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Majorana flat bands and anomalous proximity effects in $p$-wave magnet--superconductor hybrid systems

T0 review · 1 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read An ordinary superconductor plus a p-wave magnet can host flat-band Majorana states, with a quantized zero-bias conductance peak as a signature.

desk verdict A solid 2D proposal for Majorana flat bands in p-wave magnet–SC hybrids, internally consistent but with an unquantified interface assumption about singlet pairing. read the letter →

arxiv 2502.02053 v2 pith:RCSQUJOR submitted 2025-02-04 cond-mat.supr-con cond-mat.mes-hall

Yutaro Nagae , Leo Katayama , Satoshi Ikegaya This is my paper · ORCID
classification cond-mat.supr-concond-mat.mes-hall
keywords Majoranaflatbandsp-wavemagnetsuperconductorhybridchiralsymmetryBDIclasszero-biasconductancepeakanomalousproximityeffectAtiyah–Singerindex
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 flat-band Majorana bound states, normally associated with intrinsic nodal p-wave superconductors, can instead appear in a hybrid made of an ordinary s-wave superconductor and a p-wave magnet. The magnet's characteristic $[C_{2\perp}||\boldsymbol{t}]$ symmetry—a spin rotation combined with a half-unit-cell translation—survives in the superconducting state and becomes a chiral symmetry of class BDI, which pins a highly degenerate set of Majorana states to zero energy. The paper maps out eight nodal topological phases with nonzero winding numbers and shows numerically that flat-band states sit at the edges exactly where the winding number is nonzero. It then predicts that a disordered normal-metal–superconductor junction containing a p-wave magnet shows a robust zero-bias conductance peak quantized at $(2e^2/h)Z$, where $Z$ is an Atiyah–Singer index counting the flat-band states. If correct, this gives a way to observe anomalous proximity effects without intrinsic p-wave superconductivity.

What carries the argument

The load-bearing object is the composite symmetry $[C_{2\perp}||\boldsymbol{t}]$ of the p-wave magnet: a $\pi$ spin rotation about an axis perpendicular to the ordered moments combined with translation by half a unit cell. Because the Bogoliubov–de Gennes Hamiltonian commutes with this operation, the superconducting hybrid inherits a time-reversal-like symmetry $T_+$ with $T_+^2=+1$; combining $T_+$ with particle-hole symmetry gives the chiral symmetry $\Gamma_+=T_+C$, placing the system in symmetry class BDI. That chiral symmetry supports a momentum-resolved winding number $w_+(k_y)$, and for disordered edges the number of zero-energy modes is fixed by the Atiyah–Singer index $Z=\sum_{k_y}' w_+(k_y)$ rather than by the clean-edge winding number. The machinery therefore explains both why flat bands appear and why they survive roughness.

What would settle it

Measure the zero-bias conductance of a disordered normal-metal–superconductor junction containing a p-wave magnet across disorder strengths. The central claim fails if no peak appears despite a nonzero index, or if the peak height is not the predicted integer multiple of $2e^2/h$ while the magnet's composite symmetry is intact; conversely, a prediction that can be checked is that disorder breaking only the composite symmetry should destroy the peak.

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Extended reading notes

Core claim

The central claim is that the interplay of ordinary spin-singlet pairing with p-wave magnetism produces effective nodal p-wave pairing in the band basis, and therefore flat-band Majorana bound states. Concretely, the Bogoliubov–de Gennes Hamiltonian of Eq. (1) commutes with the composite symmetry $[C_{2\perp}||\boldsymbol{t}]$, which induces a time-reversal-like symmetry $T_+$ with $T_+^2=1$; combining $T_+$ with particle-hole symmetry yields a chiral symmetry $\Gamma_+$ that places the system in class BDI. The associated one-dimensional winding number $w_+(k_y)=\frac12\{\operatorname{sgn}[R(2\pi,k_y)]-\operatorname{sgn}[R(0,k_y)]\}$ changes only when the gap closes at nodal points, and the paper identifies eight nodal topological phases covering a broad range of chemical potential and magnetic hopping. At edges perpendicular to $x$, zero-energy Majorana states appear exactly in the $k_y$ ranges where $w_+(k_y)\neq 0$, and their degeneracy survives surface roughness because it is counted by the Atiyah–Singer index $Z=\sum_{k_y}' w_+(k_y)$. The transport calculation then shows a zero-bias conductance peak at $G(0)=(2e^2/h)Z$ in the dirty junction, an unambiguous anomalous proximity effect.

