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This paper claims that placing a conventional spin-singlet s-wave superconductor in contact with a p-wave unconventional magnet—a magnet with a noncollinear spin texture and zero net magnetization—effectively produces spin-triplet p-wave su

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2026-08-03 14:59 UTC pith:XSXMRQWE

load-bearing objection Solid extension of known PUM-superconductor flat-band physics to Josephson transport; the first-harmonic survival claim is plausible but not fully shielded against inter-sublattice pairing. the 2 major comments →

arxiv 2512.18636 v4 pith:XSXMRQWE submitted 2025-12-21 cond-mat.supr-con cond-mat.mes-hall

p-wave superconductivity and Josephson current in p-wave unconventional magnet/s-wave superconductor hybrid systems

Yuri Fukaya , Keiji Yada , Yukio Tanaka This is my paper
classification cond-mat.supr-con cond-mat.mes-hall
keywords p-wave unconventional magnetspin-singlet s-wave superconductorzero-energy flat bandsodd-frequency spin-triplet pairingJosephson currentsurface density of statesBogoliubov-de Genneseffective p-wave superconductivity
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper studies hybrids of a p-wave unconventional magnet (PUM) and a conventional spin-singlet s-wave superconductor. It claims that the noncollinear spin structure of the PUM changes how the s-wave pair potential acts: the quasiparticle spectrum develops a nodal, spin-triplet p-wave form, and zero-energy flat bands appear at the [100] edge when the magnetic exchange exceeds the pair potential. At that edge, odd-frequency spin-triplet even-parity pairing is strongly enhanced, while even-frequency singlet pairing survives. The survival of the singlet amplitude has a direct transport consequence: in a Josephson junction between the hybrid and an s-wave superconductor, the first harmonic of the current does not vanish, unlike in a pure p-wave/s-wave junction. The paper therefore proposes that an s+p-wave-like superconducting state can be generated in these hybrids and that its edge and Josephson signatures are observable.

Core claim

The central claim is that in a PUM/s-wave hybrid, a purely spin-singlet s-wave pair potential, combined with the p-wave unconventional magnetic order, behaves as an effective spin-triplet p-wave superconducting gap. The noncollinear spin structure along the relevant direction gives the quasiparticle dispersion the p_x-wave or p_y-wave form, and for J > Δ zero-energy flat bands appear at the [100] edge. Analyzing the pair amplitude, the paper finds that odd-frequency spin-triplet even-parity pairing is strongly induced in the presence of these flat bands, while even-frequency spin-singlet even-parity pairing remains. That remaining singlet component is what couples to the s-wave side of a Jos

What carries the argument

The load-bearing object is the effective two-dimensional tight-binding model of a p-wave unconventional magnet: H(k) = ε(k) + (t_x sin k_x + t_y sin k_y)s₃ + J s₁ in spin-sublattice space, whose noncollinear spin configuration gives p-wave-like band splitting. Coupling this to intra-sublattice spin-singlet s-wave pairing Δ produces a Bogoliubov-de Gennes Hamiltonian that separates into two sublattice sectors. Within each sector the effective gap has an odd-parity p-wave structure; analytically, the even-frequency spin-triplet odd-parity pair amplitude is proportional to ΔJ t_x sin k_x, and the odd-frequency spin-triplet even-parity amplitude to ΔJ ω_n. This identity—spin-singlet pairing plus

Load-bearing premise

The central assumption is that the superconducting order is a purely intra-sublattice spin-singlet s-wave pairing, so the Bogoliubov-de Gennes Hamiltonian separates into two independent sublattice sectors; if a realistic interface induces inter-sublattice or spin-triplet pairing, or if J is not larger than Δ, the effective p-wave gap and flat bands would be altered or absent.

What would settle it

Measure the surface density of states at the [100] edge of a PUM/s-wave hybrid while tuning the magnetic exchange J relative to the pair potential Δ: the paper predicts zero-energy flat bands (a zero-bias peak) only for J > Δ and no peak for J < Δ. Alternatively, in a low-transparency PUM-s-wave/s-wave Josephson junction the first harmonic I₁ should remain finite; observing I₁ → 0 would refute the singlet-coupling mechanism.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • At the [100] edge of a PUM/s-wave hybrid, zero-energy flat bands appear when J > Δ, showing up as a zero-bias peak in the surface density of states.
  • Odd-frequency spin-triplet even-parity pairing is strongly enhanced exactly where flat bands exist, linking edge topology to odd-frequency correlations.
  • In PUM/s-wave–s-wave Josephson junctions, the first harmonic I₁ remains nonzero because the singlet even-parity pair amplitude on the PUM side couples to the s-wave side; a pure p-wave/s-wave junction would lose I₁.
  • The temperature dependence of the maximum Josephson current can be tuned by the chemical potential, because the chemical potential controls whether flat bands exist and how much they resonate across the junction.
  • PUM/PUM junctions show skewness or φ-junction behavior depending on the p_x/p_y orientation and on the presence of flat bands, modified by the residual singlet coupling.

