Page, M. Beznogov, and D. J. Patnaude, Mon. Not. R. Astron. Soc. 484, 974 (2019)

    work page 2019

  17. [17]

    W. C. G. Ho, Y. Zhao, C. O. Heinke, D. L. Kaplan, P. S. Shternin, and M. J. P. Wijngaarden, Mon. Not. R. Astron. Soc. 506, 5015 (2021), ISSN 0035-8711

    work page 2021

  18. [18]

    Posselt and G

    B. Posselt and G. G. Pavlov, Astrophys. J. 932, 83 (2022)

    work page 2022

  19. [19]

    T. Alm, G. R¨ opke, A. Sedrakian, and F. Weber, Nucl. Phys. A 604, 491 (1996)

    work page 1996

  20. [20]

    R. N. Manchester, G. B. Hobbs, A. Teoh, and M. Hobbs, Astron. J. 129, 1993 (2005)

    work page 1993

  21. [21]

    M. E. Gusakov and E. M. Kantor, Mon. Not. R. Astron. Soc. 428, L26 (2013)

    work page 2013

  22. [22]

    L. B. Leinson, Mon. Not. R. Astron. Soc. 470, 3374 (2017)

    work page 2017

  23. [23]

    L. B. Leinson, Mon. Not. R. Astron. Soc. 479, 3778 (2018)

    work page 2018

  24. [24]

    Chamel and V

    N. Chamel and V. Allard, Phys. Rev. C 100, 065801 (2019)

    work page 2019

  25. [25]

    Allard and N

    V. Allard and N. Chamel, Phys. Rev. C 103, 025804 (2021)

    work page 2021

  26. [26]

    Allard and N

    V. Allard and N. Chamel, Universe 7, 470 (2021), ISSN 2218-1997, URL https://www.mdpi. com/2218-1997/7/12/470

    work page 2021

  27. [27]

    R. H. Parmenter, RCA Rev. (United States) 23, 323 (1962)

    work page 1962

  28. [28]

    Vollhardt and K

    D. Vollhardt and K. Maki, J. Low Temp. Phys. 31, 457 (1978)

    work page 1978

  29. [29]

    Bulgac, Physica Status Solidi B Basic Research 256, 1800592 (2019)

    A. Bulgac, Physica Status Solidi B Basic Research 256, 1800592 (2019)

    work page 2019

  30. [30]

    Blaizot and G

    J. Blaizot and G. Ribka, Quantum Theory of Finite Systems (MIT Press, Cambridge, MA, 1986)

    work page 1986

  31. [31]

    Schunck, ed., Energy Density Functional Methods for Atomic Nuclei , 2053-2563 (IOP Pub- lishing, 2019), ISBN 978-0-7503-1422-0, URL https://dx.doi.org/10.1088/2053-2563/ aae0ed

    N. Schunck, ed., Energy Density Functional Methods for Atomic Nuclei , 2053-2563 (IOP Pub- lishing, 2019), ISBN 978-0-7503-1422-0, URL https://dx.doi.org/10.1088/2053-2563/ aae0ed. 27

    work page doi:10.1088/2053-2563/ 2053

  32. [32]

    J. W. Negele and D. Vautherin, Phys. Rev. C 5, 1472 (1972)

    work page 1972

  33. [33]

    R. J. Furnstahl, Eur. Phys. J. A 56, 85 (2020)

    work page 2020

  34. [34]

    Bender, P.-H

    M. Bender, P.-H. Heenen, and P.-G. Reinhard, Reviews of Modern Physics 75, 121 (2003)

    work page 2003

  35. [35]

    Dobaczewski, H

    J. Dobaczewski, H. Flocard, and J. Treiner, Nucl. Phys. A 422, 103 (1984)

    work page 1984

  36. [36]

    de Gennes, Superconductivity of Metalls and Alloys (Westview Press, 1999)

    P.-G. de Gennes, Superconductivity of Metalls and Alloys (Westview Press, 1999)

    work page 1999

  37. [37]

    Bulgac and Y

    A. Bulgac and Y. Yu, Phys. Rev. Lett. 88, 042504 (2002)

    work page 2002

  38. [38]

    Chamel, J

    N. Chamel, J. Margueron, and E. Khan, Phys. Rev. C 79, 012801(R) (2009)

    work page 2009

  39. [39]

    Chamel, S

    N. Chamel, S. Goriely, J. M. Pearson, and M. Onsi, Phys. Rev. C 81, 045804 (2010)

    work page 2010

  40. [40]

    Goriely, N

    S. Goriely, N. Chamel, and J. M. Pearson, Phys. Rev. C 88, 024308 (2013)

    work page 2013

  41. [41]

    J. M. Pearson, N. Chamel, A. Y. Potekhin, A. F. Fantina, C. Ducoin, A. K. Dutta, and S. Goriely, Mon. Not. R. Astron. Soc. 481, 2994 (2018)

    work page 2018

  42. [42]

    J. M. Pearson, N. Chamel, and A. Y. Potekhin, Phys. Rev. C 101, 015802 (2020)

    work page 2020

  43. [43]

    J. M. Pearson and N. Chamel, Phys. Rev. C 105, 015803 (2022)

    work page 2022

  44. [44]

    Perot, N

    L. Perot, N. Chamel, and A. Sourie, Phys. Rev. C 100, 035801 (2019)

    work page 2019

  45. [45]

    L. G. Cao, U. Lombardo, C. W. Shen, and N. V. Giai, Phys. Rev. C 73, 014313 (2006)

    work page 2006

  46. [46]

    Chamel, Phys

    N. Chamel, Phys. Rev. C 82, 014313 (2010)

    work page 2010

  47. [47]

    Bardeen, L

    J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957)

    work page 1957

  48. [48]

    Combescot, M

    R. Combescot, M. Y. Kagan, and S. Stringari, Phys. Rev. A 74, 042717 (2006)

    work page 2006

  49. [49]

    Bardeen, Rev

    J. Bardeen, Rev. Mod. Phys. 34, 667 (1962)

    work page 1962

  50. [50]

    A. A. Abrikosov, Fundamentals of the theory of metals (North Holland, 1988)

    work page 1988

  51. [51]

    Pastore, N

    A. Pastore, N. Chamel, and J. Margueron, Mon. Not. R. Astron. Soc. 448, 1887 (2015)

    work page 2015

  52. [52]

    Allard and N

    V. Allard and N. Chamel, Phys. Rev. Lett. (in press) (2024). 28

    work page 2024