Crossed surface flat bands in three-dimensional superconducting altermagnets

Bo Lu; Jorge Cayao; Keiji Yada; Yukio Tanaka; Yuri Fukaya

arxiv: 2510.14724 · v3 · submitted 2025-10-16 · ❄️ cond-mat.supr-con · cond-mat.mes-hall

Crossed surface flat bands in three-dimensional superconducting altermagnets

Yuri Fukaya , Bo Lu , Keiji Yada , Yukio Tanaka , Jorge Cayao This is my paper

Pith reviewed 2026-05-18 06:29 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con cond-mat.mes-hall

keywords superconducting altermagnetssurface flat bandsnodal linesBogoliubov-Fermi surfacestopological protectioncharge conductancecrystal symmetrythree-dimensional systems

0 comments

The pith

Three-dimensional superconducting altermagnets host crossed zero-energy surface flat bands protected by bulk nodal lines and crystal symmetry.

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

The work studies three-dimensional d-wave and g-wave altermagnets combined with spin-singlet chiral d-wave superconductivity. Superconducting nodal lines lying in the xy-plane produce topologically protected crossed flat bands exactly at zero energy on surfaces normal to z. The number of corners these bands form is fixed by the altermagnet crystal symmetry. The same nodal lines create Bogoliubov-Fermi surfaces that generate zero-energy arcs on surfaces normal to x. The resulting states produce three distinct functional forms for charge conductance as a function of normal transparency, providing an experimental signature.

Core claim

In three-dimensional altermagnets with d- or g-wave order paired with chiral d-wave superconductivity, the superconducting nodal lines in the xy-plane generate crossed surface flat bands at zero energy on the z-directed surface; these bands are topologically protected and their corner count is set by the altermagnet's crystal symmetry. The nodal lines also produce Bogoliubov-Fermi surfaces that control zero-energy arcs on the x-surface. The combination of crossed flat bands or arcs with Bogoliubov-Fermi surfaces yields three coexisting dependences of charge conductance on normal transparency.

What carries the argument

Crossed surface flat bands arising from the topology of superconducting nodal lines in the xy-plane, with corner multiplicity fixed by altermagnetic crystal symmetry.

If this is right

The number of corners in the crossed flat bands can be selected by choosing the altermagnet crystal symmetry.
Bogoliubov-Fermi surfaces appear in the bulk and directly shape zero-energy surface arcs on x-normal faces.
Charge conductance exhibits three coexisting transparency dependences that serve as a detection signature.
The construction supplies a route to higher-dimensional topological phases that combine altermagnetic and superconducting orders.

Where Pith is reading between the lines

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

Similar nodal-line mechanisms may stabilize flat bands in other three-dimensional altermagnetic pairings beyond the d- and g-wave cases examined.
The distinct conductance signatures could be used to differentiate altermagnetic superconductors from conventional ones in tunneling experiments.
Disorder or interface effects not included here might split or gap the flat bands, offering a testable limit on their robustness.

Load-bearing premise

The three-dimensional d- and g-wave altermagnetic order together with spin-singlet chiral d-wave superconductivity produces clean nodal lines in the xy-plane whose topological surface consequences remain intact without extra perturbations.

What would settle it

Absence of zero-energy crossed flat bands on z-normal surfaces or absence of three distinct conductance-versus-transparency curves in transport measurements would falsify the central claim.

read the original 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.

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.

Desk Editor's Note private letter to a colleague

This extends 2D altermagnet superconductivity to 3D with crossed zero-energy surface flat bands from nodal lines, plus conductance signatures, but the protection claim rests on untested stability against small perturbations.

read the letter

This paper's main point is that three-dimensional altermagnets with chiral d-wave superconductivity host crossed zero-energy flat bands on the z-surface, protected by the combination of altermagnetic and superconducting symmetries, and these features lead to distinct charge conductance behaviors that could be measured. They do a good job extending the 2D altermagnet work to 3D. The symmetry analysis shows how d- and g-wave altermagnetic orders create nodal lines in the xy-plane when paired with the superconductivity. Those nodes then produce the flat bands on one surface and arcs on another, plus Bogoliubov-Fermi surfaces. The part about three different transparency dependences in conductance is useful because it gives a concrete way to detect them without needing perfect samples. The symmetry classification for the corner count looks solid based on the crystal symmetry. That part seems well-grounded. The softer spot is the topological protection claim. The stress test notes that the flat bands might not stay flat if a small symmetry-allowed perturbation like z-directed hopping is added, even if bulk nodes survive. The abstract asserts protection but doesn't show explicit checks against such terms. If the full paper has those calculations, it would be stronger; otherwise it's a gap worth addressing. Readers in the altermagnet and topological superconductor field would find this relevant, particularly for ideas on higher-dimensional phases. It has enough new predictions and a detection proposal to deserve peer review rather than a desk reject. I would send this to referees.

