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REVIEW 3 major objections 5 minor 4 cited by

Crossed surface flat bands in three-dimensional superconducting altermagnets

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper claims that in three-dimensional superconducting altermagnets, the nodal lines of chiral d-wave superconductivity force crossed zero-energy flat bands to appear on the [001] surface, with the number of corners fixed by the alterma

desk verdict A solid 2D-embedded model with clean conductance signatures, but the '3D generic' claim is overbroad and the paper's own supplement supplies the counterexample. read the letter →

arxiv 2510.14724 v4 pith:5FHJAHJZ submitted 2025-10-16 cond-mat.supr-con cond-mat.mes-hall

Yuri Fukaya , Bo Lu , Keiji Yada , Yukio Tanaka , Jorge Cayao This is my paper
classification cond-mat.supr-concond-mat.mes-hall
keywords altermagnetismchirald-wavesuperconductivitycrossedsurfaceflatbandsBogoliubov-FermisurfacestopologicalstatestunnelingconductanceSr2RuO4nodal-linesuperconductor
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 work extends the study of superconducting altermagnets from two to three dimensions and argues that the combination of chiral d-wave superconductivity and altermagnetic spin splitting produces a new kind of topological surface state: crossed flat bands pinned at zero energy. The mechanism is geometric: the xy-plane nodal lines of the chiral d-wave order parameter guarantee zero-energy states on the [001] surface, while the nodes of the altermagnet decide how many corners the crossed bands have. The same bulk nodal lines become Bogoliubov-Fermi surfaces when altermagnetism is present, reshaping the surface arcs on the [100] face. If correct, these features would give tunneling experiments three distinct conductance power laws in one material, offering a concrete route to detect the topological phases.

What carries the argument

The central objects are the altermagnetic order term M^α_k in the Bogoliubov–de Gennes Hamiltonian and the chiral d-wave pair potential ψ(k) = Δ sin k_z (sin k_x + i sin k_y). The nodal lines of ψ(k) in the kx–ky plane are what guarantee zero-energy flat bands on the [001] surface; the nodes of M^α_k (the spin-degenerate lines of the altermagnet) set the number of corners of the crossed flat bands. Topological protection comes from a pseudo-magnetic mirror symmetry (pMMS) — the product of pseudo-time-reversal and mirror symmetry — which survives despite broken time-reversal, and a winding number defined on the mirror-invariant lines. This is what distinguishes the crossed flat bands from mer

What would settle it

A tunneling measurement on the [001] surface of a material whose altermagnetic order is of dz(x+y) type would be decisive: the paper's own supplement predicts no crossed flat bands for this symmetry, so observing them there would contradict the core claim. Conversely, for dxy or gxy(x2−y2) altermagnets, looking for a 4- or 8-corner crossed pattern in zero-bias conductance would test the predicted connection between altermagnetic nodes and corner count.

Watch

Extended reading notes

Core claim

The central claim is that a three-dimensional spin-singlet chiral d-wave superconductor with d- or g-wave altermagnetism hosts topologically protected crossed surface flat bands at zero energy on the [001] surface. These bands are the 3D analogue of zero-energy Andreev bound states; their zero-energy character is enforced by the nodal lines of the chiral pairing in the xy-plane, and their corners are fixed by the nodes of the altermagnetic order. The same nodal lines, once altermagnetism breaks the degeneracy, produce Bogoliubov-Fermi surfaces in the bulk, which modify the surface arcs on the [100] face. The protection is shown through a pseudo-magnetic mirror symmetry and a winding number d

Load-bearing premise

The main-text model uses a two-dimensional altermagnetic term M^α(kx,ky) embedded in a three-dimensional superconductor, and the existence of crossed flat bands in a real three-dimensional altermagnet depends on that momentum dependence surviving with the right node structure (the supplement shows dz(x+y)-wave altermagnetism does not produce them).