Where Pith is reading between the lines

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

  • If the effective p-wave gap is topologically protected by winding numbers, as the paper suggests, these hybrids are a plausible platform for Majorana zero modes built from a spin-singlet superconductor and an unconventional magnet, though the paper does not propose braiding or qubit operations.
  • A clean experimental test would be a tunneling-spectroscopy scan at the [100] edge as the ratio J/Δ is varied: the paper predicts a zero-bias conductance peak only for J > Δ, vanishing for J < Δ.
  • Because the singlet pairing survives, the Josephson current is not a clean probe of pure p-wave order; a more discriminating experiment would compare the first-harmonic amplitude across chemical potentials that switch the flat bands on and off.
  • The model assumes purely intra-sublattice singlet pairing; a real interface with inter-sublattice or self-consistently inhomogeneous pairing could gap the flat bands, so engineering the interface to preserve the sublattice structure may be essential for the predicted signatures.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The manuscript studies a two-dimensional tight-binding model of a p-wave unconventional magnet (PUM) in proximity to a conventional spin-singlet s-wave superconductor. Starting from the Brekke et al. PUM Hamiltonian (Eq. (1)) and assuming intra-sublattice spin-singlet pairing (Eq. (5)), the authors show in Appendix A that the BdG Hamiltonian separates into two sublattice sectors; for J>Δ this yields nodal quasiparticle spectra and zero-energy flat bands at the [100] edge that mimic p_x/p_y-wave superconductivity. They compute surface DOS by recursive Green's functions, classify edge pair amplitudes into ESE/ETO/OSO/OTE, and present analytical bulk formulas (Appendix B). They then calculate Josephson currents in PUM/s-wave and PUM/PUM junctions in both high- and low-transparency regimes. The central claim is that the residual spin-singlet even-parity component at the edge couples to the s-wave side, so I1 survives in PUM/s-wave junctions even though the bulk behaves as an effective p-wave superconductor, and the μ-dependent flat bands control the low-temperature enhancement of Ic in PUM/PUM junctions.

Significance. If the central assumptions hold, the paper offers a concrete path to effective p-wave superconductivity and zero-energy flat bands from a conventional s-wave superconductor and an unconventional magnet, with analytical bulk pair amplitudes and standard recursive-Green's-function numerics. Strengths include the explicit derivation of the η-sector spectrum in Appendix A, the analytical bulk anomalous Green's functions in Appendix B, and the absence of parameter fitting: all transport curves follow from the stated Hamiltonian. The paper also connects the edge pair-amplitude classification (OTE enhancement) to the Josephson harmonics in a useful way. The main weakness is that the sublattice-diagonal form of the pairing is assumed rather than justified at a real interface, and the temperature-dependent Josephson results rely on a non-self-consistent BCS gap ansatz.

