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Microscopic description of axisymmetric vortices in $^{3}P_{2}$ superfluids

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Pith's one-line read The paper claims that in a strong magnetic field the o vortex is the most stable axisymmetric vortex in a $^3P_2$ superfluid and that it binds two zero-energy Majorana fermions in its core.

desk verdict First microscopic Eilenberger-plus-BdG calculation for 3P2 vortices; the axisymmetric o-vortex/Majorana result is solid, but the neutron-star stability conclusion still waits on nonaxisymmetric vortices. read the letter →

arxiv 1908.06215 v2 pith:TF5ETY5W submitted 2019-08-17 cond-mat.supr-con astro-ph.HEnucl-th

Yusuke Masaki , Takeshi Mizushima , Muneto Nitta This is my paper · ORCID
classification cond-mat.supr-conastro-ph.HEnucl-th
keywords 3P2superfluidneutronstarvortexMajoranazeromodeBogoliubov-deGennesequationEilenbergerospin-tripletpairing
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that quantized vortices in the spin-triplet $^3P_2$ superfluid thought to exist inside neutron-star cores should be described by a fully microscopic theory, not just by Ginzburg–Landau order-parameter profiles. Solving the Eilenberger equation self-consistently and then the Bogoliubov–de Gennes equation, the authors identify a family of axisymmetric vortex solutions whose stability depends on magnetic-field strength. Their central result is that in a strong magnetic field the so-called o vortex is the most stable axisymmetric configuration, and it binds two zero-energy Majorana fermions in its core, protected by a discrete magnetic rotation symmetry called P3. The paper also finds that the magnetization profile around the core computed microscopically differs sharply from earlier order-parameter-only estimates. If correct, this identifies the vortex microphysics that governs neutron-star cores in strong fields and gives a concrete astronomical setting where topologically protected Majorana zero modes could occur.

What carries the argument

The work combines the quasiclassical Eilenberger equation with the self-consistent gap equation to determine the axisymmetric order parameter $A(R) = \sum_{M=-2}^{2} \gamma_M(\rho) e^{i(\kappa-M)\theta} \Gamma_M$, and the Bogoliubov–de Gennes equation to obtain quasiparticle eigenenergies and core magnetization. The o vortex is defined as the axisymmetric configuration that preserves the P3 magnetic $\pi$-rotation symmetry, meaning the components $\gamma_0$ and $\gamma_{\pm 2}$ are real and $\gamma_{\pm 1} = 0$; this symmetry, combined with particle-hole conjugation, defines a chiral operator $\check{\Gamma}$ and a one-dimensional winding number $w_{\rm 1d} = 2$, which guarantees two zero-energy Majorana modes at $k_3 = 0$. Stability is compared through the Luttinger–Ward free-energy functional, and the BdG spin density is contrasted with the Ginzburg–Landau order-parameter formula.

What would settle it

Solve the order-parameter equations without imposing the axisymmetric ansatz of Eq. (18), allowing elliptic or double-core deformations, and compare free energies at fields $V_Z \approx 0.9$–$1.5\,T_c$; if any such nonaxisymmetric solution has lower free energy than the D4-BN-o2 vortex, the most-stable claim fails. A direct numerical check that the two $\ell = 0$ modes at $k_3 = 0$ remain exactly degenerate under arbitrary small P3-breaking perturbations would also test the protection claimed for the zero modes.

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Extended reading notes

Core claim

Within axisymmetric vortex configurations in $^3P_2$ superfluids, the paper demonstrates that the o vortex—the configuration preserving all three discrete symmetries P1, P2, and P3—is energetically the most stable vortex in the presence of a strong magnetic field, specifically in the D4-BN phase. The o vortex contains two spin-degenerate zero-energy Majorana bound states at $k_3 = 0$, protected by the P3 symmetry, which yields a one-dimensional winding number $w_{\rm 1d} = 2$. In contrast, the v vortex spontaneously breaks P3, so its would-be zero modes mix and split away from zero energy. The paper further shows that self-consistent microscopic order parameters include induced components with $M = \pm 2$ that are absent from the earlier Ginzburg–Landau ansatz, and that the local spin density computed from Bogoliubov–de Gennes quasiparticles is finite at the o-vortex core where the order-parameter-only picture gives zero. These results constitute the first microscopic calculation of a single vortex in a multicomponent superfluid with a finite Zeeman field.