Editorial extensions

If this is right

  • Crossed flat bands become a generic topological surface phenomenon in three-dimensional nodal superconductors with altermagnetism, not a fine-tuned feature of a single model.
  • The number of corners of the crossed flat bands reveals the crystal symmetry of the altermagnet: four corners for dxy or dx2−y2 altermagnetism and eight for gxy(x2−y2) altermagnetism.
  • Zero-bias tunneling conductance along the [001] direction shows a zero-bias peak originating from the crossed flat bands; its height is reduced but survives at strong altermagnetic strengths.
  • Along the [100] direction, conductance reflects surface arcs modified by Bogoliubov–Fermi surfaces, with direction-dependent transport and, for dx2−y2 altermagnetism, spin-split arcs producing two resonances.
  • The coexistence of three power laws in momentum-resolved conductance offers a direct way to detect flat bands and Bogoliubov–Fermi surfaces in tunneling spectroscopy, for example via Doppler-shift measurements.

Reading between the lines

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

  • The paper's main-text model embeds a strictly two-dimensional altermagnetic term in a three-dimensional superconductor; the supplement shows that a genuine 3D dz(x+y)-wave altermagnetism does not produce crossed flat bands. We infer that the effect is symmetry-specific: candidate materials must be screened for the correct altermagnetic node structure, not merely for the presence of altermagnetism.
  • The symmetry-based protection suggests that crossed flat bands should be robust to moderate disorder and surface roughness, making them a practical target for scanning tunneling microscopy; this robustness is not explicitly tested in the paper.
  • The demonstration that the barrier potential sharpens the conductance of the crossed flat bands implies that low-transparency junctions are the most favorable regime for experimental detection; this emphasis is our inference.
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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

3 major / 5 minor

Summary. The paper studies a 3D BdG model (Eq. 1) combining a spin-singlet chiral d-wave superconductor with 2D d- or g-wave altermagnetic terms M^α_k that depend only on kx, ky. The authors show analytically that zero-energy BFSs form when [M^α_k]^2 = ε^2(k)+|ψ(k)|^2 (Eq. 3 and surrounding text), and they argue that projection onto the [001] surface produces crossed zero-energy flat bands whose corners are set by the AM nodes, while the [100] surface hosts modified surface arcs. They further derive conductance formulas and present recursive Green's function numerics for junctions along z and x, reporting three coexisting power laws (2σ̄_N^0, σ̄_N^1, σ̄_N^2) in the momentum-resolved zero-bias conductance. The topological-protection claim is based on a pseudo-magnetic mirror symmetry and a chiral operator on mirror planes, with a winding number stated to be definable but not computed. The supplementary material extends the analysis to genuine 3D altermagnetic forms (dz(x+y), gzx(x2−3y2), gyz(3x2−y2)) and reports that crossed flat bands appear for the two g-wave forms but not for the dz(x+y) form.

Significance. If correctly scoped, the work is a valuable extension of flat-band and BFS physics from 2D to 3D superconducting altermagnets. The analytic BFS condition is clean and parameter-free within the model, and the conductance fingerprints (three distinct power laws) are concrete, falsifiable predictions that could guide tunneling experiments in Sr2RuO4-type materials. The numerical transport calculations are carefully described with all parameters disclosed, and the supplement makes a good-faith effort to test the 3D generalization. However, the paper's headline claim that crossed flat bands are a generic phenomenon in all 3D superconducting altermagnets is contradicted by the authors' own supplementary calculation for the dz(x+y) 3D d-wave altermagnet. The topological protection argument is also incomplete because no nonzero winding number is computed. These issues are load-bearing for the central 'generic, topologically protected 3D phenomenon' claim, but they are fixable by re-scoping the claim to the AM forms that actually exhibit the effect and by supplying the missing winding-number evaluation.