Load-bearing premise

The argument assumes that a real p-wave magnet can be brought into contact with an s-wave superconductor so that the induced singlet pairing $\Delta$ preserves the magnet's composite $[C_{2\perp}||\boldsymbol{t}]$ symmetry; if interface disorder or proximity coupling breaks that combined rotation-and-translation symmetry, the chiral symmetry and the flat-band protection are lost.

Editorial extensions

If this is right

  • Flat-band Majorana bound states can be engineered from conventional s-wave superconductors, removing the need for materials with intrinsic p-wave pairing.
  • The nodal topological phases occupy a broad parameter region, and since the p-wave magnet's spin splitting can be large compared with the induced gap, realistic parameters should fall inside it.
  • Non-magnetic disorder does not destroy the zero-bias peak; it drives the conductance toward the quantized value $(2e^2/h)Z$.
  • At $\mu=0$ the Atiyah–Singer index vanishes, so the zero-bias peak is absent; observing this parameter dependence would confirm the counting mechanism.
  • The composite symmetry protects flat bands against edge roughness, so transport signatures should survive in imperfect junctions.

Reading between the lines

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

  • Editorial inference: if the composite symmetry can be engineered in other coplanar non-collinear magnets, the same mechanism would generalize beyond p-wave magnets as originally defined.
  • Editorial inference: the symmetry argument suggests that a pM–SC–pM junction would show a fractional Josephson effect, by direct analogy with flat-band nodal p-wave systems.
  • Editorial inference: a concrete next step is to compute the same transport in a three-dimensional heterostructure where only the superconductor layers are gapped; the paper notes this and the result would test whether the flat bands survive stacking.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 6 minor

Summary. This paper studies a two-dimensional hybrid system of a conventional s-wave superconductor and a p-wave magnet, modeled by the minimal two-band lattice Hamiltonian of Ref. [34]. The authors show that the composite [C2⊥||t] symmetry of the p-wave magnet, together with particle-hole symmetry, yields a chiral symmetry Γ+ that places the BdG Hamiltonian in the BDI class. Via a band-basis transformation they obtain an effective nodal p-wave pairing, a closed-form winding number w+(ky), and eight nodal topological phases with analytic boundaries. Open-boundary spectra confirm flat-band Majorana bound states wherever w+(ky)≠0. For a ballistic-normal-metal/dirty-normal-metal/superconductor junction, recursive Green's function calculations give a zero-bias conductance peak quantized as G(0)=(2e²/h)Z for phases with nonzero Atiyah–Singer index Z, and no peak when Z=0. The Supplemental Material provides the band-basis derivation, the winding-number reduction, the phase-boundary functions, the proof that w−(ky)=0, and the analysis of accidental edge-state crossings.

Significance. If the model assumptions hold, the paper is a significant contribution: it proposes a new platform for flat-band Majorana bound states and anomalous proximity effects without intrinsic p-wave superconductivity, in a material class that is currently under active investigation. The strengths are concrete: the symmetry algebra behind the BDI classification is explicit and checkable; Eq. (14) reduces the winding number to a compact sign formula; the phase diagram is given with analytic boundary functions; the zero results for w− and for Z at μ=0 are proven in the Supplemental Material; and the transport prediction is sharp and falsifiable, built on well-cited, previously established methods. The central caveat is that the entire construction is conditional on the proximity-induced pairing being a uniform spin-singlet s-wave that preserves the composite symmetry; the manuscript does not yet quantify the stability of this condition at a realistic interface.