Load-bearing premise

The calculations only consider vortices that keep circular symmetry around the vortex line, with boundary conditions fixed by the uniform UN or D4-BN phases; if a nonaxisymmetric vortex has lower free energy, the o vortex would not be the global most-stable state and the P3 symmetry protecting the two Majorana zero modes could be broken.

Editorial extensions

If this is right

  • In strong magnetic fields of the size relevant to magnetars, the stable vortex state is the o vortex, so each vortex line there should carry a topologically protected pair of Majorana zero modes.
  • The v vortex, which has no protected zero modes, is stable only at weaker fields, so the presence or absence of Majorana fermions depends on magnetic-field strength through the vortex configuration.
  • Core magnetization computed from fermionic quasiparticles differs drastically from Ginzburg–Landau estimates, so neutron-star cooling and vortex dynamics calculations that use order-parameter-only profiles miss a substantial contribution.
  • At sufficiently strong fields the $M = \pm 1$ components of the v vortex vanish, the vortex continuously recovers the P3 symmetry, and the two zero modes reappear, making the strongly magnetized regime the natural place to look for these modes.
  • If these vortices exist in neutron-star cores, the fermion bound states contribute to the spectral-flow force and therefore to vortex unpinning and glitch dynamics, not only to the static structure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves nonaxisymmetric vortices open; if the symmetric-traceless tensor structure of the $^3P_2$ order parameter indeed suppresses the double-core vortex that destabilizes the o vortex in $^3$He-B, then the o vortex with its two Majorana modes could be the global ground-state vortex in magnetar interiors, not merely the best axisymmetric one.
  • Two Majorana zero modes per vortex could support non-Abelian statistics beyond the single-mode case; the paper raises this question but does not answer it, so a concrete next step is to compute braiding properties of the o-vortex pair.
  • A testable extension is to feed the self-consistent vortex profiles and BdG spectra into a vortex-dynamics calculation and ask whether the spectral-flow force they produce changes predicted glitch relaxation timescales compared with Ginzburg–Landau-based estimates.
  • The same microscopic machinery could be applied to the cyclic and ferromagnetic $^3P_2$ phases, where 1/3-quantized non-Abelian vortices have been predicted, to check whether those cores also host zero-energy fermions.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper presents a microscopic study of singly quantized axisymmetric vortices in a $^{3}P_{2}$ superfluid, modeled by a zero-range spin-triplet $p$-wave interaction with Zeeman coupling. The authors solve the Eilenberger equation and the gap equation self-consistently for several boundary conditions corresponding to UN, D2-BN, and D4-BN uniform phases, classify the resulting vortices as $o$ or $v$ according to whether the magnetic $\pi$-rotation symmetry $P_{3}$ is preserved, and compare their free energies at $T=0.4T_{c}$ with and without a magnetic field. They then solve the Bogoliubov-de Gennes (BdG) equation on the self-consistent order-parameter profiles to obtain quasiparticle spectra, spin-polarized bound-state branches, and local spin densities. The main results are that within the axisymmetric ansatz the $o$ vortex becomes the most stable configuration in a strong magnetic field, that it hosts two spin-degenerate zero-energy Majorana bound states at $k_{3}=0$ protected by a $P_{3}$-symmetry winding number, and that the BdG magnetization profiles differ substantially from the GL-order-parameter-based estimates.