major comments (3)
  1. [SM S3.1, Eq. (S78), Figs. S5–S6; main text Abstract and Conclusions] The main text and abstract state that crossed flat bands emerge in 'three-dimensional d- and g-wave altermagnets' as a 'generic topological phenomenon' (Eq. 1 effectively uses 2D AM terms, with a note that the findings 'remain in 3D AMs [65], see S3'). However, SM S3.1 introduces a genuine 3D d-wave altermagnet, dz(x+y) (Eq. S78), and SM S3.2 and Figs. S5–S6 report that this form does not produce crossed flat bands on the [001] surface and leaves the projected DOS and conductance essentially unchanged. Since dz(x+y) is itself a 3D d-wave altermagnet included in the study, the genericity claim is not supported. The phenomenon evidently depends on the specific AM node structure and its surface projection, not merely on the coexistence of 3D chiral d-wave superconductivity with altermagnetism. This is a load-bearing overgeneralization and should be corrected by restricting the claim (e.g.,
  2. [Main text 'Topological origin' and SM S1.3] The paper claims the crossed flat bands and surface arcs are 'topologically protected'. The argument establishes a pseudo-magnetic mirror symmetry and a chiral operator Γ_k on mirror planes (SM S1.3, Eqs. S24–S29), and states that a winding number can be defined on symmetric lines. However, no winding number is actually computed for any of the models, and the text only says that a nonzero value 'implies' protection. Moreover, footnote [70] explicitly concedes that protection does not hold where BFSs exist. Thus the topological-protection claim is conditional and unsupported by an explicit topological invariant. I recommend either computing the winding number on the mirror planes (a finite calculation for the 2D AM forms) or softening the language to 'symmetry-protected in the regions where the BFS is absent'.
  3. [Main text, second paragraph and Fig. 1(c,d)] The paper says the 'xy-plane nodal lines of chiral d-wave superconductivity ensure that the crossed flat bands appear at zero energy' and that 'the number of corners is determined by the altermagnetic nodes'. This is demonstrated for the 2D AM forms and for the g-wave 3D forms, but the dz(x+y) counterexample shows that the zero-energy condition is not sufficient: the AM term must also have the right node structure when projected onto the [001] surface. The statement should be qualified to avoid implying that any nodal superconducting AM will produce the effect. This is related to Major Comment 1 but deserves its own formulation because it concerns the physical mechanism claimed in the abstract.
minor comments (5)
  1. [Abstract and Introduction] The phrase 'generic topological surface phenomenon' and 'number of corners determined by the crystal symmetry' is imprecise: the corner count is not determined by symmetry alone but by the specific nodal structure of the AM form. Rephrase to say 'determined by the altermagnetic nodes' as done later in the text.
  2. [Eq. (8) and SM S2.2] The notation σ̄_N^0, σ̄_N^1, σ̄_N^2 is used in the main text and End Matter, but the definitions of σ̄_N and the superscripts are only implicit in the SM. Please define them explicitly at first use in the main text.
  3. [Reference [66]] The reference to the Supplemental Material is a placeholder ('See Supplemental Material for details.,.'). This should be completed with the actual citation details.
  4. [Fig. 1 caption] In panels (f,g), the caption says 'crossed flat bands emerge (green) with their corners defined by the nodes of the 2D spin-polarized altermagnetic Fermi surfaces', but the main text says the corners are defined by the AM nodes. Clarify the relationship between the Fermi-surface nodes and the AM nodes, especially since the Fermi surfaces shown are for a 3D system projected onto the [001] surface.
  5. [SM S3.3, Fig. S7] The discussion of the gzx(x2−3y2) AM says 'point nodes mainly become BFS' but the figure shows a line node at kz = 0. Please reconcile the terminology (line node vs point node) for consistency.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction found; the results follow analytically from the model Hamiltonian and are not fitted predictions.

full rationale

The paper is a self-contained model study, not a data-fitting or benchmark-derived claim. The BdG Hamiltonian in Eq. (1), the altermagnetic terms in Eq. (2), and the bulk energy spectrum in Eq. (3) define the model; the surface flat bands, arcs, BFSs, and conductance are then computed from this same Hamiltonian via the recursive Green's function method described in SM S2 and the mirror/chiral symmetry construction in S1.3. No parameter is fitted to a target quantity, and no predicted observable is equal by construction to an input. The statement that the crossed flat band corners are set by the altermagnetic nodes is an explanatory identification from the projected Fermi surfaces, not a tautological redefinition. The paper does contain self-citations to earlier flat-band and topological-ABS work (e.g. Refs. [46,49,63,64,67]), but these are not load-bearing: the chiral operator and pseudo-magnetic mirror symmetries are derived explicitly in the present text and supplement. Two genuine caveats exist, but they are correctness/generality issues rather than circularity: (i) the winding number is asserted to protect the states but is not explicitly evaluated, and (ii) SM S3.2 shows that the genuine 3D dz(x+y) altermagnet does not produce crossed flat bands, so the headline 'generic 3D' claim is broader than the demonstrated cases. Neither caveat makes the derivation equivalent to its inputs.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new particles or entities. Its inputs are a model Hamiltonian, symmetry conditions, and numerical transport parameters; none are fitted to experimental data. The main hidden load is the nodal structure of the chiral d-wave pairing and the survival of mirror symmetries at the surface.