major comments (2)
  1. [Sec. II and Appendix A (Eqs. (5), (A1))] The block-diagonal structure that separates the BdG Hamiltonian into η=± sublattice sectors, and hence the zero-energy flat bands in Figs. 2(b)-(c), relies on the spin-singlet pairing Δ[s0⊗σ0]i s2 commuting with σ3. At a physical PUM/s-wave interface, the proximity-induced pair potential need not be sublattice-diagonal: a term Δ12[s0⊗σ1]i s2 (or σ2) couples the two η sectors. Since the η=± spectra in Eq. (A2) have nodal/zero-energy features at different k, such a term generically opens a gap and would destroy the flat bands, the OTE enhancement in Sec. IV, and the I1/I2 ratios presented in Secs. V-VI. The topological-protection argument cited from Ref. [111] does not cover a model with σ1/σ2 pairing. Please either provide a symmetry argument that inter-sublattice pairing is exactly forbidden at the interface, or include Δ12≠0 and demonstrate that the flat bands, edge pair amplitudes, and
  2. [Sec. VI, Eq. (20)] The temperature-dependent Josephson current is computed with a fixed BCS order parameter Δ(T)=Δ0 tanh(1.74 sqrt((Tc-T)/T)), Δ0=1.76Tc, on both sides rather than a self-consistent BdG solution. For J=t and Δ=0.01t the exchange energy is two orders of magnitude larger than the gap, and the bulk spectrum is gapless for J>Δ (Appendix A, Fig. 14). The magnitude and T-dependence of the induced order parameter in the PUM layer are not guaranteed to follow the bulk s-wave BCS ansatz. Since Sec. VI's central claim—that Ic can be tuned by μ through the generation of zero-energy flat bands—depends on Δ(T), please test self-consistency at least for representative parameters (μ, J/t, tint) or explicitly state this as a limitation of the transport predictions.
minor comments (4)
  1. [Sec. III] Several figure callouts appear swapped: the p_x-wave results are cited as Fig. 3(b)/(c) instead of Fig. 2(b)/(c), and conversely. Please check all cross-references between Figs. 2 and 3, and also in the captions of Figs. 6-8 where panel letters do not match the text.
  2. [Sec. IV] Figure labels such as |F^{↑↑,↓↓}_{OTE}| are hard to parse; please define this notation explicitly in the caption (equal-spin components) and use a consistent typesetting for all pair-amplitude components.
  3. [Sec. VII] The statement that zero-energy flat bands are realized 'independent of the chemical potential μ' is overstated: Figs. 2(b) and 2(c) show that the momentum-space extent of the flat bands changes strongly with μ, and Sec. VI itself relies on this difference. Rephrase as 'robust to changes in μ within the studied range' or similar.
  4. [Sec. V-VI] The high-transparency and low-transparency regimes are studied with tint=1.0 and tint=0.1 respectively, but no convergence checks (e.g., in the number of Matsubara frequencies or in the k_y grid) are reported. Adding a brief convergence statement would strengthen the numerical claims.

Circularity Check

0 steps flagged

No significant circularity: the flat-band, odd-frequency, and Josephson results are direct outputs of a stated BdG Hamiltonian, with no fitted parameters; the only self-citations are methodological or contextual.

full rationale

The paper's central predictions are obtained by solving an explicit BdG Hamiltonian, Eqs. (4)-(5), built from the externally published PUM model of Ref. [51]. The bulk pair amplitudes, Eqs. (17)-(18) and (B2)-(B10), are analytic outputs rather than imposed symmetries, and the edge SDOS and Josephson current are computed from recursive Green's functions using the standard current formula Eq. (21). No parameter is fitted to the reported flat bands, OTE enhancement, or current harmonics, and the 'effective p-wave' statement is supported by explicit spectral and Green's-function algebra (Appendix A and Appendix B) rather than by definition. The main caveats are conditional assumptions, not circularity: the effective p-wave nodal structure requires J > Δ (Appendix A, Fig. 14), and the pairing is assumed to be purely intra-sublattice spin-singlet s-wave, Eq. (5). Inter-sublattice proximity pairing is not modeled, but that absence is a stated model limitation, not a circular step. Same-author citations such as Ref. [75] for the Josephson-current method are methodological and not load-bearing; the topological interpretation of the flat bands is assigned to external Refs. [111,113]. Therefore no prediction reduces by construction to a fitted input or to a self-citation chain.

Axiom & Free-Parameter Ledger

8 free parameters · 5 axioms · 0 invented entities

All model parameters are hand-picked in units of t; none are fitted to experimental data. The p-wave superconducting state is an emergent effective description of the fixed model, not a new particle, force, or dimension. The central claims depend on the effective PUM model and on the intra-sublattice s-wave pairing assumption.