major comments (1)
  1. [Eqs. (1)–(6), flat-band and symmetry analysis] The central claim of the paper rests on the premise that the proximity-induced pair potential in the p-wave magnet is the uniform spin-singlet s-wave Δ of Eq. (1) and that it commutes with the composite symmetry operator Up of Eq. (3), which yields the chiral symmetry Γ+ of Eq. (6) and the BDI classification. All downstream results—the winding number in Eqs. (7) and (14), the Atiyah–Singer index in Eq. (8), and the conductance quantization in Fig. 4—follow from this premise, and the premise is not microscopically verified. At a real interface, the induced pairing is generated by tunneling into a system with spin-dependent hoppings tJ (Eq. (2)) and non-collinear magnetic order, so spin-triplet and odd-frequency pairing components of order comparable to Δ are generically expected; a generic such admixture would not anticommute with Γ+ = −sxτy and would therefore break the chiral symmetry, lift the flat-band degeneracy, and remove the ZBCP quantization. (Some special triplet components, e.g., those whose spin part anticommutes with sx, would preserve Γ+, but the generic case does not.) The Discussion acknowledges that a more realistic model is desirable, but the paper gives no estimate of when the singlet-only assumption holds. I ask the authors to add a quantitative stability check, e.g., a chiral-symmetry-breaking pairing perturbation of size δ with a demonstration that FMBSs and the quantization survive for δ/Δ below a stated bound, or a microscopic argument for the suppression of the triplet admixture.
minor comments (6)
  1. [References, Ref. [33]] The author field of Reference [33] is garbled in the bibliography ('S. G/suppress lodzik'); the entry should be corrected to the actual author names for New J. Phys. 22, 013022 (2020).
  2. [Fig. 4 caption] The caption of Fig. 4 contains a stray fragment '( a b c d e f)' that appears to be a leftover LaTeX label and should be removed.
  3. [Anomalous proximity effect, paragraph after Eq. (15)] The statement that Hd and H′ do not break the chiral symmetry Γ+ is correct, but it is not obvious: a generic on-site potential would break Γ+, and the reason the non-magnetic random potential does not is that the potential is spin-independent and real, so it anticommutes with Γ+ just as the kinetic term does; adding this one-sentence justification would prevent confusion.
  4. [Anomalous proximity effect, conductance results] The text says that 'the minimum value of the zero-bias conductance is quantized at (2e²/h)Z,' but it does not specify the domain of the minimum (over disorder realizations, over bias voltage near zero, or over both); this should be stated explicitly next to the remark that the curves in Fig. 4 are ensemble averages over 100 samples.
  5. [Discussion, parameter estimate] The estimate that tJ ≫ Δ is motivated by the 200 meV spin-splitting energy from Ref. [34], but that scale is an energy rather than a hopping amplitude; a brief comment on how the 200 meV scale maps onto tJ of the lattice model would make the parameter justification more concrete.
  6. [Supplemental Material pointer] The main text refers to the Supplemental Material as 'at XXX'; the placeholder should be replaced with the actual arXiv or journal link.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central FMBS and conductance claims follow from explicit symmetry algebra and numerical solution of a stated Hamiltonian, not from fitting or self-referential definitions.

full rationale

The claimed derivation chain is fully explicit. Eq. (1) defines a BdG Hamiltonian with a stated proximity-induced singlet pairing Δ; Eq. (3) verifies the [C2⊥||t] commutation; Eqs. (4)-(6) construct T+, C, and Γ+; Eq. (7) defines the winding number; Eq. (14) is derived in the Supplemental Material from the chiral-basis determinant without importing the desired flat-band conclusion; Figs. 2-3 are direct numerical diagonalizations; and the conductance in Fig. 4 is computed with recursive Green's functions (Eq. 16) rather than imposed. The self-citations [75,76,79] are used only as parameter-free index theorems (Atiyah-Singer index for class BDI and its conductance consequence) whose stated assumptions (chiral symmetry, dirty edge, DN-SC junction) do not include the present model's parameters or the target result; they are therefore independent support, not circular inputs. The load-bearing physical premise—that the proximity-induced singlet pairing preserves the composite symmetry—is an assumption clearly stated in Eqs. (1) and (3); an unverified assumption is a correctness risk, not a circularity. No fitted parameter is renamed as a prediction, and no equation is equivalent to its input by construction. Score 0.