free parameters (5)
  • chemical potential μ = -4.5t
    Chosen to set a large Fermi surface; flat-band width, BFS location, and conductance magnitudes depend on it.
  • altermagnetic strength tα = tα = Δ with Δ = 0.01t
    Varies flat-band size and ZBCP height; corner structure depends on AM type and strength.
  • pair amplitude Δ = 0.01t
    Weak-coupling choice; the nodal structure is more important than magnitude, but all conductance numbers are computed at this value.
  • barrier potential U_b = 5t
    Chosen to sharpen conductance features; zero-bias peak visibility depends on transparency.
  • smearing δ = 0.01Δ
    Broadening in Green's functions; SM S2.3 notes the σ_N^2 law in gapped regions is only approximate because of δ.
assumptions (4)
  • domain assumption The BdG mean-field Hamiltonian in Eq. (1) with ε(k), M^α_k, and H_SC is an adequate model of 3D superconducting altermagnets.
    The entire calculation is performed on this model; it is not derived from a microscopic material-specific Hamiltonian.
  • domain assumption Chiral d-wave pairing has an exact xy-plane nodal line (H_SC ∝ sin kz) and a phase φ_k = tan^{-1}(sin ky/sin kx) removable by local gauge transformation.
    Zero-energy flat bands rely on this node structure; if the pairing differs, flat bands need not sit at zero energy.
  • domain assumption The pseudo-magnetic mirror symmetries in Eqs. (4)-(5) survive on the [001] and [100] surfaces after truncation.
    Topological protection of the surface states assumes the open boundary preserves the relevant mirror symmetries.
  • standard math The Lee-Fisher formula with recursive Green's functions gives the physical tunneling conductance.
    This is a standard condensed-matter transport formalism; the numerical implementation is described but not independently verified here.

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Cite this review

Pith. "Pith review of Crossed surface flat bands in three-dimensional superconducting altermagnets." pith.science (2026). https://pith.science/paper/5FHJAHJZ

@misc{pith2026251014724,
  author       = {Pith},
  title        = {Pith review of: Crossed surface flat bands in three-dimensional superconducting altermagnets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5FHJAHJZ}},
  note         = {Machine review of arXiv:2510.14724}
}
abstract

Superconducting altermagnets have proven to be a promising ground for emergent phenomena, but their study has involved two-dimensional systems. In this work, we investigate three-dimensional $d$- and $g$-wave altermagnets with spin-singlet chiral $d$-wave superconductivity and show the formation of crossed surface flat bands due to the interplay between superconducting and altermagnetic symmetries. We find that these crossed flat bands are topologically protected, appear at zero energy in the surface along $z$ due to the superconducting nodal lines in the $xy$-plane, and their number of corners is determined by the crystal symmetry of altermagnets. We also show that the superconducting nodal lines give rise to Bogoliubov-Fermi surfaces, which then affect the appearance of zero-energy arcs in the surface along $x$. Moreover, we demonstrate that the crossed flat bands or surface arcs, and Bogoliubov-Fermi surfaces give rise to the coexistence of three distinct dependences of the charge conductance on the normal transparency, hence offering a solid way for their detection and paving the way for realizing higher-dimensional topological phases using altermagnets.

Figures

Figures reproduced from arXiv: 2510.14724 by the authors.

Figure 1
Figure 1. FIG. 1. (a,b) Sketches of the studied junctions along [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a,c) Zero-bias conductance along the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a,c) Zero-bias conductance along the [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Zero-bias conductance along the [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a-c) Projected zero-energy DOS on the [001] surface [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Zero-bias conductance along the [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Normalized zero-bias conductance along [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]
Figure 7
Figure 7. Figure 7: FIG. 7. (a-c) Projected zero-energy DOS on the [100] surface [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

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