Significance. The result, if it holds, is a meaningful advance: it appears to be the first microscopic (quasiclassical plus BdG) description of vortex cores in $^{3}P_{2}$ superfluids, and it identifies a concrete strong-field regime in which topologically protected Majorana zero modes can exist in the cores of neutron-star vortices. The paper earns credit for doing genuine self-consistent Eilenberger calculations, for checking the zero modes both numerically and through the $P_{3}$ winding-number argument without parameter fitting, and for comparing BdG magnetizations against earlier GL estimates. The main caveat is the explicit restriction to axisymmetric configurations in Eq. (18); the paper itself lists nonaxisymmetric vortices as important future work. Therefore the stability and zero-mode statements are rigorously established only within that subspace, and the physical relevance to vortex matter in neutron stars remains conditional on the absence of a lower-energy nonaxisymmetric vortex such as the double-core vortex known in $^{3}$He-B.

minor comments (4)
  1. [Sec. III.A, Eq. (39)] The BdG spin density in Eq. (39) is computed with an energy cutoff of $15T_{c}$, but no convergence check with respect to this cutoff is reported; please add a sentence confirming convergence or a short convergence test, since the continuum contribution is important for the total magnetization profile.
  2. [Sec. IV and reference list] The citation number 82 in Sec. IV corresponds to an unnumbered footnote placed at the end of the reference list; this should be formatted as a proper numbered reference, or the in-text citation should be changed to the footnote marker, to avoid confusion.
  3. [Sec. IV] The word 'nonaxisymetric' in the first paragraph of Sec. IV should be corrected to 'nonaxisymmetric'.
  4. [Abstract and Sec. IV] The abstract and conclusion are properly careful about the axisymmetric restriction, but the neutron-star framing in the introduction suggests a global stability claim; adding one sentence in the introduction stating explicitly that global stability against nonaxisymmetric perturbations is not addressed would make the scope fully transparent.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the vortex free-energy comparison and Majorana zero-mode analysis are genuinely self-consistent, with the P3 argument applied to computed solutions rather than used to define the result.

full rationale

The paper's central claims are (i) the free-energy comparison of axisymmetric vortex solutions at finite magnetic field and (ii) the existence of two zero-energy Majorana states in the o-vortex core. Neither claim reduces to its inputs. The order parameters are obtained by self-consistently solving the Eilenberger equation (11) with the gap equation (15) under the axisymmetry condition (18), and the free energy is then computed from the Luttinger–Ward functional in Eqs. (25)-(28); this is a variational calculation within an explicitly stated ansatz, not a fit to the target result. The o-vortex is defined by the P3 symmetry with gamma_{M=±1}=0 (Sec. III.A), and the zero modes follow from the explicitly computed winding number w1d=2 in Eqs. (41)-(42), based on the chiral symmetry of the BdG Hamiltonian; the numerical BdG spectra in Fig. 4 and Fig. 7(f) independently show two ℓ=0 zero modes. The topological argument cites Refs. [64] (Teo–Kane) and [65] (Tsutsumi et al.), which are external, and Ref. [66] (Mizushima et al.), which includes a co-author; because the winding number is computed in the present paper, this self-citation is not load-bearing. The phase diagram and boundary conditions cite prior work, including Refs. [30], [50], and [52] from the same groups, but those are background inputs rather than the predicted quantities. The axisymmetric restriction is explicitly acknowledged in Sec. III.B and Sec. IV as an open question; a nonaxisymmetric vortex with lower free energy would alter the physical conclusion, but that is a scope limitation, not circularity. No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported to forbid alternative vortex configurations.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

No new particles, fields, or conserved quantities are introduced. Majorana zero modes are quasiparticle states in an existing superfluid, not new entities. The free parameters listed are numerical choices for the computation, not fits to experimental data.