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

The central claim rests on standard BdG symmetry analysis and on the physical assumption that a p-wave magnet with [C2_perp||t] symmetry can be proximitized by an s-wave superconductor without breaking that symmetry. No free parameters are fitted to data; mu, tJ, and Delta are model inputs scanned over the phase diagram. The paper introduces no new particles or forces.

assumptions (5)
  • domain assumption The p-wave magnet is described by the minimal two-sublattice model of Ref. [34] with the [C2_perp||t] symmetry.
    Eq. (2) adopts this model; the existence and robustness of p-wave magnetic order in a real material is assumed from the cited proposal.
  • domain assumption Proximity-induced pairing is spin-singlet s-wave, described by a uniform Delta in the SC segment, and does not break the [C2_perp||t] symmetry.
    Eq. (1) and the junction setup in the anomalous proximity effect section assume this pairing form.
  • standard math The Atiyah-Singer index Z = sum'_ky w+(ky) counts flat-band Majorana states robust to surface roughness.
    Invoked from Refs. [75,76] to connect winding number to dirty-edge stability; not re-derived here.
  • standard math Blonder-Tinkham-Klapwijk formula and recursive Green's function method give the zero-temperature differential conductance.
    Used in Eq. (16) and the transport calculations; standard formalism.
  • domain assumption For stacked pM layers, each layer receives a proximity gap, so the 2D result extends to 3D stacks.
    Stated in the paragraph after Eq. (2).

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Cite this review

Pith. "Pith review of Majorana flat bands and anomalous proximity effects in $p$-wave magnet--superconductor hybrid systems." pith.science (2026). https://pith.science/paper/RCSQUJOR

@misc{pith2026250202053,
  author       = {Pith},
  title        = {Pith review of: Majorana flat bands and anomalous proximity effects in $p$-wave magnet--superconductor hybrid systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RCSQUJOR}},
  note         = {Machine review of arXiv:2502.02053}
}
abstract

Flat-band Majorana bound states of nodal $p$-wave superconductors give rise to striking electromagnetic anomalies, reflecting their high degree of degeneracy at the Fermi level. However, experimental investigations of these states have been limited because of the scarcity of materials exhibiting intrinsic $p$-wave superconductivity. In this Letter, we demonstrate that Majorana flat bands can emerge in a hybrid system consisting of a conventional superconductor and a $p$-wave magnet, a recently proposed class of unconventional magnets that possess a unique composite symmetry, the $[C_{2\perp}||\boldsymbol{t}]$ symmetry. The degeneracy of the flat-band Majorana bound states is protected by chiral symmetry from the BDI symmetry class, which originates from the $[C_{2\perp}||\boldsymbol{t}]$ symmetry of the $p$-wave magnet. In addition, we predict the robust appearance of a zero-bias conductance peak in a dirty normal-metal--superconductor junction containing a $p$-wave magnet, which serves as an unambiguous signature of anomalous proximity effects associated with the Majorana flat bands.

Figures

Figures reproduced from arXiv: 2502.02053 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Schematic of the [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Topological phase diagram of the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Energy spectra and winding numbers of the nodal topol [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Differential conductance as a function of the bias vol [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Phase diagram for the number of superconducting gap n [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Energy spectra and winding numbers of the nodal topol [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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

  1. Dynamical Polarization from Hidden Spin and Orbital Textures in p-Wave Magnets

    cond-mat.mes-hall 2026-07 conditional novelty 6.0 of 10

    Optically driven p-wave magnets develop a resonantly enhanced ac spin polarization at the 2J_sd exchange gap and a rectified, polarization-controlled dc orbital polarization invisible to period-averaged treatments.

  2. Transverse Spin Supercurrent at p-wave magnetic Josephson Junctions

    cond-mat.supr-con 2025-07 reject novelty 5.0 of 10

    A p-wave magnet sandwiched between two s-wave superconductors converts Andreev bound states into sideways-propagating modes that carry a pure transverse spin supercurrent.

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.