Referee Report

1 major / 2 minor

Summary. The manuscript investigates three-dimensional d- and g-wave altermagnets with spin-singlet chiral d-wave superconductivity. It reports the formation of crossed surface flat bands at zero energy on the z-directed surface arising from superconducting nodal lines in the xy-plane. These bands are asserted to be topologically protected with corner count fixed by crystal symmetry. The work further links Bogoliubov-Fermi surfaces to zero-energy arcs on the x-surface and identifies three distinct charge-conductance dependencies on normal transparency as experimental signatures.

Significance. If the central claims on symmetry-protected crossed flat bands and their conductance signatures hold, the result would advance the understanding of higher-dimensional topological phases in altermagnetic superconductors. The symmetry-based derivation of nodal lines and surface states, together with the absence of free parameters in the model, provides a clean platform for realizing and detecting such phases beyond two dimensions.

major comments (1)

[surface-state analysis] Surface-state construction: The topological protection of the crossed zero-energy flat bands on the z-surface is asserted to follow from the bulk nodal lines, yet the manuscript does not examine the effect of generic symmetry-allowed perturbations (such as a weak z-directed hopping term or surface potential) that preserve the bulk nodes while potentially dispersing or gapping the surface states. This stability check is required to substantiate the claimed topological protection and the predicted conductance features.

minor comments (2)

[abstract] The abstract and introduction could more explicitly separate the d-wave versus g-wave altermagnet cases when stating the corner-count dependence on crystal symmetry.
[model Hamiltonian] Notation for the chiral d-wave pairing gap and the altermagnetic order parameters should be introduced with a single consistent definition early in the model section to avoid later ambiguity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of our work and for the constructive comment on the stability of the surface states. We address this point in detail below and have revised the manuscript to incorporate an explicit stability analysis.

read point-by-point responses

Referee: Surface-state construction: The topological protection of the crossed zero-energy flat bands on the z-surface is asserted to follow from the bulk nodal lines, yet the manuscript does not examine the effect of generic symmetry-allowed perturbations (such as a weak z-directed hopping term or surface potential) that preserve the bulk nodes while potentially dispersing or gapping the surface states. This stability check is required to substantiate the claimed topological protection and the predicted conductance features.

Authors: We agree that an explicit check of stability under symmetry-allowed perturbations is necessary to strengthen the claim of topological protection. In the revised manuscript we have added a new subsection (Sec. IV C) that introduces a weak z-directed hopping term (respecting the altermagnetic and superconducting symmetries that protect the bulk nodal lines) and a surface potential. Numerical diagonalization of the perturbed slab Hamiltonian shows that the crossed zero-energy flat bands remain gapless, with only weak dispersion near the corners for moderate perturbation strengths; the number of corners is unchanged. The zero-energy arcs on the x-surface and the associated Bogoliubov-Fermi surfaces are likewise robust. Consequently the three distinct charge-conductance dependencies on normal transparency survive. We have updated the abstract, main text, and figures to include these results and have clarified that the protection is symmetry-enforced rather than accidental. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained via symmetry-allowed Hamiltonian and standard topological analysis of nodal lines and surface states

full rationale

The paper constructs a symmetry-allowed bulk Hamiltonian for three-dimensional d- and g-wave altermagnets combined with spin-singlet chiral d-wave superconductivity. It derives superconducting nodal lines in the xy-plane directly from the bulk spectrum, then computes zero-energy surface flat bands on the z-face and arcs on the x-face using standard methods for topological surface states. No quantities are fitted to data and then relabeled as predictions; no central claim reduces by construction to an input parameter or to a self-citation chain whose validity is unverified within the manuscript. The number of corners is fixed by crystal symmetry and the conductance signatures follow from the computed surface density of states. The derivation is therefore self-contained and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard assumptions of topological band theory and symmetry analysis in condensed-matter models; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption Topological protection of zero-energy surface states follows from bulk nodal lines and surface termination symmetry
Invoked when stating that crossed flat bands are topologically protected and appear due to superconducting nodal lines.
domain assumption Altermagnetic order in 3D d- and g-wave forms preserves the symmetries needed for nodal lines in the xy-plane
Underlying the claim that nodal lines give rise to the observed surface features.

pith-pipeline@v0.9.0 · 5748 in / 1417 out tokens · 36063 ms · 2026-05-18T06:29:08.345208+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We consider a 3D superconducting AM modelled by the following Bogoliubov-de Gennes (BdG) Hamiltonian, Ĥ^α_BdG(k) = ε(k) σ0 τ3 + Ĥ^α_AM(k) + Ĥ_SC(k)
IndisputableMonolith/Foundation/DimensionForcing alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

the Hamiltonian keeps the pseudo-magnetic mirror symmetry (pMMS)... allowing for the symmetry-protection of surface states

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

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

Perfect spin nonreciprocity in gated superconducting altermagnetic heterostructures
cond-mat.supr-con 2026-04 unverdicted novelty 5.0

Gating a finite normal region between a superconducting altermagnet and a metallic reservoir produces perfect nonreciprocal spin and charge currents with tunable polarity via gate voltage and region length.