free parameters (5)
  • kF xi0 = 4
    Quasiclassical parameter used for the BdG calculation; sets the conversion |ell| = kF b and controls the quasiclassical validity. The Eilenberger results themselves are in the kF xi0 -> inf limit.
  • T/Tc = 0.4
    Temperature for all self-consistent calculations, chosen in the superfluid phase below Tc.
  • omega_c = 10 Tc
    Energy cutoff in the quasiclassical gap equation.
  • R0/xi0 = 80
    Radial box size for the BdG eigenvalue problem; the Eilenberger integration uses Rc ~ 50-100 xi0.
  • BdG energy cutoff = 15 Tc
    Cutoff for the eigenvalue sum in the spin density Eq. (39).
assumptions (6)
  • domain assumption Axisymmetric vortex ansatz with total angular momentum kappa, Eq. (18): A(R)=sum_M gamma_M(rho) e^{i(kappa-M)theta} Gamma_M.
    All solutions are restricted to this form, excluding nonaxisymmetric vortices that may be more stable.
  • domain assumption Quasiclassical (Eilenberger) approximation with Fermi-surface average, valid for kF xi0 >> 1.
    Standard weak-coupling quasiclassical theory used to derive Eq. (11); the BdG input kF xi0 = 4 is only moderately large.
  • domain assumption Zero-range attractive 3P2 pairing interaction with interaction strength g, Eq. (3).
    The microscopic model Hamiltonian for neutron matter 3P2 pairing.
  • domain assumption Boundary conditions at rho = Rc are the uniform UN or D4-BN phases with specified direction of the maximum eigenvalue (Fig. 1).
    Selects the vortex species; different boundary conditions give different vortex families (3, rho, theta, D4-BN).
  • domain assumption BdG magnetization is computed with a fixed order parameter, without feedback of the magnetization into the gap.
    The authors note this explicitly after Eq. (44): 'we need to take account of the feedback effect of the magnetization profile to the order parameter profile.'
  • standard math P3-symmetry-based winding number for Majorana zero modes, Eqs. (41)-(42), from Refs. 64-66.
    Topological argument imported from prior literature that guarantees two zero-energy states when P3 is preserved.

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Cite this review

Pith. "Pith review of Microscopic description of axisymmetric vortices in $^{3}P_{2}$ superfluids." pith.science (2026). https://pith.science/paper/TF5ETY5W

@misc{pith2026190806215,
  author       = {Pith},
  title        = {Pith review of: Microscopic description of axisymmetric vortices in $^3P_2$ superfluids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TF5ETY5W}},
  note         = {Machine review of arXiv:1908.06215}
}
abstract

We study quantized vortices in ${}^{3}P_{2}$ superfluids using a microscopic theory for the first time. The theory is based on the Eilenberger equation to determine the order parameters and the Bogoliubov-de Gennes (BdG) equation to obtain the eigenenergies and the core magnetization. Within axisymmetric vortex configurations, we find several stable and metastable vortex configurations which depend on the strength of a magnetic field, similar to a $v$ vortex and $o$ vortex in $^3$He superfluids. We demonstrate that the $o$ vortex is the most stable axisymmetric vortex in the presence of a strong magnetic field, and we find two zero-energy Majorana fermion bound states in the $o$-vortex core. We show that the profiles of the core magnetization calculated using the BdG equation are drastically different from those calculated using only the order parameter profiles known before.

Figures

Figures reproduced from arXiv: 1908.06215 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic images of boundary conditions using the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Order parameter profiles of (a), (c), (e) [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) Free-energy densities for vortices with max. EV p [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Eigen spectra obtained by solving BdG equations. Eac [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Profiles of spin densities. Each panel shows the spin d [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Magnetic field dependence of free energy for several v [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Order parameter profiles of (a) the case at [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. (a), (b) Eigen spectra, (c), (d) LDOS, and (e) – (f) spi [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Critical endpoint and universality class of neutron $^3P_2$ superfluids in neutron stars

    nucl-th 2019-08 conditional novelty 5.0 of 10

    The critical endpoint between two nematic phases of neutron 3P2 superfluids shows critical exponents (α≈0.6, β≈0.4, γ≈0.5, δ≈2.3) that the authors interpret as evidence of a new universality class.

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