free parameters (8)
  • J = t
    Local s-d exchange in Eq. (1); set equal to hopping t throughout; the effective p-wave gap requires J > Δ.
  • tx, ty = t or 0
    Spin-dependent hoppings; tx=t,ty=0 for px-wave; tx=0,ty=t for py-wave.
  • μ = -4t, -2t
    Chemical potentials chosen to show one/two Fermi surfaces and different flat-band regimes.
  • Δ0 = 1.76 Tc
    BCS gap amplitude used in Eq. (20) for the temperature dependence.
  • Tc = 0.01 t
    Critical temperature sets the temperature scale; chosen small relative to t.
  • t_int = 1.0 (high), 0.1 (low)
    Interface transparency amplitude in Appendix C; controls the high- and low-transparency regimes.
  • δ = 0.01 Δ
    Infinitesimal broadening in the retarded Green's function Eq. (6).
  • μN, μs = -3.5t, -1.5t
    Chemical potentials in the normal metal and s-wave superconductor for the Josephson junctions.
axioms (5)
  • domain assumption The effective PUM Hamiltonian Eq. (1) from Ref [51] captures the essential noncollinear spin structure of p-wave unconventional magnets.
    All results are derived from this model; if real p-wave magnets are not described by it, the predictions fail.
  • domain assumption The proximity-induced pair potential in PUM-SC is a constant intra-sublattice spin-singlet s-wave pairing, Eq. (5), with no inter-sublattice or self-consistent correction.
    This allows the BdG Hamiltonian to separate into sublattice sectors (Appendix A); any inter-sublattice pairing would break this and could gap the flat bands.
  • domain assumption Quasiparticle energies and Josephson currents are computed in mean-field Bogoliubov-de Gennes theory with Δ(T) given by the BCS formula Eq. (20).
    The gap is not determined self-consistently; the temperature dependence is imposed.
  • standard math The recursive Green's function method of Refs [128,129] is a valid tool for semi-infinite lattice Green's functions and Josephson currents.
    The method is standard but not re-derived in this paper.
  • domain assumption The interface is described by the local tunneling matrices in Appendix C, with no disorder, spin-flip scattering, or Fermi-surface reconstruction at the interface.
    The Josephson current results depend on this simplified interface model.

pith-pipeline@v1.3.0-alltime-deepseek · 555 in / 10926 out tokens · 338224 ms · 2026-08-03T14:59:16.180817+00:00 · methodology

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read the original abstract

We study the surface density of states in $p$-wave unconventional magnet-spin-singlet $s$-wave superconductor hybrid systems ($p$-wave unconventional magnetic superconductors). Owing to the noncollinear spin structure in $p$-wave unconventional magnets, the spin-singlet $s$-wave pair potential behaves as the spin-triplet $p$-wave superconductivity. As a result, zero-energy flat bands can emerge at the edge. Analyzing the pair amplitude at the edge, odd-frequency spin-triplet even-parity pairing is induced in the presence of zero-energy flat bands, while even-frequency spin-singlet even-parity remains. We also demonstrate the Josephson current in superconducting junctions with $p$-wave unconventional magnet-spin-singlet $s$-wave superconductor hybrid systems. By the cooperation of spin-singlet $s$-wave pair potential and the $p$-wave unconventional magnetic order, the coupling of the spin-singlet even-parity pairings in junctions generates the first harmonics of the Josephson current. In addition, the temperature dependence of the maximum Josephson current can be tuned by the chemical potential, which determines the generation of zero-energy flat bands. Our results indicate that $s+p$-wave-like superconducting state is generated in $p$-wave unconventional magnet-$s$-wave superconductor hybrid systems.

Figures

Figures reproduced from arXiv: 2512.18636 by Keiji Yada, Yukio Tanaka, Yuri Fukaya.

Figure 1
Figure 1. Figure 1: FIG. 1. (a)(b) Schematic illustration of the two-dimensional [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Schematic illustration of PUM- [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Schematic illustration of PUM- [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Absolute value of the pair amplitude for (a)(b) [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Absolute value of the pair amplitude for (a)(b) [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Schematic image of PUM-SC/spin-singlet [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Current phase relation in PUM-SC/spin-singlet [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Temperature dependence of the maximum Joseph [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Current phase relation in PUM-SC/PUM-SC [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗
Figure 9
Figure 9. Figure 9: At T = 0.001Tc, for (tx, ty, µ) = (t, 0, −4t) (px-wave UM-SC) [ [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Temperature dependence of the maximum Joseph [PITH_FULL_IMAGE:figures/full_fig_p010_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. The lowest quasiparticle energy dispersion [PITH_FULL_IMAGE:figures/full_fig_p012_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Schematic illustration of Josephson junctions in the [PITH_FULL_IMAGE:figures/full_fig_p013_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Schematic illustrations of tunneling processes [PITH_FULL_IMAGE:figures/full_fig_p013_16.png] view at source ↗

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Forward citations

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